text:
# Desmos Exponents Rules

All log a rules apply for log. Exponential functions are function where the variable x is in the exponent. You should get 5 and 2^4 or 5x^4. Exponent Battleship. In the Desmos activity, the task was to enter an equation of a parabola that would go through a set of three points. Example 6 Find the. To add an exponent symbol after a number or letter when using a Windows operating system, the alt key should be held down and the code 0176, 0185, 0178 or 0179 should be pressed. The calculator, helps you finds the roots of a second degree polynomial of the form ax^2+bx+c=0 where a, b, c are constants, a eq 0. Inspiration Negative exponents can be a confusing topic for many students. Students will simplify exponent problems using the product rule, power rule, and quotient rule. 2 # 1-3 (what you need), 4, 6, 11 5. Representing functions as rules and graphs; Exploring real numbers. Study, study, study; practice, practice, practice!. Desmos is a Non-profit Foundation that serves as a hub between donors and beneficiary organizations since 2012. As a result, students become confused on how the negative exponent effects … Continue reading →. Students will understand the key differences between linear and. Inequality Rules. b-n = 1 / b n. The same rules applied for transformations of exponentials as did for the parent functions that students became familiar with in the previous unit. You'll find all six in the graphic on this post. Another alternative sees the exponents and base number having something in common. In summary, the rules of exponents streamline the process of working with algebraic expressions and will be used extensively as we move through our study of algebra. Exponents Games: Exponents Less than Greater than - Practice Properties of Exponents Introduction to Exponents Quiz Introduction to Exponents Multiplication with Exponents Evaluate. I really like that the activity is designed to use patterns as a way for students to make sense of zero and negative exponents rules, rather than just arbitrary rules that they have to memorize. r = the growth rate. Partners were chosen by the computer with one selecting among 16 graphs of a parabola and the other asking questions trying to determine the chosen parabola. Sign up today!. Indeterminate Forms and L'Hospital's Rule What does $\frac{0}{0}$ equal? Indeterminate Differences Indeterminate Powers Three Versions of L'Hospital's Rule Proofs Optimization Strategies Another Example Newton's Method The Idea of Newton's Method An Example Solving Transcendental Equations When NM doesn't work Anti-derivatives Anti-derivatives. 1: 8-114 Transforming Cosine Functions (Desmos) Lesson 8. A simple guide to LaTeX - Step by Step Learn about LaTeX in short lessons with full code examples. Desmos Reviews. If Y is a vector, the x -coordinates range from 1 to length (Y). 8 The Rules of Exponents: Rule 4: If the base is powered by a negative exponent, then "flip" the numerator/denominator to become the denominator/numerator. You can hide students using the gear button in the teacher dashboard if you want to highlight or screenshot awesome answers, but make sure to not forget about those hidden students in following weeks!. With inequalities, you can add colored shading to your Desmos graph. This game can be played with or without an interactive whiteboard or projector to practice the rules of exponents with increasing complexity, including negative exponents. Other students will learn exponent rules later in the year, but students who start learning them now will advance to more difficult problems. As specified by latex, exponentiation with a power that is more than one character (e. 3 Adding and Subtracting Polynomials 5. √464=4 +2 2 31. Integers and real numbers can be represented on a number line. OBJECTIVE:Multiply decimals by 1000, 100, 10, 1, 0. The same rules applied for transformations of exponentials as did for the parent functions that students became familiar with in the previous unit. My big things to work on for 2014-2015 are classroom management / effective group work. Step 1 Take the natural logarithm of both sides. Desmos Parabola Opens Up or Down A, C, D, E are integers. A riff off an Andrew Stadel "Exponent Mistakes" classic. Powered by Create your own unique website with customizable templates. 3B Practice Rational Exponents; No HW; Optional Practice RPJ 6. Note that, even though Desmos geometry has tools to create perpendicular and parallel lines we won't be using them in order to mimic what would be done in the real construction. Indeterminate Forms and L'Hospital's Rule What does $\frac{0}{0}$ equal? Indeterminate Differences Indeterminate Powers Three Versions of L'Hospital's Rule Proofs Optimization Strategies Another Example Newton's Method The Idea of Newton's Method An Example Solving Transcendental Equations When NM doesn't work Anti-derivatives Anti-derivatives. These exponent rules are fundamental building blocks for your success in Math 2 and beyond. Rules of Exponents: “Used to Think” Method Lesson Objectives For this lesson, students are learning to combine rules of exponents to simplify expressions. The general form of a linear function is , where m is the slope and b is the y-axis intercept. Understanding square roots. Then you will work through Desmos Marbleslides Lines at student. 1 • Average. Square root of decimal. Use Desmos to investigate the beautiful world of integral calculus. 5; while −23. Representing and Comparing Numbers (N) Part A (Lessons 1–7) Topics include representing and comparing positive rational numbers (integers, fractions, and decimals), finding multiples and factors of positive integers, and determining the least common multiple (LCM) and the greatest common factor (GCF) of a pair of positive integers. By this convention, if the fractional part of x is 0. Calculator okay, study guide below. EXPONENTS: Laws of Exponents: “Ex-Exponents Math Rap” by Mr. Exponent Rules Worksheets Exponents Games: Exponents Less than Greater than - Practice Properties of Exponents Introduction to Exponents Quiz Introduction to Exponents Multiplication with Exponents Evaluate Exponents Division with Exponents Exponents with Decimals Exponents with Fractions Exponents Quizzes: Order of Operations with Exponents Quiz. The places where lines representing a range of values cross these axes are called intercepts. A quick analogy: You: Geometry is about shapes, lines, and so on. The SCC Rules have been. In surprise he turned his head and saw nobody. Math Algebra 1 Exponents & radicals Radicals. Use the rule log b ( N x) = x log b ( N), where b > 0, to bring down the negative exponent. Examples: Taking a root of a powers. When dividing functions, you will also need to work with exponents of different powers. Classroom rules. General Form of Parabolas that open up and down. Those are the reasons why expressions with negative exponents equal their reciprocals. Tuesday - 7-1,2 Exponent Rules Review (Multiplication, Division, Negative, and Zero) Use Desmos to create a Graph of your System of Equations. A brief review of the basics follows. Understanding how we can map one set of vectors to another set. Add a new note: type " in an empty expression. All log a rules apply for ln. Team Desmos November 16, 2020 13:05. All log a rules apply for ln. com is undoubtedly the ideal place to stop by!. Let's look at you exponent in a factored form. where $$n$$ is the largest exponent in the numerator and $$m$$ is the largest exponent in the denominator. Translating into words 1. Office hours Monday 6/1 Mrs. 1) 81^3/4 / 81^1/4 2. + (Addition) - (Subtraction) * (Multiplication) / (Division) ^ (Exponent: "raised to the power") sqrt (Square Root) (Example: sqrt(9)). 5d - Graphing Systems of Linear Inequalities in Two Variables with Desmos-This is a Word document. The Calculator automatically determines the number of correct digits in the operation result, and returns its precise result. Rules for Adding Integers Rule 1: If the signs are the same then add the numbers. Learn how to use the Algebra Calculator to solve equations. (added 9/18/16) Exponent Mistakes A riff off an Andrew Stadel "Exponent Mistakes" classic. Desmos Activities for Middle School ********* Please note ********* I stopped updating this resource sometime in 2018. Rewrite each rational exponent as a radical then solve. We are working on the traffic and server issues. I just like the St. b-n = 1 / b n. Tic Tac Times (Pam Wilson) Adding and Subtracting Polynomials Activity (Sarah Hagan) Polynomial Station Activities (Pam Wilson) Factoring. M P b M P log b = log log 100 3 =3⋅log 100 =3⋅2 =6 Think: Raise an exponential expression to a power and multiply the exponents together. Rules of Exponents: Positive Integers. Before you start, f(0. If a < b and b < c, then a < c. Setting 4 x + 2 = 0 gives the vertical line x = -1/2. However, I will focus on the topic of exponential functions in this report in order to give a precise description and comparison. indd 316snb_alg1_pe_0604. 2 x 10 ) : 3. Desmos Math Art Competition; Exponential Function Videos. M N N M log b =log b −log b log 8 1 7 56 log 8 56 log 8 7 log 8 = 8 = − = Think: Divide two numbers with the same base, subtract the exponents. No logarithmic (ln, log) or trigonometric (sine, cosine, tangent) functions. This is a little node. When I teach laws of exponents in algebra, I love using activities and interactive notebook foldables. The Modulo Calculator is used to perform the modulo operation on numbers. log b (x / y) = log b x - log b y. Practice: Square roots. 5 becomes −24, as does −24. Rules or Laws of Logarithms In this lesson, you’ll be presented with the common rules of logarithms, also known as the “log rules”. We could also simplify this b times b squared over here. And, that’s what desmos provided us. In the Desmos activity, I asked if people could think of a way to improve my rough draft. When assigning electrons to orbitals, we must follow a set of three rules: the Aufbau Principle, the Pauli-Exclusion Principle, and Hund's Rule. Power Rule: When raising monomials to powers, multiply the exponents. The exponent of a number says how many times to use the number in a multiplication. Desmos Parabola Opens Up or Down A, C, D, E are integers. People should only leave home for essential reasons such as caring responsibilities, essential. Before you begin working with monomials and polynomials, you will need to understand the laws of exponents. I love them. (2) A-SSE _ALG. We’ve been putting together a reference booklet for all of the rules, and practicing as we go. Derivatives as Rates of Change. Problems: 1. As specified by latex, exponentiation with a power that is more than one character (e. Find the vertical asymptotes. Step 5 Substitute y equals 2x^4 + 1, all raised to the exponent tangent x. Purpose of use To help me understand x and y tables better Comment/Request For it to be easier to get the answers without filling in the long bottom bar and just filling in the table and receiving the answer. Factor all numerators and denominators. Method 2 : To get the number of factors of 48, first we have to find the factors. The interface is specifically optimized for mobile phones and small screens. The Modulo Calculator is used to perform the modulo operation on numbers. 6 # 1-3 SPICY 5. Calculate the derivative of a function. You will review the five rules of exponents that were covered in Math 1. Newton-Raphson Method Calculator. Topic A: Exponent Rules, Expressions, and Radicals. A complex number is a number of the form a+bi, where a,b — real numbers, and i — imaginary unit is a solution of the equation: i 2 =-1. Powered by Create your own unique website with customizable templates. A compound inequality contains at least two inequalities that are separated by either "and" or "or". Negative exponents rule. Expressions 1. Try it yourself: 3 4 = 3 × 3 × 3 × 3 = 81. This now completes all six of the exponent rules that you need to memorize and be able to use to solve and simplify a variety of problems. When you do regular division with numbers and the division "comes out even", it means that the number you divided by is a factor of the number you're dividing. If you are multiplying by 10, 100 or 1000… then move the decimal to theRIGHT the same number as there are zeroes. number exponents through proof by induction, if you know how that works. Build a set of equations from the table such that q ( x) = a x + b. Simply input the repeating part of the decimal (the repetend) and its non-repeating part (where applicable). Chain Rule: (f(g(x))0 = f0(g(x))g0(x) Common Derivatives Trigonometric Functions d dx (sinx) = cosx d dx (cosx) = sinx d dx (tanx) = sec2 x d dx (secx) = secxtanx d dx (cscx) = cscxcotx d dx (cotx) = csc2 x. Let’s review how we used factoring to solve the quadratic equation. Rule ii) we have seen in the previous Topic. The Des Moines Register is the number one source for Des Moines and Iowa breaking news, jobs, real estate, photos, videos and blogs. Example 5 Find the limit Solution to Example 5: As x gets larger x + 1 gets larger and e^(1/(x+1)-1) approaches 0 hence an indeterminate form infinity. Compute dw dq given that w 2q3 = sin(w2) + q5. Step 5 Substitute y equals 2x^4 + 1, all raised to the exponent tangent x. Find all the GRE exponent practice you need here, along with expert answers Questions. This activity was created to reveal student misconceptions about exponential expressions, and to cause students to critically …. Use the rule y = –2x + 5 to answer the questions below. Rules of Exponents: “Used to Think” Method Lesson Objectives For this lesson, students are learning to combine rules of exponents to simplify expressions. Check your work here: Study Guide – Key. Find all the GRE exponent practice you need here, along with expert answers Questions. Focus Question: Is 00 = 0? The expressions below are NOT equivalent. The interface is specifically optimized for mobile phones and small screens. ^ Power/Exponent operator Parentheses; Constants. "Enhanced" level four rules. Quick Check on Bayes Rule in Terms of Sets. Rule 6: Log of Exponent Rule The logarithm of an exponential number where its base is the same as the base of the log equals the exponent. Instead of x"3, of course, you can also enter xxx. Students will simplify exponent problems using the product rule, power rule, and quotient rule. a ma n= a + 2. These operators follow standard precedence rules, and can operate on any kind of expression. The chessboard problem is a great one. This now completes all six of the exponent rules that you need to memorize and be able to use to solve and simplify a variety of problems. First things first, prioritize major topics with our printable compilation of 8th grade math worksheets with answer keys. practice structures MTBos Blaugust hybrid learning activities objectives based grading resources #teach180 Sunday Funday geometry algebra 2 desmos EdPuzzle Kagan goals ozobot conditional statements logarithms #MTBoS angle pairs lessons #MTBoSBlogsplosion ExploreLearning Warm-ups biconditionals blogging color by number exponent rules first day. Unit 4: Exponent Rules & Polynomial Add/Sub Unit 2: Linear Functions Unit 1: Equations and Word Problems FM Unit 6: Exponent Rules & Polynomial Add/Sub Unit 6: Quadratic Functions/Projectile Motion Unit 5: Multiplying & Factoring Polynomials FM Unit 1: Expressions and Equations FM Unit 2: Literal Equations and Inequalities. Writing large numbers sometimes becomes tedious. 2: Left Endpoint Rectangle eTool (Desmos) Lesson 2. Undo: ctrl + z. A complex number is a number of the form a+bi, where a,b — real numbers, and i — imaginary unit is a solution of the equation: i 2 =-1. For more complex math equations that require the rules of order of operations or PEMDAS, use the Math Equation Solver. Divide by , and. Either multiply the denominators and numerators together or leave the solution in factored form. Study, study, study; practice, practice, practice!. Exponents are simple to learn, but you must know a few rules. Move the original graph y = x3 to the right 2 units. 1-12 Exponent Rules. When operators on the same level of precedence, as a thumb rule, work from left to right. Make math learning fun and effective with Prodigy Math Game. Example 5 Find the limit Solution to Example 5: As x gets larger x + 1 gets larger and e^(1/(x+1)-1) approaches 0 hence an indeterminate form infinity. x3 = x · x · x The patterns that you have been using during this section of the book are called the laws of exponents. 1 10/14-10/15 Rules of Integer Exponents Rules of Exponents Study Guide ALEKS Rational Exponent Quiz 10/20 10/22 Rational. Where: t = time (number of periods) P (t) = the amount of some quantity at time t. These operators follow standard precedence rules, and can operate on any kind of expression. Vanier College Sec V Mathematics Department of Mathematics 201-015-50 Worksheet: Logarithmic Function 1. Let me write it as, let me write it as 10 times t over 10. Four variables - percent change, time, the amount at the beginning of the time period, and the amount at the end of the time period - play roles in exponential functions. This involves the Greek letter sigma, Σ. People should only leave home for essential reasons such as caring responsibilities, essential. Laws of Exponents. desmos quiz, Day 2-Recreate with Desmos: Using Desmos , recreate your Roller Coasters using your knowledge of functions and their transformations. No logarithmic (ln, log) or trigonometric (sine, cosine, tangent) functions. 48 = 1 x 48. ((12 −2) 6 4 −3 2) A common misconception that students may have is to simplify the integer coefficient before applying the power rule. 1) 81^3/4 / 81^1/4 2. Then Rule i) is f(g(x)) = x. I love them. 2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Read about Day 1 here. Desmos Math Art Competition; Exponential Function Videos. You'll find all six in the graphic on this post. Tic Tac Times (Pam Wilson) Adding and Subtracting Polynomials Activity (Sarah Hagan) Polynomial Station Activities (Pam Wilson) Factoring. Rule 2: Simplify all exponents, working from left to right. Simplifying Radicals. here are a few. Loading Complete puzzles related to rules of exponents. Before learning how to add, subtract, multiply and divide polynomials we need to understand the rules of exponents. Like Terms are terms whose variables (and their exponents such as the 2 in x 2) are the same. All you need to remember here is that when you raise an exponent to another exponent, you can multiply them: 10 2. We've prepared a brief introductory video and a series of keyboard shortcut graphing challenges to help you level up your graphing game. Mathispower4u 39 views. where b is a positive real number not equal to 1, and the argument x occurs as an exponent. The handout has a table with some sections already completed so students can complete the rest by noticing patterns and discussing them. For Example 1 the output for LOGEST(B6:B16, A6:A16, TRUE, TRUE) is as in Figure 4. Example D: Write. Log-normal distribution is a statistical distribution of random variables that have a normally distributed logarithm. 1) f(x) = - 2 x + 3 + 4 1) A) domain of f: ( - Q , Q ); range of f: ( - 4, Q ); horizontal asymptote: y = 4 B) domain of f: ( - Q , Q ); range of f: ( - Q , 4); horizontal asymptote: y = 4 C) domain of f: ( - Q , Q ); range of f: ( - Q , - 4); horizontal asymptote: y = - 4 D) domain of f: ( - Q. To do this we simply need to remember the following exponent property. With inequalities, you can add colored shading to your Desmos graph. Includes an introduction to exponents and reading and writing simple exponents. 4) - Negative and zero exponent rules. Nearly all of my students (this year and in the past) who struggle with this topic have almost no trouble with square roots in radical form, some trouble with cube roots in radical form, and (if they have any. Here is a screenshot typical of a first round game. I just observed it yesterday. 5 Exponent Laws part 2 Lesson HW: p126 #8f-i, 9, 10, 12*, 19* Wed Feb 12 1. 7 x 10 ) x (6. Desmos Graphing Site Derivative Rules. DESMOS Partner Activity (Exponents) Radical Expressions Worksheet January 14 Quiz on Exponents Radical Exponents-Practice January 13 Radical Exponents/Roots Operations January 10 Radical Exponents/Roots Operations-Notes January 8/9 Exponent Rules-Practice January 7 Exponent Rules December 17-minimum day Finals Period 4, 5, 6 December 16. Press the "Log" button on your calculator. 3: 10-43 Student. ) Write the product in exponential form. A biased exponent makes that work, while an exponent represented in two's complement would make the comparison more involved. My big things to work on for 2014-2015 are classroom management / effective group work. For example, if you want the number 100000000 as a scientific notation, you enter 10, then click SN, then 7, and then equals (10eX10^7=100000000). Graph a Piecewise Defined Function Using Desmos (3 parts) - Duration: 4:58. Pursue conceptual understanding of topics like number systems, expressions and equations, work with radicals and exponents, solve linear equations and inequalities, evaluate and compare functions, understand similarity and congruence, know and apply the Pythagorean Theorem. It helps to find best approximate solution to the square roots of a real valued function. While things like SIN, square roots, inverse exponents, and pi are often either represented by keys or secondary text above keys, you can find more advanced functions (e. The SCC Rules; the Arbitration Rules and the Rules for Expedited Arbitrations, entered into force on 1 January 2017. d dx (ab) = 0 This is a constant d dx (xn) = nxn−1 Power Rule d dx (ax) = axlna Derivative of an exponential function d dx (xx) = xx(1+lnx) Logarithmic Differentiation d d x ( a b) = 0 This is a constant d d x ( x n) = n x n − 1 Power Rule d d x ( a x) = a x ln. Don't Break the Quotient Rule or Our Friendly Giant will Stick You Between His Couch Cushions that Never get Vacuumed! I've been meaning to put these up for a while. The rule that applies (found in the properties of limits list) is: lim x→a [ f(x) ± g(x) ] = lim 1 ± lim 2. Second: Exponents = 100 ÷ 10 − 4(−6) + 5 Third: Multiplication and Division (from left to right) = 10 + 24 + 5 Fourth: Addition and Subtraction (from left to right) = 39 Evaluate the expression. Power Rule: When raising monomials to powers, multiply the exponents. All exponents in these problems are either positive or zero. The Calculator automatically determines the number of correct digits in the operation result, and returns its precise result. Press the "Log" button on your calculator. Sine waves confused me. xxm mn n Example 3: (x2y3)4 = x2 4 y3 4 = x8y12 Example 4: (2x3yz2)3 = 23 x3 3 y3 z2 3 = 8x9y3z6 Quotient Rule: When dividing monomials that have the same base, subtract the exponents. You'll find all six in the graphic on this post. All log a rules apply for log. In the given function | x| 3, u sing chain rule, first we have to find derivative for the exponent 3 and then for |x|. Note that, even though Desmos geometry has tools to create perpendicular and parallel lines we won't be using them in order to mimic what would be done in the real construction. The rules outlined above apply to the UK and USA. My big things to work on for 2014-2015 are classroom management / effective group work. These fall under the category of classroom management that I just haven't quite figured out yet. Determine the domain, range, and horizontal asymptote of the function. Click on Submit (the blue arrow to the right of the problem) and click on Write the Absolute Value as Piecewise to see the answer. Implicit Differentiation. Occasionally you may find yourself wanting to type superscript (see the adjacent graphic) or subscript (10 2). We are talented in algebra. Use Desmos to investigate the beautiful world of integral calculus. Supported integration rules and methods. dividing zero exponents, integer exponents, negative exponents, fractional exponents, decimal exponents. Today, after completing Day 128 of Estimation 180, we briefly reviewed the first four questions of the handout from Day 1 of exponent mistakes. The rule that applies (found in the properties of limits list) is: lim x→a [ f(x) ± g(x) ] = lim 1 ± lim 2. (Word) / PDF Version; A. Study, study, study; practice, practice, practice!. GeoGebra Classroom Activities. desmos quiz, Day 2-Recreate with Desmos: Using Desmos , recreate your Roller Coasters using your knowledge of functions and their transformations. If you are entering the integral from a mobile phone, you can also use ** instead of ^ for exponents. x (a - b) = x a / x b. Exponential functions tell the stories of explosive change. Definitions. This online calculator implements Euler's method, which is a first order numerical method to solve first degree differential equation with a given initial value. Math Plane ---->>>FLIGHT DELAYS!! Recently, Mathplane has been experiencing slow page loads. If signs are the same keep the signs and add the numbers. Rules of Exponents. (2) A-SSE _ALG. Finally, the product of factors is expressed as the sum of factors. This may indicate the student does not understand the order of applying the laws of exponents when raising a. Learn about linear equations using our free math solver with step-by-step solutions. Anti-logarithm calculator. The calculator, helps you finds the roots of a second degree polynomial of the form ax^2+bx+c=0 where a, b, c are constants, a eq 0. Note: Both numbers are negative and therefore have the same sign. Find the slope of the line tangent to the curve x3 + y3 = xy2 + 5 at the point (1,2). Remember, Desmos is a cloud-based application after all, which means that every time you save a graph and publish it somewhere online, you are in effect contributing to an ever-growing database of Desmos modules — all in the name of science and technology!. Tic Tac Times (Pam Wilson) Adding and Subtracting Polynomials Activity (Sarah Hagan) Polynomial Station Activities (Pam Wilson) Factoring. Practice: Square roots. Move the original graph of the circle x2 + y2 = 9 to the right 2 units. As specified by latex, exponentiation with a power that is more than one character (e. 2 Introduction to Polynomials and Evaluating 5. Sections: 4. I would do this by using post-its to replace the 10’s with 2’s, then 5’s, and eventually rewriting with x’s. It is NOT necessary to use the product rule. Step 1, Isolate the logarithm. If signs are the same keep the signs and add the numbers. I think that correctly mastering exponent rules is a gateway skill. ^ Power/Exponent operator Parentheses; Constants. 2) Use your laptop to go to wolfram alpha to check each of your answers. Joining your favorite partners, like Desmos, PhET, iCivics, and Readworks, are our new video partners, including CrashCourse, SciShow Kids, Professor Dave Explains and more. Alg 1 Lesson Ideas by Lea Ann Smith 1. Occasionally, you'll want to modify an existing activity to make it more personalized, add or delete content, or change. As a result, students become confused on how the negative exponent effects … Continue reading →. Since we can model many physical problems using curves, it is important to obtain an understanding of the slopes of curves at various points and what a slope means in real applications. Given two numbers, a (the dividend) and n (the divisor), a modulo n (abbreviated as a mod n) is the remainder from the division of a by n. Rules or Laws of Logarithms In this lesson, you’ll be presented with the common rules of logarithms, also known as the “log rules”. Learn about linear equations using our free math solver with step-by-step solutions. This is a little node. 3) For each answer you got wrong originally, make up 3 similar problems that use the same rule(s), answer them and check them using Wolfram Alpha. If p(x) p ( x) has degree n n, then it is well known that there are n n roots, once one takes into account multiplicity. Fill in the boxes using digits 1 through 9 to make the biggest number. Step 5 Substitute y equals 2x^4 + 1, all raised to the exponent tangent x. I wanted to change this outcome. You can also browse problems by concept (e. Step 4 Multiply by Y on both sides. Come to Polymathlove. by the GCF (). You should be able to complete this activity purely on the functions you learned from Part 1 of this Unit. Linear functions 2. 452 #4ab, 6bcd, 7b) due Jan 11 Geometric Series (p. Properties of Exponents and Logarithms Exponents Let a and b be real numbers and m and n be integers. Therefore, after dividing by the GCF, the expression is. I really like how Desmos makes this activity much less low-risk -- students can try a function and adjust iteratively until they find the right linear equation/domain, and since they get three chances to graph a side of the square they have a chance to figure out more efficient ways to identify the function and identify patterns and connections. Try Open Omnia Today. In large mathematical expressions, they occupy. Using Desmos "Sketch", students generate patterns to explore zero as an exponent and negative exponents. In a semilogarithmic graph, one axis has a logarithmic scale and the other axis has a linear scale. Week 1 [9/14 - 9/18] Desmos Activity (code on Schoology) B-DAY - Radical Functions 2 - All course materials on Schoology. + (Addition) - (Subtraction) * (Multiplication) / (Division) ^ (Exponent: "raised to the power") sqrt (Square Root) (Example: sqrt(9)). ACT MATH ONLINE TEST. Then you will work through Desmos Marbleslides Lines at student. Solve 5 2 x = 25 3 x + 2. By (date), when given a problem involving evaluating rational exponents for an exponential model (e. The Hypotenuse Calculator makes it easy to find the length of any hypotenuse. Test Review (Replay the Knowledgehook gameshow we did in class if you didn't get perfect). TRANSFORMATIONS OF FUNCTIONS. Tap for more steps Solve a = 2 - b for a. Newton-Raphson Method is a root finding iterative algorithm for computing equations numerically. Now, divide. It went over really well. Polymathlove. For example, if the transmission rate is 6 percent, use 1 + 6/100 =. After you've recorded it, go to your blog at Blogger. Fox and I will send out a weekly plan that. Notice that number lines are spaced equally, or proportionately (see Figure 1). x (a/b) = b th root of (x a) = ( b th (x) ) a. blog b (y a) = ya = (y)a = blog b y a = balog b y Again, if this is true for arbitrary base b, then the exponents on the far left. This involves the Greek letter sigma, Σ. Estimate products of whole numbers. 6 Operations with Fractions Day 1 Lesson HW: 1. Given: $$\left(x^2\right)^5$$ apply the power of a product rule to simplify the expression. (Word) / PDF Version; A. No constants like π or e. Students had a few. 8 Exponents Review Lesson + Finish Summary Card (Exponents). Example 6 Find the. There are ideas for introducing properties and rules of exponents, the dreaded negative exponents, and multiplying and dividing exponents. Desmos calculator: https://www. Now, I have moved to Desmos, where the students play with graphs to learn what the transformations do without ever seeing any “rules”. Team Desmos November 16, 2020 13:05. Practice taking exponents of whole numbers. 4,820 views. Notes on Solving Word Problems of Systems of Inequalities by Graphing (Desmos) *HW 5-10: Solving Systems by Graphing: 12 Review for Test: 13 Test - Systems of Equations and Inequalities: 14 Notes on Exponent Rules *HW 6-1: Simplifying Expressions: 15 Notes on Exponent Rules *HW 6-2: Power Rule. Extend the power rule to functions with negative exponents. Exponent Block Game (Maria Anderson) Scientific Notation Foldable (Julie Reulbach) Properties of Exponents- Around the Room (Miss Rudolph) Polynomials. It is the second degree equation in which one variable contains the variable with an exponent of 2. x 1 + x 2 = a 1 × 10 b1 + a 2 × 10 b2. We require a common language if we want to communicate mathematical concepts So what exactly is Exponential Notation? Well, exponential notation consists of two parts. The exponent of a number says If you find it hard to remember all these rules, then remember this: you can work them out when you. For example, if you want the number 100000000 as a scientific notation, you enter 10, then click SN, then 7, and then equals (10eX10^7=100000000). Exponents power rules Power rule I (a n) m = a n⋅m. When a term with an exponent is raised to a power, we multiply the exponents, so (x 2) 2 becomes x 4. Desmos Parabola Opens LEFT a, h, and k integers. Tuesday - 7-1,2 Exponent Rules Review (Multiplication, Division, Negative, and Zero) Use Desmos to create a Graph of your System of Equations. ^ Used for exponent or to Raise to Power ; sqrt Square root operator; Pi : Represents the mathematical Constant pi or \pi What’s more or what are the limitations of the online algebra calculator. Here are the basic rules with examples: Student Homework & Skeleton-Notes Packet Page 2. When a logarithm is written "ln" it means natural logarithm. All you have to do to use this free online Hypotenuse Calculator is to just enter in the length of side 1 and side 2 and then press the calculate button - that's it!. And Rule ii) is g(f(x)) = x. 12B; Totaling Functions; Function Notation Foldable; A. This Algebra 2 Desmos Graphing Project Activities & Project is suitable for 9th - 12th Grade. Reflection through x -axis. Submit when you are done. Get step-by-step solutions. Topic 5: Multiplying binomials. Desmos isn't here to tell you that we've solved this problem, Scientific notation is useful for a number of things. Hagan’s Book of Exponent Rules from Math = love. ) However, it is entirely possible that the exponent a is neither positive nor a whole number. org Monument Mountain Regional High School 600 Stockbridge Road Great Barrington, MA 01230 (413) 528 - 3346 The 411 from B11, Mrs. Explain why not. Unit 6: Exponents and Exponential Functions. 1 • Average. 00000863 in scientific notation. If a > b and b > c, then a > c. Lessons, Videos and worksheets with keys. Determine if each statement is true or fase. The first one is done for you. If a factor is repeated multiple times, then the product can be written in exponential formAn equivalent expression written using a rational exponent. ” Would you consider a name change to “Two Truths and a Lie (Circle Graphs)”? I use Desmos activities with students and teachers in adult numeracy, where we like to keep the focus on the content adults need, without reminding them what level they are working on. Use the power rule to simplify expressions with exponents raised to powers. Lessons, Videos and worksheets with keys. I already had a couple of activities build for each chapter, and I added a couple of more too. I definitely think you should try it. algebra applets area asymptotes books Byrne calculus christmas cosine derivatives Desmos division domain and range end behavior Euclid exponential functions exponents factoring functions Geogebra geometry graphs group students integration inverse functions JSXGraph kinematics learning limits linear equations logarithms manipulatives Mathematica. Focus Question: Is 00 = 0? The expressions below are NOT equivalent. probability solution-verification. There are ideas for introducing properties and rules of exponents, the dreaded negative exponents, and multiplying and dividing exponents. Change the exponent value to 2. Try a complete lesson on Combining Exponent Rules, featuring video examples, interactive practice Combining Exponent Rules. OlaWilliam. What is an Infinite Discontinuity? An infinite discontinuity has one or more infinite limits—values that get larger and larger as you move closer to the gap in the function. This video shows the method to write a logarithm as a sum or difference of logarithms. Its general form is. The handout has a table with some sections already completed so students can complete the rest by noticing patterns and discussing them. I wanted to change this outcome. Prerequisite: MFM2P or MPM2D In this course, students will study quadratic, trigonometric and exponential functions. A quick analogy: You: Geometry is about shapes, lines, and so on. com/quiz/6983719. As usual, in solving these equations, what we do to one side of an equation we must do to the other side as well. , Desmos) and tracing the plot of the exponential factor, (name) will correctly solve (4 out of 5) problems. By using this website, you agree to our Cookie Policy. Answers and Explanations. ) Write the product in exponential form. Scientific notation converter Scientific notation calculations. Translating into words 1. Recall that in dividing an algebraic expression with the same base, we have to subtract their exponents. Inequalities on the Number Line. Direct Instruction. Derivative Rules 1. Notes on Solving Word Problems of Systems of Inequalities by Graphing (Desmos) *HW 5-10: Solving Systems by Graphing: 12 Review for Test: 13 Test - Systems of Equations and Inequalities: 14 Notes on Exponent Rules *HW 6-1: Simplifying Expressions: 15 Notes on Exponent Rules *HW 6-2: Power Rule. Desmos Files (these files investigate the following) Using Power Rule to determine the derivative at a point; Using Negative Exponents to use Product Rule. Or you can get really small numbers by using negative powers like 10^-5=0. In this section we will be looking at vertical and horizontal shifts of graphs as well as reflections of graphs about the x and y-axis. This involves the Greek letter sigma, Σ. Exponents are used to showing repeated multiplication of a number by itself. Office hours Monday 6/1 Mrs. As usual, in solving these equations, what we do to one side of an equation we must do to the other side as well. properties of exponents. log b (x / y) = log b x - log b y. Starting from the 16th-century, mathematicians faced the special numbers' necessity, also known nowadays as complex numbers. Product Rule. Translating into words 1. Within Desmos, you called the activity, “Two Truths and a Lie (Primary). All exponents in these problems are either positive or zero. Hence, the number of factors of 48 is 10. If signs are the same keep the signs and add the numbers. In other words, terms that are "like" each other. Implicit multiplication (5x = 5*x) is supported. log ( A B − 1) = + The second term on the right side has a value with a negative exponent. EX: log (10 / 2) = log (10) - log (2) = 1 - 0. Due to time constraints, I’ve focused more on the rules in lieu of introducing negative exponents conceptually. com/calculator Algebra tiles from the National Library of Virtual Manipulatives for factoring polynomials:. It explains how to do so with the natural. To do this we simply need to remember the following exponent property. The chessboard problem is a great one. November 02, 2017 Today's Agenda: 1) Correct 5. The exercises covered in this module include finding common multiples, finding the least common multiple for a set of numbers and much more!. We have already solved some quadratic equations by factoring. Rule ii) we have seen in the previous Topic. Therefore, The second equation above is a consequence of this result. The graph of a compound inequality with an "and" represents the intersection of the graph of the inequalities. Explore math with our free online graphing calculator and networked classroom activities!. [Linear and quadratic]. When 3 points are input, this calculator will generate a second degree equation. Vertical translation. Boy was I wrong. The input coordinates are sent to a computer and run through a program that applies the x and y values to those two equations. Comparing and ordering integers' worksheets have a variety of activities to compare the integers. 6ab - Writing Equations of Lines-This is a Word document. If you are entering the derivative from a mobile phone, you can also use ** instead of ^ for exponents. Example: The graph below depicts g (x) = ln (x) and a function, f (x), that is the result of a transformation on ln (x). 5 – Logarithms, Exponents & Other Functions; 6 – Differential Equations; 7 – Applications of Integrals; AP Exam Resources; AP Calculus AB Links; Summer Assignment; YouTube Channel. It will also evaluate the composition at the specified point, if needed. We have already solved some quadratic equations by factoring. McCullough rules that PA preliminary ELECTION CERTIFICATION injunction was PROPERLY ISSUED and should be upheld. For this, we use the property of exponents that (bx)a = bax = bxa. Software Company in San Francisco, California. There are many other tools useful when dealing with trigonometry problems. Pass out the activity for today and allow students one minute to read the directions and look at the first few questions on their own - no talking yet. You may only use a number once. a m a n = a m n, a 6= 0 5. Usage of brackets varies wildly across regions. The largest exponent or the largest sum of exponents of a term within a polynomial Polynomial Degree of Each Term Degree of Polynomial -7m3n5 -7m3n5 → degree 8 8 2x+ 3 2x → degree 1 3 → degree 0 1 6a3 + 3a2b3 – 21 6a3 → degree 3 3a2b3 → degree 5 -21 → degree 0 5. We begin with Polygraph Parabolas from Desmos. Don't have an account yet? Ask your teacher for a code and enter it above. As specified by latex, exponentiation with a power that is more than one character (e. By using this website, you agree to our Cookie Policy. Change the exponent value to 2. In log-log graphs, both axes have a logarithmic scale. Desmos on the Interactive Whiteboard When teaching exponential functions to Year 12, I used Desmos on my IWB to show the graphs of functions such as y = 2 x and y = 3 x. Exponent Rules Homework Handout. I love this game so much, I can't wait to try it again. 5 – Logarithms, Exponents & Other Functions; 6 – Differential Equations Desmos; Desmos Activity; IXL; Calculus Online Textbook 2. Inverse Function Calculator inverts function with respect to a given variable. The resultant graph is y = (x ' 2)3. 1-12 Exponent Rules. Communicating a sense of the size of the numbers involved is one of those things. But when I first came to work at my current school, none of my high schoolers knew their exponent rules. Write your answers using only positive exponents. When dividing functions, you will also need to work with exponents of different powers. Move the original graph y =sin (x) to the right 2 units. Power of a Product Single Term in Parentheses Rule $\left(a^m\right)^n=a^{m \cdot n}$ When raising an exponent to another exponent, such as $$\left(a^m\right)^n=a^{m \cdot n}$$, you multiply the exponents. 5 - Rational Functions and Asymptotes. If a > b and b > c, then a > c. The Slope of a Tangent to a Curve (Numerical Approach) by M. Pre-Algebra. Learn math the simple way. Extend properties of exponents to rational exponents, properties of rational and irrational numbers. 3B Practice Rational Exponents; No HW; Optional Practice RPJ 6. 5 – Logarithms, Exponents & Other Functions; 6 – Differential Equations; 7 – Applications of Integrals; AP Exam Resources; AP Calculus AB Links; Summer Assignment; YouTube Channel. There are a few shortcuts that we can take when multiplying polynomials. This website uses cookies to improve your experience while you navigate through the website. Show your work/thinking. The interface is specifically optimized for mobile phones and small screens. Apply the sum and difference rules to combine derivatives. d dx (ab) = 0 This is a constant d dx (xn) = nxn−1 Power Rule d dx (ax) = axlna Derivative of an exponential function d dx (xx) = xx(1+lnx) Logarithmic Differentiation d d x ( a b) = 0 This is a constant d d x ( x n) = n x n − 1 Power Rule d d x ( a x) = a x ln. 1 a n = a − n 1 a n = a − n. Sines, cosines, and tangents, oh my! 3. Rules or Laws of Logarithms In this lesson, you’ll be presented with the common rules of logarithms, also known as the “log rules”. 75 = ab 0 or a = 1. Where t is the time (total number of periods), P (t) is the amount of a quantity at given time t, P0 is the initial among at the time t = 0, and r is taken as the growth rate. It has 20 questions and it works great as a review before the test. 1 # Anything you think you need (this is precalc 12 and early limit review) 5. Thanks to the ever-innovative-and-tuned-in-to-what-teachers-would-love-to-have Desmos team, I've created an online version of a card sort that I originally developed in paper form to help precalculus students think carefully about the meaning of the parameters in various forms of exponential equations. indd 316snb_alg1_pe_0604. You'll find all six in the graphic on this post. Day 2-Recreate with Desmos: Using Desmos, recreate your Roller Coasters using your knowledge of functions and their transformations. Example: log3(x + 5) + 6 = 10 log3(x + 5) + 6 - 6 = 10 - 6 log3(x + 5) = 4Step 2, Rewrite the equation in exponential form. Exponents are used to showing repeated multiplication of a number by itself. Expressions Wednesday 2/19/20: Practice Exponent Rules; No HW Thursday 2/20/20: 6. Exponents apply only to the values to the immediate left. EX: log (2 6) = 6 × log (2) = 1. Product of Powers Property of Exponents Discovery Activity Complete the following. Get step-by-step solutions. What is an exponent. Answers and Explanations. A quick analogy: You: Geometry is about shapes, lines, and so on. Now we can get rid of the parentheses in the term with the exponents by using the exponent rules we learned earlier. Using the result of an Euler's method approximation to find a missing parameter. Use the rule y = –2x + 5 to answer the questions below. Use inverse operations to move any part of the equation that is not part of the logarithm to the opposite side of the equation. Product, Quotient, and Power Rule for Exponents. You can hide students using the gear button in the teacher dashboard if you want to highlight or screenshot awesome answers, but make sure to not forget about those hidden students in following weeks!. In other words, terms that are "like" each other. Starting from the 16th-century, mathematicians faced the special numbers' necessity, also known nowadays as complex numbers. Let's look at you exponent in a factored form. From simplify exponential expressions calculator to division, we have got every aspect covered. y = f (x + c): shift the graph of y= f (x) to the left by c units. e^{2x}) require curly braces around the exponent. Desmos Graphing Calculator. e Euler's Number (2. , factorials) in the MATH menu. (Word) / PDF. a n × a m = a (n+m) EX: 2 2 × 2 4 = 4 × 16 = 64 2 2 × 2 4 = 2 (2 + 4) = 2 6 = 64 When an exponent is negative, the negative sign is removed by reciprocating the base and raising it to the positive exponent. Desmos Classroom Activities Loading. To complete this. lim x→2 [ (x 2 + √ 2x) ] = lim x→2 (x 2) + lim x→2 (√ 2x). Inequality Rules. Remember, Desmos is a cloud-based application after all, which means that every time you save a graph and publish it somewhere online, you are in effect contributing to an ever-growing database of Desmos modules — all in the name of science and technology!. More Math Symbols. No, they need better proof. Intro to square roots. Exponents worksheets for grade 5. I have a desmos calculator that has complex exponents, and you can make functions in it that my computer sort of stuggles with thinks like z^(1+10i). base "e" exponential functions. This is a double sided PDF foldable that explains the Properties of Exponents. State the constant, constant multiple, and power rules. 1 # Anything you think you need (this is precalc 12 and early limit review) 5. MIT grad shows how to find derivatives using the rules (Power Rule, Product Rule, Quotient Rule, etc. [email protected] The students usually already know the exponent rules by then, but the lesson starts by having them extend them to rationals by asking stuff like: Rewrite 2*4 = 8 only twos as bases and whatever exponents are needed, so (2^1) * (2^2) = (2^3) And then more complicated stuff like. To do this we simply need to remember the following exponent property. McCullough rules that PA preliminary ELECTION CERTIFICATION injunction was PROPERLY ISSUED and should be upheld. P 0 = initial amount at time t = 0. Negative Exponent Rule: , this says that negative exponents in the numerator get moved to the denominator and become positive exponents. practice structures MTBos Blaugust hybrid learning activities objectives based grading resources #teach180 Sunday Funday geometry algebra 2 desmos EdPuzzle Kagan goals ozobot conditional statements logarithms #MTBoS angle pairs lessons #MTBoSBlogsplosion ExploreLearning Warm-ups biconditionals blogging color by number exponent rules first day. For example, the equation $2n+6p=12$ expresses a functional relationship between [latex]n[/latex. Step 3 : Multiplying the modified exponents, we get 5 x 2 = 10. 5 Exponent Laws part 2 Lesson HW: p126 #8f-i, 9, 10, 12*, 19* Wed Feb 12 1. Free for students, parents and educators. Multiply terms with exponents using the general rule: x a + x b = x ( a + b ) And divide terms with exponents using the rule: x a ÷ x b = x ( a – b ) These rules work with any expression in place of a and b , even fractions. Exponent Lab (Criterion B): Scholars investigate power rules through a multi-step lab where they look at exponential form, expanded form, and solutions in order to generate the rules themselves. If a factor is repeated multiple times, then the product can be written in exponential form An equivalent expression written using a rational exponent. Scholars verify their results using additional examples. com is undoubtedly the ideal place to stop by!. 1415926535897932384626433).


text:
# Why did Christopher Columbus and his crew run out of food on their voyage in 1492?

1 answer
###### Question:

Why did Christopher Columbus and his crew run out of food on their voyage in 1492?

## Answers

2 answers

### The term "Pax Mongolica" ("Mongolian Peace") is used by historians to describe

The term "Pax Mongolica" ("Mongolian Peace") is used by historians to describe...
1 answer

### SOMEEEEONEEE pls help pls and thx :) Question 1) A sculptor created a marble water basin by beginning with a cylinder and then carving out a hemisphere. The cylinder has a radius of 30 centimeters and a height of 45 centimeters. The hemisphere has a radius of 25 centimeters. What is the approximate volume of the marble that forms the water basin? A) 123,307.5 cubic centimeters B) 61,784.7 cubic centimeters C) 125,925.5 cubic centimeters D) 94,509.6 cubic centimeters Question 2) Toby made a hear

SOMEEEEONEEE pls help pls and thx :) Question 1) A sculptor created a marble water basin by beginning with a cylinder and then carving out a hemisphere. The cylinder has a radius of 30 centimeters and a height of 45 centimeters. The hemisphere has a radius of 25 centimeters. What is the approximate...
1 answer

### Medical researchers have determined that for exercise to be beneficial, a person’s desirable heart rate, R, in beats per minute, can be approximated by the formulas where a represents the person’s age. If the desirable heart rate for a man is 135 beats per minute, how old is he?

Medical researchers have determined that for exercise to be beneficial, a person’s desirable heart rate, R, in beats per minute, can be approximated by the formulas where a represents the person’s age. If the desirable heart rate for a man is 135 beats per minute, how old is he?...
1 answer

1 answer

2 answers

### Heeeeeelp pleaseeeeeee

Heeeeeelp pleaseeeeeee...
2 answers

### Henry is 4 years younger than Linda. Linda is 20 years old. After correctly carrying out the Solve step of the five-step problem-solving plan, which solution should you arrive at to find Henry’s age?

Henry is 4 years younger than Linda. Linda is 20 years old. After correctly carrying out the Solve step of the five-step problem-solving plan, which solution should you arrive at to find Henry’s age?...
1 answer

### The weight of water is 62 1/2 lb per cubic foot water that weighs 300 lb will fill how many cubic feet

The weight of water is 62 1/2 lb per cubic foot water that weighs 300 lb will fill how many cubic feet...
1 answer

### Need help rewriting the equation

Need help rewriting the equation...
1 answer

### A story in which characters experience different events but are connected by a common theme is an example of a

A story in which characters experience different events but are connected by a common theme is an example of a...
2 answers

### Write the equation of the line that contains the given point and the given slope. (-46,45), slope=2.5 please help and explain

Write the equation of the line that contains the given point and the given slope. (-46,45), slope=2.5 please help and explain...
1 answer

### What are redox half-reactions?￼ A. Equations that separate the oxidation and reduction parts of the reaction B. Equations that separate the reactants from the products in a redox reaction C. Equations that separate the electron transfer part from the atomic part D. Equations that show the oxidation potentials and the reduction potentials (Answer is A)

What are redox half-reactions?￼ A. Equations that separate the oxidation and reduction parts of the reaction B. Equations that separate the reactants from the products in a redox reaction C. Equations that separate the electron transfer part from the atomic part D. Equations that show the oxidatio...
1 answer

### X + 3y = −1 2x + 2y = 6

x + 3y = −1 2x + 2y = 6...
1 answer

### NEED HELP ASAP PLS (with work) A punt is kicked at an angle of 60 with a speed of 40 m/s, and lands at the same height from which it was kicked. How far away will the ball land? What is the maximum height reached by the ball? What is the hang time? Total

NEED HELP ASAP PLS (with work) A punt is kicked at an angle of 60 with a speed of 40 m/s, and lands at the same height from which it was kicked. How far away will the ball land? What is the maximum height reached by the ball? What is the hang time? Total...
1 answer

### Which of the following molecules has only two non binding pairs (long pairs) of electrons in its lewis structure? a)O2 b)CH4 c)NH3 d)H2s

which of the following molecules has only two non binding pairs (long pairs) of electrons in its lewis structure? a)O2 b)CH4 c)NH3 d)H2s...
2 answers

### What is the arithmetic mean of 7 and 27?

What is the arithmetic mean of 7 and 27?...
1 answer

### Why would the air temperature over New Jersey be warmer than it is in Pennsylvania?

Why would the air temperature over New Jersey be warmer than it is in Pennsylvania?...
1 answer

### Balancing Equations: _(NH4)3PO4 + _Pb(NO3)4 = _Pb3(PO4)4 + _NH4NO3 With work please.

Balancing Equations: _(NH4)3PO4 + _Pb(NO3)4 = _Pb3(PO4)4 + _NH4NO3 With work please....
1 answer

### Express the sum in the simplest form. x/x+5 + 5/x+5

Express the sum in the simplest form. x/x+5 + 5/x+5...
1 answer

-- 0.053026--

format: plain


text:
# How do you find the parametric equations for a line segment?

Sep 6, 2014

The line segments between $\left({x}_{0} , {y}_{0}\right)$ and $\left({x}_{1} , {y}_{1}\right)$ can be expressed as:
$x \left(t\right) = \left(1 - t\right) {x}_{0} + t {x}_{1}$
$y \left(t\right) = \left(1 - t\right) {y}_{0} + t {y}_{1}$,
where $0 \le q t \le q 1$.

The direction vector from $\left({x}_{0} , {y}_{0}\right)$ to $\left({x}_{1} , {y}_{1}\right)$ is
$\vec{v} = \left({x}_{1} , {y}_{1}\right) - \left({x}_{0} , {y}_{0}\right) = \left({x}_{1} - {x}_{0} , {y}_{1} - {y}_{0}\right)$.
We can find any point $\left(x , y\right)$ on the line segment by adding a scalar multiple of $\vec{v}$ to the point $\left({x}_{0} , {y}_{0}\right)$. So, we have
$\left(x , y\right) = \left({x}_{0} , {y}_{0}\right) + t \left({x}_{1} - {x}_{0} , {y}_{1} - {y}_{0}\right)$,
which simplifies to:
$\left(x , y\right) = \left(\begin{matrix}1 - t {x}_{0} + t {x}_{1} \\ 1 - t {y}_{0} + t {y}_{1}\end{matrix}\right)$,
where $0 \le q t \le q 1$.


text:
# Grav field fun!

by Phymath
Tags: field, grav
 P: 186 $$\vec{g}(x,y,z) = -kG((x^3 y^2 z^2)\hat{e_x} + (x^2 y^3 z^2)\hat{e_y} + (x^2 y^2 z^3) \hat{e_z})$$ given this grav field (k is constant) find the mass density of the source of this field, and what is the total mass in a cube of side 2a centered about the origin? hmmm well we all know...$$\int \int \vec{g} \bullet d\vec{a} = 4 \pi G m_{enclosed}$$ and density $$p = \frac{m}{V}$$ at least the overall density of it is (non-differential) sooooo...$$\frac{1}{4 \pi G}\int\int\int (\nabla \bullet \vec{g}) dV = m_{source/enclosed}$$ now the limits i made a cube of side 2a, because the flux through a box is easier when g is given in cart coords....any way i get... $$m_{source} = \frac{-2}{3 \pi}G k a^9$$ how do I get a neg mass (unless this is dark matter which it very well could be) and I'm thinking i missed something about density cause why would it ask that first and then the mass enclosed second?...a lil help?
 Sci Advisor HW Helper P: 11,928 Grav field fun! You forgot about the minus in the RHS of Gauss's theorem for the gravitostatic field... $$\oint\oint_{\Sigma} \vec{g}\cdot d\vec{S}=\mbox{-}4\pi G m_{\mbox{enclosed by}\ \Sigma}$$ Daniel.
 P: 186 ok if $$\frac{1}{4 \pi G}\int\int\int (\nabla \bullet \vec{g}) dV = m_{source/enclosed}$$ then...ill just do it and show you $$\nabla \bullet \vec{g} = -9kGx^2y^2z^2$$ $$\frac{9k}{4 \pi }\int^a_{-a}\int^a_{-a}\int^a_{-a} (x^2y^2z^2) dxdydz = \frac{2}{3}k \pi a^9 = m_{source/enclosed}$$ so as my "flux cube" getts larger obviously it isn't limiting at some value as the function is ever increasing so, the mass should always be increasing yes?


text:
# Stanton number

The Stanton number, St or CH, is a dimensionless number that measures the ratio of heat transferred into a fluid to the thermal capacity of fluid. It is used to characterize heat transfer in forced convection flows.

$\mathrm{St} = \frac{h}{c_p \rho V}$

where

It can also be represented in terms of the fluid's Nusselt, Reynolds, and Prandtl numbers:

$\mathrm{St} = \frac{\mathrm{Nu}}{\mathrm{Re}\cdot\mathrm{Pr}}$[1]

where

The Stanton number arises in the consideration of the geometric similarity of the momentum boundary layer and the thermal boundary layer, where it can be used to express a relationship between the shear force at the wall (due to viscous drag) and the total heat transfer at the wall (due to thermal diffusivity).

The Stanton number is named after Thomas Edward Stanton (1865–1931).[2]

## References

ca:Nombre de Stanton


text:
# NCERT Solutions for Class 9 Science Chapter 4 Structure of the Atom

These NCERT Solutions for Class 9 Science Chapter 4 Structure of the Atom Questions and Answers are prepared by our highly skilled subject experts to help students while preparing for their exams.

## Structure of the Atom NCERT Solutions for Class 9 Science Chapter 4

### Class 9 Science Chapter 4 Structure of the Atom InText Questions and Answers

Question 1.
What are canal rays?
Canal rays are positively charged radiations. These rays consist of positively charged particles known as protons. They were discovered by Goldstein in 1886.

Question 2.
If an atom contains one electron and one proton, will it cany any charge or not?
An electron is a negatively charged; particle, whereas a proton is a positively charged; particle. The magnitude of their charges is equal. Therefore, an atom containing one electron and one proton will not carry any charge. Thus, it will be a neutral atom.

Question 3.
Name the three sub-atomic particles of an atom.
The three sub-atomic particles of an atom are:

1. Protons
2. Electrons
3. Neutrons

Question 4.
Helium atom has an atomic mass of 4 u and two protons in its nucleus. How many neutrons does it have?
A helium atom has two neutrons. The mass of an atom is the sum of the masses of protons and neutrons present in its nucleus. Since helium atom has two protons, mass contributed by the two protons is (2 × 1) u = 2 u. Then, the remaining mass (4 – 2) u = 2 u is contributed $$\frac{2 \mathrm{u}}{1 \mathrm{u}}$$ = 2 by neutrons.

Question 5.
Draw a sketch of Bohr’s model of an atom with three shells.
Bohr’s model of an atom with three shells

Question 6.
What do you think would be the observation if the a-particle scattering experiment is carried out using a foil of a metal other than gold?
If the a-scattering experiment is carried out using a foil of a metal rather than gold, there would be no change in the observation. In the a- scattering experiment, a gold foil was taken because gold is malleable and a thin foil of gold can be easily made. It is difficult to make such foils from other metals.

Question 7.
Write the distribution of electrons in carbon and sodium atoms?
The total number of electrons in a carbon atom is 6. The distribution of electrons in carbon atom is given by:
First orbit or K-shell = 2 electrons
Second orbit or L-shell = 4 electrons
Or, we can write the distribution of electrons in a carbon atom as 2, 4.

The total number of electrons in a sodium atom is 11. The distribution of electrons in sodium atom is given by:
First orbit or K-shell = 2 electrons
Second orbit or L-shell = 8 electrons
Third orbit or M-shell = 1 electron
Or, we can write the distribution of electrons in a sodium atom as 2, 8, 1.

Question 8.
If K and L shells of an atom are full, then what would be the total number of electrons in the atom?
The maximum number of electrons that can occupy K and L-shells of an atom are 2 and 8 respectively. Therefore, if K and L-shells of an atom are full, then the total number of electrons in the atom would be (2 + 8) = 10 electrons.

Question 9.
If number of electrons in an atom is 8 and number of protons is also 8, then (i) what is the atomic number of the atom and (ii) what is the charge on the atom?
(i) The atomic number is equal to the number of protons. Therefore, the atomic number of the atom is 8.
(ii) Since the number of both electrons and protons is equal, therefore, the charge on the atom is 0.

Question 10.
With the help of Table 4.1, find out the mass number of oxygen and sulphur atom.
Mass number of oxygen = Number of protons + Number of neutrons
= 8 + 8
= 16
Mass number of sulphur=Number of protons + Number of neutrons
= 16 + 16
= 32

Question 11.
For the symbol H, D and T tabulate three sub-atomic particles found in each of them.

 Symbol Proton Neutron Electron H 1 0 1 D 1 1 1 T 1 2 1

Question 12.
Write the electronic configuration of any one pair of isotopes and isobars.
Two isoto of carbon are $$\frac {12}{6}$$C and $$\frac {14}{6}$$C.
The electronic configuration of $$\frac {12}{6}$$C is 2, 4.
The electronic configuration of $$\frac {14}{6}$$C is 2, 4.
[Isotopes have the same electronic configuration]
$$\frac {40}{20}$$Ca and $$\frac {40}{18}$$Ar are a pair of isobars
The electronic configuration of $$\frac {40}{20}$$Ca is 2, 8, 8, 2.
The electronic configuration of $$\frac {40}{18}$$Ar is 2, 8, 8.

### Class 9 Science Chapter 4 Structure of the Atom Textbook Questions and Answers

Question 1.
Compare properties of electrons, protons and neutrons.

 Electron Proton Neutron (i) Electrons are present outside the nucleus of an atom (i) Protons are present in the nucleus of an atom. (i) Neutrons are present in the of an atom (ii) Electrons are negatively charged (ii) Protons are positively charged (ii) Neutrons are neutral (iii) The mass of an electron is considered to negligible (iii) The mass of a proton is approximately 2000 times as times as the mass of an of an electron (iii) The mass of neutron is nearly equal to the mass of a proton

Question 2.
What are the limitations of J.J. Thomson’s model of the atom?
According to J.J. Thomson’s model of an atom, an atom consists of a positively charged sphere with electrons embedded in it. However, it was later found that the positively charged particles reside at the centre of the atom called the nucleus, and the electrons revolve around the nucleus.

Question 3.
What are the limitations of Rutherford’s model of the atom?
According to Rutherford’s model of an atom, electrons revolve around the nucleus in fixed orbits. But, an electron revolving in circular orbits will not be stable because during revolution, it will experience acceleration. Due to acceleration, the electrons will lose energy in the form of radiation and fall into the nucleus. In such a case, the atom would be highly unstable and collapse.

Question 4.
Describe Bohr’s model of the atom.
Bohr’s model of the atom:
Niels Bohr proposed the following postulates regarding the model of the atom.

1. Only certain orbits known as discrete orbits of electrons are allowed inside the atom.
2. While revolving in these discrete orbits, the electrons do not radiate energy.

These discrete orbits or shells are shown in the following diagram.

The first orbit (i.e., for n = 1) is represented by letter K. Similarly, for n = 2, it is L – shell, for n = 3, it is M – shell and for n = 4, it is N – shell. These orbits or shells are also called energy levels.

Question 5.
Summarize the rules for writing of distribution of electrons in various shells for the first eighteen elements.
The rules for writing of the distribution of electrons in various shells for the first eighteen elements are given below.
(i) The maximum number of electrons that a shell can accommodate is given by the formula ‘2n2‘, where ‘n’ is the orbit number or energy level index (n = 1, 2,3..).

The maximum number of electrons present in an orbit of n = 1 is given by 2n2 = 2 × 12 = 2
Similarly, for second orbit, it is 2n2 = 2 × 22 = 8
For third orbit, it is 2n2 = 2 × 32 = 18
And so on ……
(ii) The outermost orbit can be accommodated by a maximum number of 8 electrons.
(iii) Shells are filled with electrons in a stepwise manner i.e., the outer shell is not occupied with electrons unless the inner shells are completely filled with electrons.

Question 6.
Define valency by taking examples of silicon and oxygen.
The valency of an element is the combining capacity of that element. The valency of an element is determined by the number of valence electrons present in the atom of that element.

If the number of valence electrons of the atom of an element is less than or equal to four, then the valency of that element is equal to the number of valence electrons. For example, the atom of silicon has four valence electrons. Thus, the valency of silicon is four.

On the other hand, if the number of valence electrons of the atom of an element is greater than four, then the valency of that element is obtained by subtracting the number of valence electrons from eight. For example, the atom of oxygen has six valence electrons. Thus, tire valency of oxygen is(8 – 6) i.e., two.

Question 7.
Explain with examples (i) Atomic number (ii) Mass number (iii) Isotopes and (iv) Isobars. Give any two uses of isotopes.
(i) Atomic number:
The atomic number of an element is the total number of protons present in the atom of that element. For example, nitrogen has 7 protons in its atom. Thus, the atomic number of nitrogen is 7.

(ii) Mass number:
The mass number of an element is the sum of the number of protons and neutrons present in the atom of that element. For example, tire atom of boron has 5 protons and 6 neutrons. So, the mass number of boron is 5 + 6 = 11.

(iii) Isotopes:
Isotopes are atoms of the same element having the same atomic number, but different mass numbers. For example, hydrogen has three isotopes. They are protium $$\left({ }_{1}^{1} \mathrm{H}\right)$$, deuterium $$\left({ }_{1}^{2} \mathrm{H}\right)$$, and tritium $$\left({ }_{1}^{3} \mathrm{H}\right)$$.

(iv) Isobars:
Isobars are atoms having the same mass number, but different atomic numbers i.e., isobars are atoms of different elements having the same mass number. For example, $${ }_{20}^{40} \mathrm{Ca}$$ and $${ }_{18}^{40} \mathrm{Ar}$$ andare isobars.

Two uses of isotopes are:

1. One isotope of uranium is used as a fuel in nuclear reactors.
2. One isotope of cobalt is used in the treatment of cancer.

Question 8.
Na+ has completely filled K and L shells. Explain.
An atom of Na has a total of 11 electrons. Its electronic configuration is 2,8,1. But, Na+ ion has one electron less than Na atom i.e., it has 10 electrons. Therefore, 2 electrons go to K-shell and 8 electrons go to L-shell, thereby completely filling K and L shells.

Question 9.
If bromine atom is available in tile form of, say, two isotopes $${ }_{35}^{79} \mathrm{Br}$$ (49.7%) and $${ }_{35}^{81} \mathrm{Br}$$ (50.3%), calculate the average atomic mass of bromine atom.
It is given that two isotopes of bromine are $${ }_{35}^{79} \mathrm{Br}$$(49.7%) and $${ }_{35}^{81} \mathrm{Br}$$ (50.3%). Then, the average atomic mass of bromine atom is given by:

= 80.006 u
= 80 u (approx)

Question 10.
The average atomic mass of a sample of an clement X is 16.2 u. What are the
percentages of isotopes $${ }_{8}^{16} \mathrm{X}$$ and $${ }_{8}^{18} \mathrm{X}$$ in the sample?
It is given that the average atomic mass of the sample of element X is 16.2 u.
Let the percentage of isotope $${ }_{8}^{18} \mathrm{X}$$ be y%. Thus,
the percentage of isotope $${ }_{8}^{16} \mathrm{X}$$ will be (100 – y) %.

Therefore,

⇒ 18y + 1600 = 1620
⇒ 2y + 1600 = 1620
⇒ 2y +1620 – 1600
⇒ 2y = 20
⇒ y = 10
Therefore, the percentage of isotope $${ }_{8}^{18} \mathrm{X}$$ is 10%.
And, the percentage of isotope $${ }_{8}^{16} \mathrm{X}$$ is (100 – 10) % = 90%.

Question 11.
If Z = 3, what would be the valency of the element? Also, name the element
By Z = 3, we mean that the atomic number of the element is 3. Its electronic configuration is 2, 1. Hence, the valency of the element is 1 (since the outermost shell has only one electron).

Therefore, the element with Z = 3 is lithium.

Question 12.
Composition of the nuclei of two atomic species X and Y are given as under

Give the mass numbers of X and Y. What is the relation between the two species?
Mass number of X = Number of protons + Number of neutrons
= 6 + 6
= 12
Mass number of Y = Number of protons + Number of neutrons
= 6 + 8
= 14
These two atomic species X and Y have the same atomic number, but different mass numbers. Hence, they are isotopes.

Question 13.
For the following statements, write T for ‘True’ and F for ‘False’.
(a) J.J. Thomson proposed that the nucleus of an atom contains only nucleons.
(b) A neutron is formed by an electron and a proton combining together. Therefore, it is neutral.
(c) The mass of an electron is about $$\frac{1}{2000}$$ times that of proton.
(d) An isotope of iodine is used for making tincture iodine, which is used as a medicine.
(a) J.J. Thomson proposed that the nucleus of an atom contains only nucleons: (F)
(b) A neutron is formed by an electron and a proton combining together. Therefore, it is neutral (F).
(c) The mass of an electron is about $$\frac{1}{2000}$$ times that of proton. (T)
(d) An isotope of iodine is used for making tincture iodine, which is used as a medicine. (F)

Question 14.
Put tick (√) against correct choice and cross (×) against wrong choice in the following question:
Rutherford’s alpha-particle scattering experiment was responsible for the discovery of
(a) Atomic nucleus
(b) Electron
(c) Proton
(d) Neutron
Rutherford’s alpha-particle scattering experiment was responsible for the discovery of
(a) Atomic nucleus (√)
(b) Electron
(c) Proton
(d) Neutron

Question 15.
Put tick (√) against correct choice and cross (×) against wrong choice in the following question:
Isotopes of an element have
(a) the same physical properties
(b) different chemical properties
(c) different number of neutrons
(d) different atomic numbers
Isotopes of an element have
(a) the same physical properties
(b) different chemical properties
(c) different number of neutrons (√)
(d) different atomic numbers

Question 16.
Put tick (√) against correct choice and cross (×) against wrong choice in the following question:
Number of valence electrons in Cl- ion are:
(a) 16
(b) 8
(c) 17
(d) 18
Number of valence electrons in Cl- ion are:
(a) 16
(b) 8(√)
(c) 17
(d) 18

Question 17.
Which one of the following is a correct electronic configuration of sodium?
(a) 2, 8
(b) 8, 2, 1
(c) 2, 1, 8
(d) 2, 8, 1
(d) The correct electronic configuration of sodium is 2, 8, 1.

### Class 9 Science Chapter 4 Structure of the Atom Additional Important Questions and Answers

Question 1.
What are nucleons?
The subatomic particles present in the nucleus of an atom are called nucleons. These include protons and neutrons.

Question 2.
Which element does not contain any neutron?
Hydrogen

Question 3.
Who discovered the presence of the protons in an atom?
Goldstein after his discovery of anode rays.

Question 4.
What did Rutherford discover from his alpha particle scattering experiment?
Rutherford from alpha particle scattering experiment had discovered the nucleus of an atom.

Question 5.
Which subatomic particles were not known when Rutherford performed his experiment?
Neutrons

Question 6.
What is the number of protons and neutrons present in a helium atom?
In a helium atom the number of both protons and neutrons is two.

Question 7.
What is common in isotopes of an element?
Atomic number

Question 8.
What was the major drawback of Rutherford’s model of atom?
Rutherford had failed to describe the stability of an atom.

Question 9.
What are the fundamental building blocks of matter?
Atoms and molecules

Question 10.
What is the mass and charge of a proton?
The mass of a .proton is taken as one unit and so also its charge is taken as one positive unit.

Question 11.
What is the charge on the nucleus of an atom?
The nucleus of an atom is positively charged.

Question 12.
What are valence electrons?
The electrons present in the outermost shell of an atom are called valence electrons.

Question 13.
If an atom has atomic number-16, what are the number of valence electrons and what is its valency?
Electronic configuration: 2, 8, 6.
Valence electrons = 6
Valency = 2

Question 14.
Which isotope of uranium is used as nuclear fuel in nuclear reactor?
U-235.

Question 1.
How will you prove that matter is electrical in nature but neutral?
When a glass rod rubbed with silk cloth is brought near an inflated balloon, the balloon gets attracted to the rod but when a glass rod without being rubbed to silk cloth is brought close to an inflated balloon nothing happens shows that matter is electrical in nature but neutral.

Question 2.
What made the scientists to realize that atom is divisible after Dalton proposed his theory of atom in which he described atom as indivisible?
The attraction of bits of paper by a comb after combing dry hair or gluing of a balloon to a wall after rubbing made the scientists think that atom is divisble.

Question 3.
What is the charge and mass of an electron and a proton?
Charge on an electron = -1.6 x 10-19C
Mass of an electron = 1/1840 times of a proton
Charge on a proton = + 1.6 x 10-19C
Mass of an electron = mass of a hydrogen atom

Question 4.
(i) What are the mass number (A) and atomic number (Z) of an element?
(ii) How are they indicated in a symbol of an atom?
(i) Mass number refers to the sum of the numbers of protons and neutrons present in atom while the atomic number refers to the number of protons present in the nucleus of an atom.
(ii) Mass number is always written in superscript and atomic number in the subscript as shown below.

Question 5.
(i) What is the nucleus of atom?
(ii) What charge does the nucleus of an atom carry?
(i) Nucleus is a small dense region present in the centre of an atom. It is 10s times smaller than an atom. It is like a football placed in the centre of football field, (ii) It contains positively charged protons and neutral neutrons, therefore it is positively charged.

Question 6.
What did Thomson propose about the model of an atom?
Thomson proposed that:

• An atom consists of a positively charged into it.
• The negative and positive charges are equal in magnitude and hence, an atom is electrically neutral.

Question 7.
If in an element K shell is completely filled then what will be the
(a) number of valence electrons
(b) valence and
(c) name of element.
(a) Number of valence electrons = 2
(b) Valence = Zero
(c) Name of element = Helium

Question 8.
If chlorine atom is available in the form of say two isotopes 35C17(75%). 37Cl17(25%). Calculate the average atomic mass of bromine.
The average atomic mass of bromine would be

Question 9.
The average atomic mass of a sample of an element X is 12.2 u. What is the percentage of isotopes 12X6 and 14X6 in the sample?
Atomic mass of X =12 unit
Let the percentage of 12X6 = X %
Percentage of 14X6 = Y% = (100 – x)%

Question 10.
What are isotopes? State their uses.
Isotopes of an element are the atoms which have the same atomic number: different atomic mass number because of the presence of the different number neutrons. These isotopes are used in

• Generation of electricity as fuels in nuclear power stations such as uranium-235.
• In chemotherapy such as Co-60 is used in treatment of cancer.
• Isotope of iodine is used in treatment of goitre.
• Research works to find the metabolic pathways.

Question 11.
How is an isotope different from an isobar?
Isotopes are the atoms of an element which have same atomic number – differ in the atomic mass number because they carry different number of neutrons but Isobars are the atoms of different elements which have same atomic mass number but they differ in their atomic number.

Question 12.
Which of the following are the isotopes and isobars? Why?
18X8, 18Y6, 16X8
18X8, and 18Y6 are isobars because they have different atomic number but same atomic mass number.
18X8, and 16X8 are isotopes because they have same atomic number but different atomic masses.

Question 13.
How is an alpha particle different from an atom of helium?
Alpha particle is the nucleus of helium represented by the formula without any electron but an atom of helium consists of 2 electrons along nucleus having two protons and neutrons.

Question 14.
Calculate the number of electrons, protons and neutrons in the following
40X20, 14Y6, 238Z92
40X20 – Number of protons = 20, Number of electrons = 20 Number of neutrons = 20
14Y6 – Number of protons = 6, Number of electrons = 6 Number of neutrons = 8
238Z92 – Number of protons = 92, Number of electrons = 92 Number of neutrons = 146

Question 15.
Why the valence electrons are considered responsible for the chemical activity of an element?
Valence electrons are the electrons present in the outermost shell of an atom. The number of these electrons determines that how many electrons an atom need to gain or lose, to gain stability. The lesser is the number of electrons required to be lost or gained, the reactive is the element.

Question 16.
How is a discharge tube different from an ordinary tube?
A discharge is a tube with the gas under very low pressure i.e. 1 atmospheric pressure where it is possible to generate the electric discharge but in an ordinary tube has average atmospheric pressure in which an electrical discharge cannot be produced in any condition.

Question 1.
What are cathode rays? How are they produced?
When an electric discharge is produced in a discharge tube at low pressure 10-2 atmospheric pressure and the voltage of 10,000 V, the rays emitted from the cathode are called cathode rays. These rays consist of negatively charge particles called electrons because they get deflected towards the positively charged electrode when passing through an electric field.

In discharge tube, the low pressure is maintained to have less number of molecules hence, minimum interference in the path of the cathode rays emitted from cathode. The emission of these rays proved that the atoms are not indivisible as stated by Dalton in his atomic theory but contain negatively charge electrons.

Question 2.
What are anode rays? Describe their nature.
Anode rays are the positively charged rays produced in a discharge tube having a perforated cathode. These rays consist of positively charge particles called protons. These rays get deflected towards the negatively charged electrode when passing through an electric field.

The charge on a proton is equal to the charge on an electron, 1.6 × 10-19C but is resite in sign being positively charged.

Question 3.
What observations did Rutherford made in his alpha particles scattering experiment? What did he conclude from his observations?
Rutherford in his alpha particle experiment when he made the alpha particles incident on the surface of a very thin and fine gold foil observed that

• Most of the fast moving alpha particles passed through the gold foil.
• Some of the alpha particles underwent deflection in their path.

• Very few i.e. one out of 12,000 alpha particle bounced back at 180°.
From his observations Rutherford concluded and proposed that
• There is a positively charged centre in an atom. He called this positively charged centre as nucleus.
• The size of the nucleus is very small as compared to the size of an atom.
• The negatively charged electrons are present outside the nucleus revolving around it.

Question 4.
Explain Rutherford’s α-particle scattering experiment and give its observation and conclusion drawn.
Rutherford’s α-particle scattering experiment:
Fast moving α-particles were made to fall on a thin gold foil. Particles have + 2 charge and 4u mass, and considerable amount of energy.

Observations:

• Most of the α-particles passed straight through the foil.
• Some Of the α-particles were deflected by small angles by the foil.
• One out of every 12000 particles rebounded.

Conclusion from observation:

• Most of the space inside the foil is empty.
• Positive charge of atom occupies very less space.
• Mass of the atom is concentrated in the centre with all positive charge concentrated in small volume within the atom.
error: Content is protected !!


text:
# Forced Damped Vibration

This applet shows the solution of a forced damped spring-mass system: ,, , for .

You can adjust the damping constant , forcing angular frequency , forcing amplitude and initial conditions and by using the sliders, or typing a value in the box at the right and pressing Enter. The allowed ranges of these parameters are as follows: , , ,, .

The forcing term is shown in green, and the solution y in red. If you click on the graph, the coordinates [t,y] of the point where you clicked are shown at the top left of the graph.

Robert Israel
11/5/2000


text:
# How to write lagrangian terms related to only one variable in a semidefinite constraint?

I have a semidefinite problem as follows(which is nonconvex) \begin{alignat}{3} &\min_{x_{un}} \min_{t,H,w} &&t+f( w)\cr &\text{s.t. } &&\begin{bmatrix} K\odot H &1-Xg(w)+ c\\ (1-Xg(w)+c)^\mathsf{T} &t \end{bmatrix} \succeq 0 \cr &&&X=diag([x_k,x_{un}])\in R^{n*n},\cr&&&(w\in R^n, f(w)\in R, g(w)\in R^n,K,H\in R^{n*n}) \end{alignat}

$f(w)$ is a convex quadratic and $g(w)$ is linear function of $w$. I think it's correct to move optimization about $x_{un}$ to most inner problem. Is it correct?

Then how can I write Lagrangian and obtain the dual problem with respect to the most inner variable, i.e, $x_{un}$? My confusion is how to write Lagrangian terms related to two semidefinite constraints, because I don't want to write the Lagrangian terms related to the any other variable.

EDIT: After sometime, I think it may be incorrect to write lagrangian terms related to only single variable. Any hint or comment, Appreciated.

Don't quite understand your question. The primal variable does not matter when you write the Lagrangian form, the constraint matters. For the positive-semidefinite constraint, you also need a matrix as the Lagrangian variable. $$\lambda=\begin{bmatrix} \lambda_{11} & \lambda_{12} \\ \lambda_{21} & \lambda_{22} \\ \end{bmatrix}$$ And when you write the Lagrangian equation, use the trace term for the positive-semidefinite constraint. Suppose your constraint is $A \geq 0$,corresponding Lagrangian term should be: $$Trace(A\lambda)$$ Hope this can help you.


text:
# Math Help - Abstract Math proof

1. ## Abstract Math proof

If a_n <= a_(n+1) for all n, show a_i <= a_n for all n

2. Originally Posted by natester
If a_n <= a_(n+1) for all n, show a_i <= a_n for all n
As written, the statement is not true.
Is $a_{10} \le a_n$ for $\forall n$?
Well of course not!
So there must be more to the statement of this problem than you have told us.

3. sorry I input the wrong thing. Instead of show a_i <= a_n for all n, it should be show a_1 <= a_n for all n. I also need to use induction to solve for it.

4. Originally Posted by natester
sorry I input the wrong thing. Instead of show a_i <= a_n for all n, it should be show a_1 <= a_n for all n. I also need to use induction to solve for it.
Actually it occurred to me that you might have meant that.
But because it makes it such a trivial question, I though ‘surely not’.

5. Yes that made it a lot easier. But thank you anyways


text:
Intermediate Algebra (6th Edition)

$$-24$$
$$-14+(-10)$$ Add -14 to (-10) to get the answer of -24: $-14+(-10)=-24$


text:
### Author Topic: TUT0602 QUIZ2  (Read 517 times)

#### Kun Zheng

• Jr. Member
• Posts: 6
• Karma: 0
##### TUT0602 QUIZ2
« on: October 04, 2019, 02:31:08 PM »
Hi everyone, the problem is to prove (x^2)(y^3) + x(1+y^2)y'=0 is not exact, then multiply μ(x,y)=1/(xy^3) to solve the equation

First of all, we observe it follows M(x,y)+N(x,y)∂y/∂x = 0 form, we can get M(x,y)=(x^)(2y^3), N(x,y)= x(1+y^2)=x+xy^2
Then we get My=∂/∂y((x^2)(y^3))=3(x^2)(y^2) and Nx=∂/∂x(x(1+y^2))=1+y^2
As we see My is not equal to Nx, this is not an exact equation.

Second, we try multiplying the integrating factor μ(x,y)=1/(xy^3) on both sides: x+((1/y^3) + (1/y))∂y/dx=0
We do the first step again: M(x,y)=x and N(x,y)=1/y^3 + (1/y)
Derivative them: My=∂/∂y(x) =0 ; Nx=∂/∂x((1/y^3) + (1/y))=0
As we see My=Nx, now this is an exact equation.

Now we need to get the solution: ∫ M∂x+N∂x=(1/2)x^2 - (1/2)y^(-2)+ln|y|=2ln|y|+(x^2)-(1/y^2)=c
« Last Edit: October 04, 2019, 02:36:32 PM by Kun Zheng »


text:
## An application to Euler's -function

Let

as in 1.4.5.

Proposition 1.7.36   If then there is a 1-1, onto mapping between the sets

proof: If and , then define , where satisfies . This exists and is unique , by the Chinese remainder theorem, so is well-defined - provided we show . To show this, we must show that if , and that if and , then . We leave these as exercises.

is 1-1: If then and . Subtracting, we get , so and by definition of and . Therefore, is 1-1.

is onto: Let . Define and by . Then, by definition . Therefore, is onto.

The following result has been proven by more direct methods above. We given another proof as an application of the Chinese remainder theorem.

Proposition 1.7.37   If then .

proof: The number of elements in the range of is and the number of elements in the domain of is . Since is a bijection, these two must be equal.

Corollary 1.7.38   If is the prime factorization of an integer then .

Exercise 1.7.40   Verify Example 1.7.39.

Exercise 1.7.41   Define , , . Find a formula for as in the above example.

Exercise 1.7.42 (Fibonacci numbers)   Let satisfy and let . Solve for .

Exercise 1.7.43   Let satisfy and let , , . Solve for . (This is the Lucas-Perrin sequence.)

Exercise 1.7.44   An internet pyramid scheme starts with a group of friends in different states. They come up with an email which asks the recipient to forward the email to others. Find the linear recurrence relation which describes this process after iterations (assuming every email is sent to a different person). First, solve it in terms of and in general. Next, for specific values of and (say and ).

Exercise 1.7.45   An internet pyramid scheme starts with a group of friends in different states. They come up with an email which asks the recipient to add their name onto the list (which originally contains the names of the friends) and forward the email to others, where is the number of people on the list. Find the linear recurrence relation which describes this process after iterations (assuming every email is sent to a different person). First, solve it in terms of in general. Next, for specific values of (say ).

Exercise 1.7.46   Use Caeser's cipher in §1.7.7 to decode ehdw dupb''. (Hint: a commonly uttered phrase at the U. S. Naval Academy.)

Exercise 1.7.47   Find the first 20 terms of the LFSR with seed and recursion equation , .

Exercise 1.7.48   Find the first 20 terms of the LFSR with seed and recursion equation , .

Exercise 1.7.49   A sequence is eventually periodic if there are fixed (called the period) and such that ,for all . (If then we call the sequence periodic.) Are the sequences in the above exercises eventually periodic? If so, what are their periods?

Exercise 1.7.50   Using the algorithm in §1.7.9, solve for the day of the week of your birthday.

Exercise 1.7.51   Using the algorithm in §1.7.9, solve for the day of the week of today's date.

Exercise 1.7.52   Using the algorithm in §1.7.9, solve for the day of the week of your birthday.

Exercise 1.7.53   Using the algorithm in §1.7.9, solve for the day of the week of today's date.

Exercise 1.7.54   At the annual Mathematics Department Fun Run, when the joggers lined up 4 abreast there was 1 left over, when the joggers lined up 5 abreast there were 2 left over. How many were at the fun run? (Take the smallest solution to the congruences.)

Exercise 1.7.55   Solve the following problem: If eggs are removed from a basket , , , , and at a time, there remain respectively , , , , and eggs. But if the eggs are removed at a time no eggs remain. What is the least number of eggs that could have been in the basket?

Exercise 1.7.56   A politician runs for office every 6 years and cheats on his taxes every 5 years. He was just elected and cheated on his taxes last year. When is the next time he does both in the same year?

Exercise 1.7.57   At a local triathalon, when everyone lined up 4 abreast there was 1 left over, when everyone lined up 5 abreast there were 2 left over, and when everyone lined up 7 abreast there were 3 left over. How many were at the triathalon? (Take the smallest solution to the congruences.)

Exercise 1.7.58   Show that , in the notation of §1.7.11. Compute , .

david joyner 2008-04-20


text:
# Can you divide a square into 5 equal area regions

Given this shape:

# Is it possible to divide the cyan area into 5 equal area shapes

such that:

1. Each shape is the same
2. Each shape has an edge touching the red square
3. Each shape has an edge touching the outside.
4. No diagonal lines.

Its reasonably easy to perform #2 and #3 as long as you violate #1

And I don't believe its possible to actually satisfy #1, and the very fact its contained within a square suggests #1 is not satisfiable given the presence of #2.

Though its just a bit of fun really =).

Context: Was simply devising a town plan for a guild oriented town for a minecraft world, and it became possible that we might want 5 guilds, and the fun of giving each guild a fair equal area region , in conjunction with all 5 guilds having a shared space in the middle.

Diagonals are unwanted as it makes dividing the land fairly and applying region controls onerous, as most things are rectangular, regions included.

So while you can roughly approximate a diagonal with sufficiently many rectangles, the less steps, the better.

# If it is not possible

Please provide reasoning as to why not.

It would be interesting to see what sort of alternatives people can come up with, perhaps there is an optimal shape that results in all the shapes being highly similar geometrically, despite not being identical.

-
It's impossible, it is possible with 4 shapes, but not 5. For example: they need to be the same, but they have to touch the red square, so you can't divide the red squares perimeter into 5 identical pieces :) –  Gintas K Nov 7 '13 at 10:59
@GintasK But it might be possible that 5 shapes are not symmetrically located... (I'm not sure about this) –  JiminP Nov 7 '13 at 11:00
ok, the red squares perimeter is 1600u, so one piece should touch 320u of it. the red square has 4 corners, so at least one piece wont have that corner what makes them not the same :) –  Gintas K Nov 7 '13 at 11:03
When I first fell upon this problem, I didn't even think it was possible to do 5 equal-area shapes of different shapes, but I've since proven myself wrong about that =) Though I came upon my knowledge by simply doing brute-force spacial geometry, I have no Idea how somebody would throw a calculator at this problem. –  Kent Fredric Nov 7 '13 at 11:26
@GintasK: That seems to be the right intuition, but a priori it's not entirely obvious that the "lack of corner" in the fifth shape could not be fixed by fitting it into corners made by the other pieces. –  Eric Stucky Nov 7 '13 at 11:48

-
Too much free time, eh? –  ja72 Nov 8 '13 at 20:22
@ja72 Yeah, today is my day off, and I decided to take time to this exercise. It was very interesting. –  RicardoCruz Nov 8 '13 at 20:50
So I am not the only one solving problems for fun. What a relief. –  ja72 Nov 8 '13 at 21:00
That sir, is awesome =). Thats about as good as I think I could expect to get =). –  Kent Fredric Nov 10 '13 at 13:34

I don't think it possible to have all shapes identical, because the criteria of "touching the outside" and "touching the inner square" basically means there has to be at least 5 edges running from center to outside.

And because it is impossible to draw 5 edges from a square in such a way that all edges leave the square in such a way they present as being geometrically the same, the rest of the image becomes dependent on this fact.

However, if you waive the "shapes are identical rule", this is the best solution I've come up with so far.

Logic:

The Cyan area is basically: $$\begin{array}{cc} A_{cyan} & = & A_{outer} & - & A_{inner} \\ A_{cyan} & = & 4,000 \times 4,000 & - & 400 \times 400 \\ A_{cyan} & = & 15,840,000 \end{array}$$ So each must be

$$\begin{array}{cc} A_{section} & = & \frac{15,840,000}{5} \\ A_{section} & = & 3,168,000 \end{array}$$

Its apparent that at least one shape may need an edge from center to edge, so you quickly find a starting rectangle of

$$\begin{array}{cc} A_{section} & = & \frac{W_{outer}}{2} & - & \frac{W_{inner}}{2} & \times & y \\ A_{section} & = & \frac{4000}{2} & - & \frac{400}{2} & \times & y \\ A_{section} & = & 2000 & - & 200 & \times & y \\ A_{section} & = & 1800 & & & \times & y \\ y & = & \frac{3,168,000}{1800} \\ y & = & 1760 \end{array}$$

From there, it was basically a case of shape-shifting the perimeter while keeping area the same, to achieve a sort of balanced distribution, which is reasonably easy to do.

After you've created 2 shapes with that process, duplicating it was pretty simple, and you're then left with a void which will logically be the same size.

For the usecase we had, having different shapes was acceptable, and this combination of shapes was useful enough. Though somebody can probably find a combination of more similar shapes.

( You'll also see that the arrangement loosely models a pentagon, just with far less regularity )

-