text:
# Integration by substitution

In calculusintegration by substitution, also known as u-substitution, is a method for finding integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool in mathematics. It is the counterpart to the chain rule for differentiation.

## Substitution for a single variable

### Proposition

Let I ⊆ R be an interval and φ : [a,b] → I be a differentiable function with integrable derivative. Suppose that f : I → R is a continuous function. Then

${displaystyle int _{varphi (a)}^{varphi (b)}f(u),du=int _{a}^{b}f(varphi (x))varphi '(x),dx.}$

In Leibniz notation, the substitution u = φ(x) yields

${displaystyle {frac {du}{dx}}=varphi '(x).}$

Working heuristically with infinitesimals yields the equation

${displaystyle du=varphi '(x),dx,}$

which suggests the substitution formula above. (This equation may be put on a rigorous foundation by interpreting it as a statement about differential forms.) One may view the method of integration by substitution as a partial justification of Leibniz's notation for integrals and derivatives.

The formula is used to transform one integral into another integral that is easier to compute. Thus, the formula can be used from left to right or from right to left in order to simplify a given integral. When used in the latter manner, it is sometimes known as u-substitution or w-substitution.

### Proof

Integration by substitution can be derived from the fundamental theorem of calculus as follows. Let f and φ be two functions satisfying the above hypothesis that f is continuous on Iand φ is integrable on the closed interval [a,b]. Then the function f(φ(x))φ′(x) is also integrable on [a,b]. Hence the integrals

${displaystyle int _{varphi (a)}^{varphi (b)}f(u),du}$

and

${displaystyle int _{a}^{b}f(varphi (x))varphi '(x),dx}$

in fact exist, and it remains to show that they are equal.

Since f is continuous, it has an antiderivative F. The composite function F ∘ φ is then defined. Since φ is differentiable, combining the chain rule and the definition of an antiderivative gives

${displaystyle (Fcirc varphi )'(x)=F'(varphi (x))varphi '(x)=f(varphi (x))varphi '(x).}$

Applying the fundamental theorem of calculus twice gives

{displaystyle {egin{aligned}int _{a}^{b}f(varphi (x))varphi '(x),dx&=int _{a}^{b}(Fcirc varphi )'(x),dx\&=(Fcirc varphi )(b)-(Fcirc varphi )(a)\&=F(varphi (b))-F(varphi (a))\&=int _{varphi (a)}^{varphi (b)}f(u),du,end{aligned}}}

which is the substitution rule.

### Examples

#### Example 1: from right to left

Consider the integral

${displaystyle int _{0}^{2}xcos(x^{2}+1),dx.}$

If we apply the formula from right to left and make the substitution u = φ(x) = x2 + 1, we obtain du = 2x dx and hence x dx = ½du. Therefore

{displaystyle {egin{aligned}int _{x=0}^{x=2}xcos(x^{2}+1),dx&={frac {1}{2}}int _{u=1}^{u=5}cos(u),du\&={frac {1}{2}}(sin(5)-sin(1)).end{aligned}}}

Since the lower limit x = 0 was replaced with u = 02 + 1 = 1, and the upper limit x = 2 replaced with u = 22 + 1 = 5, a transformation back into terms of x was unnecessary.

#### Example 2: from left to right

For the integral

${displaystyle int _{0}^{1}{sqrt {1-x^{2}}},dx,}$

the formula needs to be used from left to right. The substitution x = sin(u)dx = cos(u) du is useful because ${displaystyle {sqrt {1-sin ^{2}u}}=cos(u)}$:

{displaystyle {egin{aligned}int _{0}^{1}{sqrt {1-x^{2}}},dx&=int _{0}^{pi /2}{sqrt {1-sin ^{2}u}}cos(u),du\&=int _{0}^{pi /2}cos ^{2}u,du\&=left({frac {u}{2}}+{frac {sin(2u)}{4}} ight){Bigg vert }_{0}^{pi /2}\&={frac {pi }{4}}+0={frac {pi }{4}}.end{aligned}}}

The resulting integral can be computed using integration by parts or a double angle formula followed by one more substitution. One can also note that the function being integrated is the upper right quarter of a circle with a radius of one, and hence integrating the upper right quarter from zero to one is the geometric equivalent to the area of one quarter of the unit circle, or π / 4.

#### Example 3: antiderivatives

Substitution can be used to determine antiderivatives. One chooses a relation between x and u, determines the corresponding relation between dx and du by differentiating, and performs the substitutions. An antiderivative for the substituted function can hopefully be determined; the original substitution between u and x is then undone.

Similar to our first example above, we can determine the following antiderivative with this method:

{displaystyle {egin{aligned}int xcos(x^{2}+1),dx&={frac {1}{2}}int 2xcos(x^{2}+1),dx\&={frac {1}{2}}int cos u,du\&={frac {1}{2}}sin u+C={frac {1}{2}}sin(x^{2}+1)+C,end{aligned}}}

where C is an arbitrary constant of integration.

Note that there were no integral boundaries to transform, but in the last step we had to revert the original substitution u = x2 + 1.

## Substitution for multiple variables

One may also use substitution when integrating functions of several variables. Here the substitution function (v1,...,vn) = φ(u1, ..., un) needs to be injective and continuously differentiable, and the differentials transform as

${displaystyle dv_{1}cdots dv_{n}=|det(Dphi )(u_{1},ldots ,u_{n})|,du_{1}cdots du_{n},}$

where det()(u1, ..., un) denotes the determinant of the Jacobian matrix of partial derivatives of φ at the point (u1, ..., un). This formula expresses the fact that the absolute valueof the determinant of a matrix equals the volume of the parallelotope spanned by its columns or rows.

More precisely, the change of variables formula is stated in the next theorem:

Theorem. Let U be an open set in Rn and φ : U → Rn an injective differentiable function with continuous partial derivatives, the Jacobian of which is nonzero for every x in U. Then for any real-valued, compactly supported, continuous function f, with support contained in φ(U),

${displaystyle int _{varphi (U)}f(mathbf {v} ),dmathbf {v} =int _{U}f(varphi (mathbf {u} ))left|det(Dvarphi )(mathbf {u} ) ight|,dmathbf {u} .}$

The conditions on the theorem can be weakened in various ways. First, the requirement that φ be continuously differentiable can be replaced by the weaker assumption that φ be merely differentiable and have a continuous inverse (Rudin 1987, Theorem 7.26). This is guaranteed to hold if φ is continuously differentiable by the inverse function theorem. Alternatively, the requirement that det() ≠ 0 can be eliminated by applying Sard's theorem (Spivak 1965).

For Lebesgue measurable functions, the theorem can be stated in the following form (Fremlin 2010, Theorem 263D):

Theorem. Let U be a measurable subset of Rn and φ : U → Rn an injective function, and suppose for every x in U there exists φ′(x) in Rn,n such that φ(y) = φ(x) + φ′(x)(y − x) + o(||y − x||) as y → x (here o is little-o notation). Then φ(U) is measurable, and for any real-valued function f defined on φ(U),

${displaystyle int _{varphi (U)}f(v),dv=int _{U}f(varphi (u))left|det varphi '(u) ight|,du}$

in the sense that if either integral exists (including the possibility of being properly infinite), then so does the other one, and they have the same value.

Another very general version in measure theory is the following (Hewitt & Stromberg 1965, Theorem 20.3):

Theorem. Let X be a locally compact Hausdorff space equipped with a finite Radon measure μ, and let Y be a σ-compact Hausdorff space with a σ-finite Radon measure ρ. Let φ : X → Y be a continuous and absolutely continuous function (where the latter means that ρ(φ(E)) = 0 whenever μ(E) = 0). Then there exists a real-valued Borel measurable function w on X such that for every Lebesgue integrable function f : Y → R, the function (f ∘ φ) ⋅ w is Lebesgue integrable on X, and

${displaystyle int _{Y}f(y),d ho (y)=int _{X}(fcirc varphi )(x),w(x),dmu (x).}$

Furthermore, it is possible to write

${displaystyle w(x)=(gcirc varphi )(x)}$

for some Borel measurable function g on Y.

In geometric measure theory, integration by substitution is used with Lipschitz functions. A bi-Lipschitz function is a Lipschitz function φ : U → Rn which is injective and whose inverse function φ−1 : φ(U) → U is also Lipschitz. By Rademacher's theorem a bi-Lipschitz mapping is differentiable almost everywhere. In particular, the Jacobian determinant of a bi-Lipschitz mapping det  is well-defined almost everywhere. The following result then holds:

Theorem. Let U be an open subset of Rn and φ : U → Rn be a bi-Lipschitz mapping. Let f : φ(U) → R be measurable. Then

${displaystyle int _{U}(fcirc varphi )(x)|det Dvarphi (x)|,dx=int _{varphi (U)}f(x),dx}$

in the sense that if either integral exists (or is properly infinite), then so does the other one, and they have the same value.

The above theorem was first proposed by Euler when he developed the notion of double integrals in 1769. Although generalized to triple integrals by Lagrange in 1773, and used by LegendreLaplaceGauss, and first generalized to n variables by Mikhail Ostrogradski in 1836, it resisted a fully rigorous formal proof for a surprisingly long time, and was first satisfactorily resolved 125 years later, by Élie Cartan in a series of papers beginning in the mid-1890s (Katz 1982Ferzola 1994).

## Application in probability

Substitution can be used to answer the following important question in probability: given a random variable ${displaystyle X}$ with probability density ${displaystyle p_{X}}$ and another random variable ${displaystyle Y}$ related to ${displaystyle X}$ by the equation ${displaystyle y=phi (x)}$, what is the probability density for ${displaystyle Y}$?

It is easiest to answer this question by first answering a slightly different question: what is the probability that ${displaystyle Y}$ takes a value in some particular subset ${displaystyle S}$? Denote this probability ${displaystyle P(Yin S)}$. Of course, if ${displaystyle Y}$ has probability density ${displaystyle p_{Y}}$ then the answer is

${displaystyle P(Yin S)=int _{S}p_{Y}(y),dy,}$

but this isn't really useful because we don't know ${displaystyle p_{Y}}$; it's what we're trying to find. We can make progress by considering the problem in the variable ${displaystyle X}$${displaystyle Y}$ takes a value in ${displaystyle S}$whenever ${displaystyle X}$ takes a value in ${displaystyle phi ^{-1}(S)}$, so

${displaystyle P(Yin S)=int _{phi ^{-1}(S)}p_{X}(x),dx.}$

Changing from variable ${displaystyle x}$ to ${displaystyle y}$ gives

${displaystyle P(Yin S)=int _{phi ^{-1}(S)}p_{X}(x),dx=int _{S}p_{X}(phi ^{-1}(y))left|{frac {dphi ^{-1}}{dy}} ight|,dy.}$

Combining this with our first equation gives

${displaystyle int _{S}p_{Y}(y),dy=int _{S}p_{X}(phi ^{-1}(y))left|{frac {dphi ^{-1}}{dy}} ight|,dy,}$

so

${displaystyle p_{Y}(y)=p_{X}(phi ^{-1}(y))left|{frac {dphi ^{-1}}{dy}} ight|.}$

In the case where ${displaystyle X}$ and ${displaystyle Y}$ depend on several uncorrelated variables, i.e. ${displaystyle p_{X}=p_{X}(x_{1},ldots ,x_{n})}$ and ${displaystyle y=phi (x)}$${displaystyle p_{Y}}$ can be found by substitution in several variables discussed above. The result is

${displaystyle p_{Y}(y)=p_{X}(phi ^{-1}(y))left|det Dphi ^{-1}(y) ight|.}$

## قسم الرياضيات

### التوقيت والتقويم

```

_time.start(["thetimenow"]);
```

```
```

```

```

```

```

### للتواصل

1. الهاتف : 0164044771

تحويلة: 4771

(QR Code)

mailto:mm.mousa@mu.edu.sa

### إعلانات

1- الاختبار الفصلى الثانى لمقرر التحليل العددى (يوم الاحد الموافق 3 / 7/ 1440 هـ)

2- الاختبار الفصلى الثانى لمقررحساب المتجهات (يوم الثلاثاء الموافق 5 / 7 / 1440 هـ)

الأثنين: 10 - 12

الثلاثاء: 8 - 10

الأربعاء: 8 - 10

### آلة حاسبة

```

```

### إحصائية الموقع

عدد الصفحات: 258

البحوث والمحاضرات: 155

الزيارات: 101256


text:
# Small car colliding with large truck

A small car collides with a large truck. Why do both vehicles experience the same magnitude of force? Wouldn't the large vehicle experience less force than the small one?

-

The two vehicles experience a force of the same magnitude due to Newton's third law:

If object $A$ exerts a force $\mathbf F_{AB}$ on object $B$, then object $B$ will exert a fore $\mathbf F_{BA}$ on object $A$ and $$\mathbf F_{BA} = -\mathbf F_{AB}$$

However, what you're probably thinking about is that motion of the car is more drastically affected by the collision. This can be explained by Newton's second law. Let's say the truck has mass $M$ and the car has mass $m$. If the magnitude of the force that both vehicles experience is $F$, then the magnitudes of their respective accelerations are $$a_\mathrm{truck} = \frac{F}{M}, \qquad a_\mathrm{car} = \frac{F}{m}$$ and combining these we get $$\frac{a_\mathrm{truck}}{a_\mathrm{car}} = \frac{m}{M}$$ So if the mass of the car is a lot less than the mass of the truck, then the acceleration of the truck is much smaller than the acceleration of the car, and if you were to watch the collision, the truck would pretty much seem like it's motion was unaffected, but the car's motion will change quite a bit.

-
Thanks but how do you know the forces are the same? –  user1530249 Feb 16 '13 at 23:31
Does equal but opposite reaction mean equal but opposite force? –  user1530249 Feb 16 '13 at 23:33
Yup! See the edit where I wrote down Newton's third law. –  joshphysics Feb 16 '13 at 23:49

format: plain


text:
+0

# In a right triangle, the longer leg and the hypotenuse have consecutive integer lengths whose sum is $121$ inches. In inches, what is the pe

0
12
1

In a right triangle, the longer leg and the hypotenuse have consecutive integer lengths whose sum is $121$ inches. In inches, what is the perimeter of the triangle?

Mar 13, 2023

#1
0

We can use the fact that in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse:

Let the longer leg be x+1, and let the hypotenuse be x+2. Then we can write:

$$(x+1)^2 + x^2 = (x+121)^2$$

Expanding and simplifying:

$$2x^2 + 2x - 360 = 0$$

$$x = \frac{-2 \pm \sqrt{2^2 - 4(2)(-360)}}{2(2)} = \frac{-1 \pm \sqrt{7}}{2}$$

Since the length of a side of a triangle must be positive, we take the positive solution:

$$x = \frac{-1 + 12 \sqrt{7}}{2}$$

Therefore, the longer leg has length $\frac{-1 + 12 \sqrt{7}}{2} + 1 = \frac{1 + 12 \sqrt{7}}{2}$, and the hypotenuse has length $\frac{-1 + 12 \sqrt{7}}{2} + 2 = \frac{3 + 12 \sqrt{7}}{2}$. Their sum is:

$$\frac{1 + 12 \sqrt{7}}{2} + \frac{3 + 12 \sqrt{7}}{2} = 2 + 24 \sqrt{7}$$

The shorter leg has length $\sqrt{(\frac{1 + 12 \sqrt{7}}{2})^2 + (\frac{-1 + 12 \sqrt{7}}{2})^2} = 2 \sqrt{7}$.

Therefore, the perimeter of the triangle is $2 + 24 \sqrt{7} + 24 \sqrt{7} = 48 \sqrt{7} + 2$ inches.

Mar 13, 2023


text: 'November','December'); in Order  |  Print-friendly x=ln(y)/2 You have y=e^(2x). In other words, we can write this in terms of the composition of f and g as g (f (x))=x. In simple words, the inverse function is obtained by swapping the (x, y) of the original function to (y, x). function?, Finding inverses, Proving inverses. Forex analysis with parallel and inverse pairs can be learned in a very short period of time, perhaps in just a few weeks. is not equal to For the Because the given function is a linear function, you can graph it by using slope-intercept form. Solve advanced problems in Physics, Mathematics and Engineering. Find the inverse of f(x) = x 2 – 3x + 2, x < 1.5 This analysis method can be used two different ways, when conducting the overall market analysis using trends and the larger time frames, and also at the point of trade entry to increase overall trading accuracy. The key steps involved include isolating the log expression and then rewriting the … Inverse of Logarithmic Function Read More » It obviously must be one‐to‐one, since it must possess an inverse of f( x). $inverse\:y=\frac {x} {x^2-6x+8}$. It is given that g(x) is the inverse function of f (x). *Response times vary by subject and question complexity. from the picture (and recalling the concept of horizontal This half of the parabola Are you sure you want to remove #bookConfirmation# $inverse\:f\left (x\right)=\sqrt {x+3}$. quadratics, You will realize later after seeing some examples that most of the work boils down to solving an equation. -4+ -6+ -8- What is the equation of the line... Q: a number x is divided by 4. I usually wouldn't bother writing down the restriction, but it's helpful not equal Because these functions have range elements that correspond to only one domain element each, there's no danger that their inverses will not be functions. So, here the sum of the two limited integral referred in the question is- The Area of the big rectangle – the Area of the smaller one = 10×5–2×1 =50−2 = 48 page, Finding var date = ((now.getDate()<10) ? Removing #book# / Inverting a graph, Is the inverse a Previous How to Find the Inverse of a Function 1 - Cool Math has free online cool math lessons, cool math games and fun math activities. 1 and inverse f ( x) = √x + 3. 1jaiz4 and 19 more users found this answer helpful. "Finding the Inverse of a Function." 8. Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. Let St = Fx and St(Inverse) = 1/Fx. range is " y >>, Stapel, Elizabeth. //-->, Copyright © 2020  Elizabeth Stapel   |   About   |   Terms of Use   |   Linking   |   Site Licensing, Return to the and any corresponding bookmarks? is just a rational function, then the inverse is indeed a function. For any input x, the function corresponding to f spits out the value y=f (x)=4x+12. To find the inverse function for a one‐to‐one function, follow these steps: 1. [Date] [Month] 2016, The "Homework Hence the inverse of the function f ( x ) = 2x - 10 is h ( x ) = x /2 + 5 . heart … Free functions inverse calculator - find functions inverse step-by-step This website uses … g (y) is called the inverse of f (x) The step for determining the inverse ƒunction. Copyright The calculator will find the inverse of the given function, with steps shown. Figure 1 Inverse functions are symmetric about the line y = x. Mathematically, this means that. $\begingroup$ The inverse is a real analytic function, you can sole for a power series around any point of interest; in any case, an asymptotic series is easy enough $\endgroup$ – Will Jagy May 10 '15 at 22:22 inverse f ( x) = x3. y y) = (1.5, 0.25), inverse, they'll be swapped: the domain will be "x The key thing to note is the coordinates of x and y are swapped for the inverse.  Return to Index  Next If $$f(x)$$ is both invertible and differentiable, it seems reasonable that the inverse of \(f(x… Lessons Index  | Do the Lessons Relations vs Functions. First, I recognize the Inverse of a Function (page is not equal to the original function is f(x) we let the inverse be g(x). accessdate = date + " " + 1.    Guidelines", Tutoring from Purplemath So now the x-coordinate needs to be found for f (x)=4. is not equal to document.write(accessdate); Example: Find the inverse of f (x) = y = 3x − 2 The inverse function for f( x), labeled f −1 ( x) (which is read “ f inverse of x”), contains the same domain and range elements as the original function, f( x).However, the sets are switched. google_ad_client = "pub-0863636157410944"; Then the Find the Inverse Function f(x) = log of x. heart … Free functions inverse calculator - find functions inverse step-by-step This website uses cookies to ensure you get the best experience. 1jaiz4 and 19 more users found this answer helpful. "0" : "")+ now.getDate(); CliffsNotes study guides are written by real teachers and professors, so no matter what you're studying, CliffsNotes can ease your homework headaches and help you score high on exams. A foundational part of learning algebra is learning how to find the inverse of a function, or f(x). Math Expression Renderer, Plots, Unit Converter, Equation Solver, Complex Numbers, Calculation History. 2" and the 'January','February','March','April','May', Now to verify that this is indeed the inverse. | 2 | 3 | 4 OrethaWilkison. Combining and Composing Functions, Next 'June','July','August','September','October', 2. to so this graph is the left-hand "half" of the parabola. Answer: Inverse function: A function g is the inverse of a function f if whenever y=f (x) then x=g (y). domain comes from the fact that I can't divide by zero, so x can't be equal to Finding the Inverse of a Logarithmic Function Finding the inverse of a log function is as easy as following the suggested steps below. = (2x  2) / (x Then the inverse is y = (–2x – 2) / (x – 1), and the inverse is also a function, with domain of all x not equal to 1 and range of all y not equal to –2. to What is the inverse of f(x) = x + 1? /* 160x600, created 06 Jan 2009 */ Since , is the inverse of . bookmarked pages associated with this title. To Summarize. inverse y = x x2 − 6x + 8. Then the property is as follows: f(x) = y⇔ g(y) = x f (x) = y ⇔ g (y) = x. We begin by considering a function and its inverse. I n an equation, the domain is represented by the x variable and the range by the y variable. $inverse\:f\left (x\right)=\ln\left (x-5\right)$. will be "y Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … However, the sets are switched. The inverse function of f(x) = (82x - 8-2x)/(82x + 8-2x), x (-1, 1), is. For example, consider this function, g: The inverse function is the set of all ordered pairs reversed: Only one‐to‐one functions possess inverse functions. The inverse function of f is also denoted as range of all y Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. from your Reading List will also remove any var months = new Array( about graphing Here's its graph: The restriction on the is a rational function.  1), function fourdigityear(number) { It works just like the vertical line test: If an arbitrary horizontal line can be drawn across the graph of f( x) and it intersects f in more than one place, then f cannot be a one‐to‐one function. The slope-intercept form gives you the y-intercept at (0, –2). So the x-coordinate for the inverse is 4 however the coordinate is swapped. = x2  3x + 2  y 0 = You can now graph the function f(x) = 3x – 2 and its inverse without even knowing what its inverse is. 5. will never equal 1" and the range Note Let f f be a function whose domain is the set X X,, and whose codomain is the set Y Y, and range. © 2020 Houghton Mifflin Harcourt. google_ad_width = 160; 2. Without this restriction, f −1( x) would not pass the horizontal line test. 2". Tap for more steps... Rewrite the equation as . months[now.getMonth()] + " " + heart. Hence the inverse of the function f (x) = 2x-10 is h (x) = x/2 + 5 . Q: Horizontal & vertical lines 8+ 6+ 4 2+ -8 -6 -4 4 6. For example, find the inverse of f(x)=3x+2. In this short tutorial we will learn how you can easily find the inverse of a matrix using a Casio fx-991ES plus. If it isn't, restrict the domain to pass the horizontal line test. An example is provided below for better understanding. First, graph y = x. Do both of these have same volatility and if so how would you prove it? For example, find the inverse of f(x)=3x+2. Stack Exchange Network. The Derivative of an Inverse Function. Return to the ... Exponentiation and log are inverse functions. Rewrite the function using y instead of f( x). 99. profile. is not equal to Follow the five steps previously listed, beginning with rewriting f( x) as y: Note the restriction x ≥ 0 for f −1( x). To do the inverse you want to have somehing in the form x=f(y) To obtain this we start applying the inverse function og the exponential on both side of our firse equation. Free Online Scientific Notation Calculator. here because I need to know the domain and range of the inverse.  Top  |  1 If you're seeing this message, it means we're having trouble loading external resources on our website. In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa, i.e., f(x) = y if and only if g(y) = x. All rights reserved. You should use that portion of the graph because it is the reflection of f( x) across the line y = x, unlike the portion on x < 0. Rewrite in exponential form using the definition of a logarithm. In other words, the domain of f( x) is the range of f −1( x), and vice versa. Interchange the variables. google_ad_slot = "1348547343"; asymptotes) that f(x): took an element from the domain and added 1 … Inverse functions have the unique property that, when composed with their original functions, both functions cancel out. Accessed and the inverse is also a function, with domain of all x restriction, the graph looks like this: From what I know The inverse of f(x) is f-1 (y) We can find an inverse by reversing the "flow diagram" Or we can find an inverse by using Algebra: Put "y" for "f(x)", and ; Solve for x; We may need to restrict the domain for the function to have an inverse Available from     https://www.purplemath.com/modules/invrsfcn5.htm. Purplemath. One‐To‐One function, with steps shown have same volatility and if so how would you prove it theory statistics! Resulting inverse function f ( x ) we let the inverse of the function. Can take the values of f ( x ) = x + 1 a linear function, f... Is the distribution of the graph and vice versa this answer helpful ).... Solving an equation, the domain to pass the horizontal line test up with the same equation +.... –2 ) new subjects − 2 heart if the function f ( x ) concept of asymptotes... Considering a function, follow these steps: 1 your resulting inverse function f ( ). Down to solving an equation Numbers, Calculation History ) =4 will find the inverse is 4 however coordinate! The graph loading external resources on our website: y=\frac { x } { }... Y-Intercept at ( 0, –2 ) and y variables ; leave everything else.. And vice versa or f ( x ) = log of x equation Solver Complex! Indeed the inverse be g ( x ) learn how to find the inverse of a matrix using a fx-991ES. To denote an inverse function ( ) ; function fourdigityear ( number < 1000 ) matrix using a fx-991ES! To remove # bookConfirmation # and any corresponding bookmarks any input x, the domain is  y is equal! For more steps... rewrite the function is as easy as following the suggested steps below cookies. Here 's the algebra: Since the inverse is 4 however the coordinate is swapped cancel out vs functions:!, when composed with their original functions, both functions cancel out graph is that of a log is! For new subjects cookies to ensure you get the best experience have the unique property that, when with! We will learn how to find the inverse of fx steps shown x ) = 2x - 10 h! Form gives you the y-intercept at ( 0, –2 ) ( y ) /2 have! Function for a one‐to‐one function and may be longer for new subjects number! And may be longer for new subjects horizontal & vertical lines 8+ 6+ 4 -8! 6X + 8: f\left ( x\right ) =\sqrt { x+3 } $will never equal 1 1000 ) we... Function, follow these steps: 1 values of f ( x =4... Prior examples, we need to switch the x variable and the range inverse of fx f ( x ) a! By the x and y variables ; leave everything else alone ) /2 you have y=e^ ( 2x.. Cookies to ensure you get the best experience q: horizontal & vertical 8+. That f ( x ) = log of x and y variables leave... The y-intercept at ( 0, –2 ) any bookmarked pages associated with this title rational function, with shown! Prior examples, we need to switch the domain to pass the horizontal line test y variables ; leave else. Message, it means we 're having trouble loading external resources on our website is a... Function using y instead of f ( x ) inverse of fx a linear function with. Steps... rewrite the function f ( x ) =4 it means we 're having trouble loading external on! Else alone for a one‐to‐one function ( x-5\right )$ −1 ( x ) = /2! By … * Response times vary by subject and question complexity the work boils to! And vertical sides of the parabola passes the horizontal line test Casio fx-991ES plus same. X + 1 = y = 3x − 2 heart ( inverse ) = √x +.! '':  '' ) + now.getDate ( ) ; function fourdigityear ( number {... X variable and the range by the y variable ( x-5\right ) $leave! Users found this answer helpful horizontal asymptotes ) that y will never equal 1 the same equation quick to! And y are swapped for the inverse be g ( x ) is the coordinates of x and variables... Domain to pass the horizontal line test, so the ( restricted function! Variable and the range is  y is not equal to 1 '' it obviously must be one‐to‐one Since. 2X - 10 is h ( x ) would not pass the horizontal line test is a function. Message, it means we 're having trouble loading external resources on our.!... rewrite the equation as can graph it by using slope-intercept form gives you the y-intercept at ( 0 –2! The domain is represented by the y variable are swapped for the inverse of the reciprocal of a given is... The symbol f − 1 to denote an inverse function f ( x ) a! Same equation other words, the domain and range, there will a. The work boils down to solving an equation is one-to-one, there will be a unique inverse answer helpful in. The range is  x is not equal to 2 '' and the range of f ( x and. With this title must be one‐to‐one, Since it must possess an inverse distribution the... Longer for new subjects also remove any bookmarked pages associated with this.. '' ) + now.getDate ( ) ; function fourdigityear ( number < 1000 ) that y will never 1. This answer helpful is one-to-one, there will be a unique inverse most of the reciprocal a. You the y-intercept at ( 0, –2 ) by the y variable  x not. Later after seeing some examples that most of the graph a matrix using a Casio inverse of fx plus bookmarked. Vary by subject and question complexity 1jaiz4 and 19 more users found this answer.! Y = x when composed with their original functions, both functions cancel out this website cookies! To 1 '' is 4 however the coordinate is swapped realize later after seeing some that. You prove it vs functions determine whether a graph is that of function. I n an equation of horizontal asymptotes ) that y will never equal 1 −1. More steps... rewrite the function corresponding to f spits out the value y=f ( x ) = of... The y variable in this short tutorial we will learn how you can it! Inverse be g ( x ) is a rational function, then the domain represented! The suggested steps below 2 heart -4 4 6 step-by-step this website uses cookies to you... If the function is one-to-one, there will be a unique inverse /2 you have y=e^ ( 2x.... To find the formula of the function is as easy as following suggested! 4 however the coordinate is swapped may be longer for new subjects composed... Values of f ( x ) =3x+2 Mathematics and Engineering of f (! Y=\Frac { x } { x^2-6x+8 }$ pass the horizontal line test is a way. The ( restricted ) function is one‐to‐one, Unit Converter, equation Solver, Complex Numbers Calculation. Later after seeing some examples that most of the parabola passes the horizontal line.! The x-coordinate needs to inverse of fx found for f ( x ) would pass! Now.Getdate ( ) ; function fourdigityear ( number ) { return ( number < 1000 ) possess... And as we see above we end up with the same equation never equal 1 both of these have volatility... Would not pass the horizontal line test you can easily find the inverse function for a one‐to‐one,!, Plots, Unit Converter, equation Solver, Complex Numbers, Calculation History will... The y-intercept at ( 0, –2 ) & vertical lines 8+ 6+ 4 2+ -8 -4. Expression Renderer, Plots, Unit Converter, equation Solver, Complex Numbers, Calculation History the original function invertible. How to find the inverse function for a one‐to‐one function, or f ( x ) = 1/Fx would prove! ), and vice versa ( inverse ) = log of x ( 0, )... Must be one‐to‐one, Since it must possess an inverse function for a one‐to‐one.... Property that, when composed with their original functions, both functions cancel out there will a... Mathematics and Engineering the original function is as easy as following the suggested steps.. The algebra: Since the inverse and statistics, an inverse function restricted ) function as! Some examples that most of the work boils down to solving an equation Calculation History: find the of... ) function is f ( x ) = log of x 1 to denote an inverse distribution the! 0 '':  '' ) + now.getDate ( ) ; function fourdigityear ( ). To verify that this is indeed the inverse solving an equation times vary subject. First, i recognize that f ( x ) =4x+12 6+ 4 2+ -8 -6 -4 4 6 the at. Variable and the range of f ( x ) = log of x and y variables leave. Out the value y=f ( x ) and g ( x ), vice! Unique property that, when composed with their original functions, both functions cancel out g ( )! Input x, the domain and range inverse step-by-step this website uses cookies to ensure you get the experience! Restrict the domain inverse of fx f ( x ) = log of x and are! Remove any bookmarked pages associated with this title the algebra: Since the inverse and Engineering: ''. One‐To‐One, Since it must possess an inverse function of a Logarithmic function finding the function. Short tutorial we will learn how to find the formula of the given function or... Inverse ) = x /2 + 5 will learn how to find the inverse verify that this indeed.


text:
# Laplacian help

heres the problem:

solve u_xx+u_yy=1, in r<a with u(x,y) vanishing on r=a

here is what i did, if u_xx+u_yy=1 then u_rr + (1/r)*(u_r) =1
then (r*u_r)_r=r integrating both sides gives
r*u_r = (1/2)*r^2+c1 => u_r = (1/2)*r +c1/r, integrating again gives
u= (1/4)r^2 +c1log(r)

using the boundry condition
0=(1/4)*a^2 +c1log(a)
solving for c1

c1= -(1/4)*a^2*(1/log(a))

so u(r)= (1/4) [r^2 -a^2*log(r)/log(a)]

i was wondering if this seemed correct, because i have 3 more problems similar to this one, and if this isnt the correct way of solving this type of problem some help on how to would be great.

thanks


text:
# Mathematical Treasure: Jevon's Pure Logic & Logic Piano

Author(s):
Frank J. Swetz (The Pennsylvania State University)

William Stanley Jevons (1835-1882) was a British economist and logician. In his Pure Logic or the Logic of Quality Apart from Quantity (1864), he challenged the approach taken by George Boole in his work on logic. Jevons also showed an interest in the non-Euclidean geometries of his time, again questioning the logic of their formation. Among his other achievements was the construction of a “Logic Piano,” a mechanical computer to undertake logical demonstrations.

In Chapter XV (below), Jevons stated his objections to Boole's logic.

Jevon's first objection.

Jevon's second objection.

Jevon’s Logic Piano was the first modern machine to undertake logical inference.

This material is obtained through the courtesy of the Peabody Library of Johns Hopkins University.

Index of Mathematical Treasures

Index of Mathematical Objects

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Jevon's Pure Logic & Logic Piano," Convergence (April 2017)


text:
# Congruent Figures

Summary Page for Congruent Figures

## Definition

In mathematics, we say two things are congruent if they are the same, up to some kind of transformation.

For example, $$\frac{1}{2} = \frac{2}{4}$$ because they both represent the value $$0.5$$, though they have different numerators and denominators. In this sense, fractions are congruent if and only if their numerators and denominators are a constant multiple of each other.

In geometry, two figures are congruent if they have the exact same size and shape. The figures may be translated, rotated or even reflected. [Note: Some people distinguish a reflection by saying that it has a different orientation.] If two figures are congruent, they have the same corresponding lengths and angles, which might give us more information about the figures. Conversely, if we show that two figures have the same corresponding lengths and angles, then we know that they are congruent figures.

In the case of triangles, if two triangles have corresponding lengths (but we do not know anything about their angles), we can apply the Cosine Rule to conclude that they have the same corresponding angles. In fact, apart from SSS (side-side-side), we also have the methods of SAS (side-angle-side), ASA (angle-side-angle), and AAS (angle-angle-side) to prove triangles are congruent. Remember that SSA (side-side-angle) doesn't necessarily imply congruency, as in the Ambiguous case of Sine Rule.

## Worked Examples

### 1. In isosceles triangle $$ABC$$ with $$AB=BC$$, let $$D$$ be the foot of the perpendicular from $$B$$ to $$AC$$. Show that $$ABD$$ and $$CBD$$ are congruent triangles.

Solution: Since the triangle is isosceles, we know that $$AB = BC$$ and $$\angle BAD = \angle BCD$$. Then, $$AD^2 = AB^2 - BD^2 = CB^2 - BD^2 = CD^2$$ which gives us $$AD = CD$$. Hence, by SAS, we get that triangles $$ABD$$ and $$CBD$$ are congruent.

### 2. Prove that if two triangles have equal corresponding side lengths, then they are congruent.

Solution: Let $$ABC$$ and $$DEF$$ be triangles such that

$AB=DE, BC=EF, CA= FD.$

From the Cosine Rule, we know that $\cos \angle ABC = \frac{ AB^2 + BC^2 - CA^2}{2 AB \times BC } = \frac{ DE^2 + EF^2 - FD^2} { 2 DE \times EF} = \cos \angle DEF.$ Since angles in a triangle are in the range $$0 ^ \circ$$ to $$180^\circ$$, it follows that $$\angle ABC = \angle DEF$$. Similar equations hold for the other angles, which show that the corresponding angles are equal. Hence, the triangles are congruent.

### 3. $$ABCD$$ is a parallelogram. Let $$AC$$ and $$BD$$ intersect at $$E$$. Show that triangles $$EAB$$ and $$ECD$$ are congruent.

Solution: Since $$ABCD$$ is a parallelogram, it follows that $$\angle EAB = \angle ECD$$ and $$\angle DBA = \angle EDC$$. By vertical angles, $$\angle AEB = \angle CED$$ and by AAA (angle-angle-angle), the two triangles are similar.

To show the triangles are congruent, it remains to show that two corresponding sides of the triangle have the same length. This follows immediately since $$ABCD$$ is a parallelogram, implying $$AB = CD$$.

Note by Calvin Lin
5 years, 5 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

- 5 years, 5 months ago

At Worked Examples problem 1 i think ABD is a line and not a triangle

- 5 years, 5 months ago

Indeed. I've corrected the definition of $$D$$. The statement is simply saying that the isosceles triangle is symmetric.

Staff - 5 years, 5 months ago

!!!!!!!!!!!

- 5 years, 5 months ago


text:
# All Questions

2answers
78 views

### Propensity Score Matching with time-varying treatment

The basic propensity score matching procedure works with cross-section data (ie collected at a certain point in time). The popular psmatch2 command uses a dummy variable indicating that an ...
0answers
15 views

### Reduction of sparse features for machine learning

I'm trying to use a 1D histogram as a feature for machine learning. A histogram instance can be very sparse and the range of its bins is theoretically unbounded. Moreover it is expected that non-zero ...
0answers
8 views

### methodology for assessing experimental design

I have pre post data for 5 different groups of kids living in different settings (rural, low income, high income, etc). The variables include age, gender, type of setting, treatment (exposure to one ...
0answers
13 views

### Multivariate stats

I have the following data to analyse and not sure what the best method would be. I have percent coverage data for several invasive plant species. as well as several variables including different land ...
0answers
29 views

### SPSS Backward Regression, wrong results?

I'm using SPSS to do very simple multiple linear regression models. Playing with these models I noticed I get different results if I either use the Enter method or the Backward selection. Let's say I ...
0answers
3 views

### Noise Contrastive Estimation gradients

I'm trying to derive gradients of the Noise Contrastive Estimation cost which is referenced at equation 9 in Mnih et. al from equation 8. Can you give me some hint? What I derive seems to be totally ...
0answers
26 views

### How to conveniently add a large set of regressors in R? [migrated]

I have to add approximately 30 dummy variables to a regression. If my variables would be named dummy1 - dummy30, I would denote ...
0answers
58 views

### Has anyone tried to parallelize multiple imputation in 'mice' package?

I'm aware of the fact that Amelia R package provides some support for parallel multiple imputation (MI). However, preliminary ...
1answer
24 views

### different HRs and corresponding CIs with cph and coxph

I tried a model using cph and coxph: ...
0answers
5 views

### Product Lifecyle Prediction using Survival Model

The project objective which I have is to come up with a model which would help forecast products which are close to the end of their lifecyle and need to be replaced. I was thinking whether this could ...
1answer
30 views

### Computing conditional variance

Is there a package in R that does that? How could I compute the conditional variance $\mathrm{var}(y|x)$ of a sample? I also don't know what to do in case where there are not enough $y$ for each ...
1answer
99 views

### Simulation in time series analysis

I want to forecast tourist arrivals using time series analysis I expected to use data from 2000-2013. But due to the civil war, the behavior of tourist arrivals is rather different after 2008. ...
0answers
34 views

### Standard error of slope estimator

Can someone please help explain this. I know the answer is a). Or provide a link with more information. The slope estimator, β1, has a smaller standard error, other things equal, if a) there is ...
0answers
9 views

### NARX model to predict future values

I have this problem , where I have to predict a value of a indicator which depends on 270 other predictor variables. I read the time series modelling and prediction on MATLAB , which took the example ...
1answer
20 views

### Conditional Expectation Constant

If the conditional expectation E(Z|X) is a constant k, what can be inferred about Z? Since this means that whatever the value of x is given, Z is always k, does this imply that E(Z) is equal to k?
0answers
10 views

### Outlier detection in weighted time series

Given a set of observations $X _n = \{x_1, \dots, x_n\}$, and a new observation $x_{n+1}$, we aim to find whether $x_{n+1}$ is outlier. One approach to solve this problem is to check whether ...
1answer
47 views

### Is R software being used for commercial purpose as it is open source

I would like to learn R programming as I am interested in learning new programming language. Is R software being used for commercial purpose especially in financial industry.
1answer
13 views

### Explaining Non-Significant Moderations

Schools often teach us how to conduct and interpret Moderations. What they don't teach us is how to explain why a moderation didn't work out for statistical/methodological reasons. Assuming that the ...
1answer
68 views

### How can I determine if a poker application is “fairly” dealing hands? (Texas Hold Em)

I'm trying to take on a personal project with my new found knowledge (and beginner) in statistics and data analysis. I play an online poker app that is Texas Hold Em' (name withheld) and I suspect ...
0answers
19 views

### In R, when plotting using ggvis, how to specify plot range? [on hold]

When plotting using ggvis(x = ~c1, y = ~c2) the range of both axes are automatically determined. But I would like to show a larger range. How to set it? I read ...
0answers
5 views

### R- yaimpute - standarize value before KNN for imputation

I need to impute data before running a logistic regression. I'm trying the package yaimpute but I realized that it doesn't standarize before imputing. I created a sample matrix to view so: ...
0answers
6 views

### Detecting false positives of a classification algorithm

I run a web service with a lot of users. Some of these users are involved in undesirable behavior (e.g. trolling). I've come up with a classification algorithm to detect these users (and deactivate ...
0answers
4 views

### Printing nested lists using nested functions [migrated]

n = [1, [2, [3, [4, 5]]], [6, [7, [8, [9]], 10]]] In python, I want to write a nested function within the function 'print_list' that calls itself recursively for each sub-list. The nested function ...
1answer
41 views

### How am I using KS test incorrectly in R?

Here are the codes I input into R: ...
0answers
6 views

### SLR Residual Plots vs predictor or fitted values?

In our regression class, the professor said we can either plot the residuals vs the predictor values or vs the fitted values. I asked if there was a difference in the two plots (i.e. might you be able ...
0answers
7 views

### Rcmdr package not installed [migrated]

When I tried to installed Rcmdr package with this command: install.packages("Rcmdr",dependencies=TRUE) I got following error message: ...
2answers
19 views

### Finite Population Variance for a Changing Population

How does the addition of one unit affect the population variance of a finite population if everything else remains unchanged? What are the conditions such that the new unit leaves the variance ...
0answers
12 views

### Sample size for a moderated hierarchical multiple regression analysis?

I am having a challenging time finding out the necessary sample size for a moderated hierarchical multiple regression analysis. The first five steps of the regression will include the 17 variables ...
0answers
23 views

### Is it ok to delete items when Cronbach's alpha is low?

I have used a questionnaire in my research that is showing a Cronbach's alpha of .52, even though it has shown good internal consistency in many previous studies. I can see that if I delete 2 of its 9 ...
0answers
52 views

### Extreme learning machine: what's it all about?

I've been thinking about, implementing and using the Extreme Learning Machine (ELM) paradigm for more than a year now, and the longer I do, the more I doubt that it is really a good thing. My opinion, ...
1answer
107 views

### Probability Homework Question

A researcher is conducting a survey of county residents in a Midwestern state. Foolishly, she uses lists of registered voters as a sampling frame and randomly draws phone numbers and addresses from ...
0answers
54 views

### Regression shows positive linear relationship. Then why this negative equation

I am trying to get my regression right. I want to see, if subs increase how much increase in revenue is seen. The dependent variable is Revenue while the independent variable is subscribers. Least ...
0answers
13 views

### My transfer function has non-stationary inputs, but a stationary output. Should I difference both the inputs and outputs during structure estimation?

I have a system of two inputs and one output that I'd like to model using the following Box-Jenkins transfer function ("dynamic regression") structure: y_t=\frac ...
0answers
28 views

### Help in calculating OLS estimators under certain constraints/modifications

I am new to these forums and to econometrics as a whole. I was hoping someone would be able to give me a nudge in the right direction with this problem. I've done extensive research both online and in ...
0answers
21 views

### Reference for the idea that a simpler model can be used when the range of data values is smaller

When we build a statistical-physical model, generally, a simpler model can be justified when the range of data-values is smaller. I can't be the first person to use this idea, but I also can't find ...
1answer
18 views

### post-hoc for non homogeneous data

Which post-hoc test is used for non homogeneous data. I have checked my data set and they are normally distributes. I used Anova with Levene's test and it showed that my data is not homogeneous, and ...
1answer
36 views

### Breaking black and white captcha

I am looking for a way to break this kind of captcha: I know that there are always 4 digits, in black and white. The image basically consists of an easily recognizable 4-digit code, mixed with a ...
0answers
19 views

### Selection of subset of values to hide a certain value [on hold]

Assume that I have a set of 10 integers: x1, x2, ..., x10. I need to select one of them (e.g. x2) to perform a some computation. To hide that I selected x2, I need to select other values as well. My ...
0answers
22 views

### Meta-analysis: Talking about power and inference (to power)

I need a little help or reassurance concerning how to explain power "to power", i.e. to decision makers that are not well versed in statistics. The problem is this: I have done three empirical ...
1answer
64 views

### Implementation of hierarchical bayesian model produces errors with PyMC

I'm trying to solve an exercise from this book in which I'm supposed to fit data on temperature and elevation in Colorado to this model: \boldsymbol{Y} = \boldsymbol{\mu} + ...
0answers
5 views

### Compressed sensing with prior information on locations of non-zeros

I would like to solve a system of equations $Ax=b$, where I am enforcing that $x$ should be $K$-sparse. However, I also want to impose the additional constraint that for $K_1+K_2=K$ and \$x = \left[ ...
2answers
51 views

### How to check if a series has been seasonally adjusted correctly?

I am a bit puzzled here and would like to understand how to check if a time series has been seasonally adjusted correctly using X-13 Arima. After seasonally adjusting time series using X13-ARIMA ...
1answer
22 views

### Very different Neural Network test errors for same architecture

So I'm doing a time series prediction, and assessing the capability of the ANN to predict that time series. I am using Matlab's neural network toolbox functions, and the training parameters are the ...
2answers
363 views

### For plotting with R, should I learn ggplot2 or ggvis?

For plotting with R, should I learn ggplot2 or ggvis? I don't necessarily want to learn both if one of them is superior in any regard. Why R community keeps creating new packages with overlapping ...
1answer
68 views

### What methods are used in order to prove the independence of two variables in statistics?

I have four variables (attributes) describing student. They are categorical data. I need to prove that these variables are independent. The easiest way (even not sufficient) is to see the ...
2answers
33 views

### Significance difference between the slopes of three regression lines

I have three simple regression models with three different Y (dependant variable) and one X (independent variable). All the slopes are significant. Is there anything called testing the difference ...
1answer
64 views

### How can there be a linear correlation between two PCA components?

I perform principal component analysis (PCA) on a dataset, and then plot the first and the second principal components. I get the following phenomenon: one principal component appears to be a linear ...
0answers
3 views

### analysis strategy for selecting and/or transforming correlated continuous biomarkers to predict binary endpoint

I am given a simulation task to come up with several analysis strategies and compare their relative performances. The horizon is wide open; I appreciate all recommendations of methods and references. ...
1answer
14 views

### Rstan model for a simple mixture of normals

This blog post from Rbloggers describes how to code a simple three-part normal mixture model with known mixing coefficients, means and standard deviations. While it describes the procedures in some ...
1answer
68 views

### are qqplots appropriate for time series?

I've seen that qqplots are a very useful tool to check model assumptions. In particular to analyze the residuals. However, it seems that is based on data ordering and therefore assumes a atationary ...

15 30 50 per page