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### Home > CALC > Chapter 2 > Lesson 2.1.1 > Problem2-7

2-7.
1. Consider the following function: Homework Help ✎

1. Sketch a graph of the function. Label any holes or asymptotes.

2. Find the inverse of f(x). Is the inverse a function? Why or why not?

3. Sketch a graph of the inverse and compare this sketch to the graph of f(x) from part (a).

No holes. Two asymptotes: x = 4 and y = 0. Label them.

$y=\frac{1}{x-4}\ \ \ \ \ \text{Rename }f(x)\text{ as }y$

$y(x-4)=1 \ \ \ \text{solve for }x$

$x-4=\frac{1}{y}$

$x=\frac{1}{y}+4$

$f^{-1}(x)=\frac{1}{x}+4 \ \ \ \text{Switch the }x \text{ and }y$

You do not need to know the inverse equation to sketch a graph of the inverse. Start by switching the horizontal and vertical asymptotes. Then plot some key points, and connect.

Use the eTool below to explore the graphs.
Click on the link to the right to view the full version of the eTool. Calc 2-7 HW eTool


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×

# Bug in Techniques Section...

I was recently attempting the Advanced Set 1 of Limits of Function under Calculus Section. While attempting the third question of this set, I had input an incorrect answer and then viewed its solution without using up the remaining two tries. Then I clicked on return to home but there Set 1 doesn't show the completed mark and my set 2 is not being enabled... Please debug this.....

Note by Rahul Nahata
4 years, 6 months ago

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paragraph 1

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print "hello world"
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# up as a code block.

print "hello world"
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Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
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a_{i-1} $$a_{i-1}$$
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Hi. This issue should be resolved now.

Please let me know if the problem persists.

- 4 years, 6 months ago

Hi Rahul,

1. "No completed mark" is a bug, and it will be fixed soon.
2. When you view the solution, you forfeit your right to answer the problem (just in case there was any confusion.)

Staff - 4 years, 6 months ago

Me Too..

- 4 years, 6 months ago

I have also experienced the same bug in "Evaluate Algebraic Expressions" problems. I request to the technical authority to debug this.

- 4 years, 6 months ago

format: plain


text:
# Math Help - Find X

1. ## Find X

Find x in the diagram below. Show how you got it please.

$cos x = \frac{RS}{RT} = \frac{RS}{2RS} = \frac{1}{2}$


text:
# PhysicsMechanics- friction

#### Shah 72

##### Member
A boy drags a sledge of mass 4kg from rest down a rough slope at an angle of 18 degree to the horizontal. He pulls it with a force of 8N for 3s by a rope that is angled at 10 degree above the parallel down the slope. After 3s the rope becomes detached from the sledge. The coefficient of friction between the slope and the sledge is 0.4. Find the total distance the sledge has moved down the slope from when the boy started dragging it until it comes to rest.
For the first part before the rope is detached
I calculated
R=40cos 18+8sin 10
Coefficient of friction =0.514
F=m×a
So I got a = 0.003m/s^2.
I then calculated s= ut +1/2 at^2 and got s1 =0.0135m.
I calculated v=0.009m/s
Part 2 when the rope is detached
R=40cos 18
By using F=m×a
I got a=0.714m/s^2.
Iam not getting the right ans for the total distance.
The ans in the textbook is 18.5m.
Pls help

#### skeeter

##### Well-known member
MHB Math Helper
during the drag ...

$ma = mg\sin(18) + 8\cos(10) - \mu[mg\cos(18) - 8\sin(10)]$

try again

#### Shah 72

##### Member
I did your way and got a= 0.35m/s^2, v=1.05m/s and s=1.575m
Now the second part when the rope is detached
R=40cos18, F=40sin18 -0.4×40cos18, F=2.84N, using F=m×a, I get a=0.71m/s^2, and s2=0.776m. I still don't get the ans 18.5m

#### skeeter

##### Well-known member
MHB Math Helper
recheck your acceleration calculation during the drag …

#### Shah 72

##### Member
recheck your acceleration calculation during the drag …

View attachment 11134
I did not understand in the second part when the rope is detached, how is it 10 sin 18 + 2 cos 10. Shouldn't it be R = 40 cos 18 and F= 40sin 18- 0.4(40cos 18)

#### skeeter

##### Well-known member
MHB Math Helper
during the drag …

$a_1 = 10\sin(18) + 2\cos(10) - 0.4[10\cos(18) - 2\sin(10)] = 1.39 \, m/s^2$

$\Delta x_1 = \dfrac{1}{2}a_1 \cdot 3^2 = 6.28 \, m$

$v_1 = 0 + a_1 \cdot 3 = 4.18 \, m/s$

after the rope is detached …

$a_2 = 10\sin(18) - 4\cos(18) = -0.714 \, m/s^2$

$\Delta x_2 = \dfrac{0^2- v_1^2}{2a_2} = 12.25 \, m$

total displacement = 6.28 + 12.25 = 18.5 m

#### Shah 72

##### Member
during the drag …

$a_1 = 10\sin(18) + 2\cos(10) - 0.4[10\cos(18) - 2\sin(10)] = 1.39 \, m/s^2$

$\Delta x_1 = \dfrac{1}{2}a_1 \cdot 3^2 = 6.28 \, m$

$v_1 = 0 + a_1 \cdot 3 = 4.18 \, m/s$

after the rope is detached …

$a_2 = 10\sin(18) - 4\cos(18) = -0.714 \, m/s^2$

$\Delta x_2 = \dfrac{0^2- v_1^2}{2a_2} = 12.25 \, m$

total displacement = 6.28 + 12.25 = 18.5 m
Thank you very much!!!

#### Shah 72

##### Member
during the drag …

$a_1 = 10\sin(18) + 2\cos(10) - 0.4[10\cos(18) - 2\sin(10)] = 1.39 \, m/s^2$

$\Delta x_1 = \dfrac{1}{2}a_1 \cdot 3^2 = 6.28 \, m$

$v_1 = 0 + a_1 \cdot 3 = 4.18 \, m/s$

after the rope is detached …

$a_2 = 10\sin(18) - 4\cos(18) = -0.714 \, m/s^2$

$\Delta x_2 = \dfrac{0^2- v_1^2}{2a_2} = 12.25 \, m$

total displacement = 6.28 + 12.25 = 18.5 m
Can you also pls explain why I don't need to take the normal contact force and the force between the slope and the sledge?

#### skeeter

##### Well-known member
MHB Math Helper
The Normal Reaction force, $N$, is perpendicular to the incline. Reference the two FBD's attached ...

during the drag by the 8N applied force,

$N_1 = mg\cos{\theta} - F\sin{\phi} \implies f_1 = \mu(mg\cos{\theta} - F\sin{\phi})$

after the dragging force is gone,

$N_2 = mg\cos{\theta} \implies f_2 = \mu mg\cos{\theta}$

Last edited:

#### Shah 72

##### Member
during the drag …

$a_1 = 10\sin(18) + 2\cos(10) - 0.4[10\cos(18) - 2\sin(10)] = 1.39 \, m/s^2$

$\Delta x_1 = \dfrac{1}{2}a_1 \cdot 3^2 = 6.28 \, m$

$v_1 = 0 + a_1 \cdot 3 = 4.18 \, m/s$

after the rope is detached …

$a_2 = 10\sin(18) - 4\cos(18) = -0.714 \, m/s^2$

$\Delta x_2 = \dfrac{0^2- v_1^2}{2a_2} = 12.25 \, m$

total displacement = 6.28 + 12.25 = 18.5 m
I understood. Thank you so so so much!!!
The Normal Reaction force, $N$, is perpendicular to the incline. Reference the two FBD's attached ...

during the drag by the 8N applied force,

$N_1 = mg\cos{\theta} - F\sin{\phi} \implies f_1 = \mu(mg\cos{\theta} - F\sin{\phi})$

after the dragging force is gone,

$N_2 = mg\cos{\theta} \implies f_2 = \mu mg\cos{\theta}$

View attachment 11136

View attachment 11136
I understood this. I was doing the silly mistake of finding a new coefficient of friction. Thank you so so much for explaining!!!


text:
# Is FACTORIZATION or PRIMES known to be in LOGSPACE

Are the integer factorization and PRIMES known to be in LOGSPACE?

Recently, it has been shown by researchers that PRIMES is in P. But this does not say anything about LOGSPACE since it is not known if LOGSPACE = P.

• Factorization definitely isn't, if it were then RSA encryption would be easily broken because factorization would be in P. – jmite Aug 1 '14 at 16:40
• jmite: But we don't know whether $\mathrm{FACTORIZATION}$ is in $\mathrm{P}$ or not. – Martin Jonáš Aug 1 '14 at 19:23
• Definitely isn't known, sorry, should have clarified. – jmite Aug 1 '14 at 21:15
• I guess the only cases that are currently being tackled by algorithms in P are such cases as close primes... – user13675 Aug 9 '14 at 20:37

Factoring is not known to be even in $\mathsf{P}$.

Primality is not known to be in any class conjectured to be smaller than $\mathsf{P}$ (AFAIK).

• "Factoring is not known to be even in" BPP. $\;$ – user12859 Aug 1 '14 at 21:10
• @Ricky, yes, I wanted to use more famous classes used in the question (and before you say, yes, BPP should be more well-known. :) – Kaveh Aug 1 '14 at 21:13

If the number is unary, yes, both are logspace. Sum up the 1's to find the number. Use trial division.

If the number is in binary, then logspace is in terms of the length of binary input. It's unknown.

Note that some semi-prime's smallest factor is approximately the square root of the number, which has O(n) digits.

Also note that logspace is in P (there are only polynomial state configurations with logspace), so factorization in logspace would imply factorization in P.


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# Model for the growth of the World Airline Network

### Abstract

We propose a probabilistic growth model for transport networks which employs a balance between popularity of nodes and the physical distance between nodes. By comparing the degree of each node in the model network and the WAN, we observe that the difference between the two is minimized for ${\alpha \approx 2}$. Interestingly, this is the value obtained for the node-node correlation function in the WAN. This suggests that our model explains quite well the growth of airline networks.

Publication
International Journal of Modern Physics C


text:
Note that this applies only to the read-write DOMMatrix Below is an example of the fixed-position. Default Sizing Algorithm CSS 2.1 px If the algorithm returns null, then there is no intrinsic aspect ratio. create a new user coordinate system. process between the SVG document fragment generating the SVG viewport, and the The position vectors clearly depend on the choice of coordinate origin. SVGTransformList object is returned that reflects the given own units set. bounding box of an element. presentation attribute on the outermost svg element. otherwise, the initial viewport's height must be the value of the height For all elements, SVG uses a coordinate system or grid system similar to the one used by canvas (and by a whole lot of other computer drawing routines). A translation moves all the points of an element in the same direction and by the same amount. Jakob Jenkov CTMparent is CTM for the transformation matrix from Note: Because SVG uses a y-down coordinate system, then by default, glyphs will often be drawn in the +x -y quadrant of the SVG coordinate system. with a particular element. See As y decreases the points move downin the coordinate system. The net force on the sledge can be expressed in the Cartesian coordinate system as vector $\overset{\to }{F}=(-2980.0\hat{i}+8200.0\hat{j})\text{N}$, where $\hat{i}$ and $\hat{j}$ denote directions to the east and north, respectively. (width, height), and an origin, measured in abstract This leaves us with a few obvious options when scaling an image to a different aspect ratio using the preserveAspectRatioattribute of an SVG: Note: Amelia Bellamy-Royds gives a great primer on SVG scaling over at CSS Tricks. A value of zero disables rendering of the translate(translate-x, translate-y) scale(scale-x, scale-y). the coordinate systems of mathematics, graphs etc. Values of vector-effect other than value as specified in the meet-or-slice constant table above for the meet-or-slice Below is an example of the non-scaling-size non-rotation fixed-position. (Figure) is valid for any vector, so we can use it to express the x – and y -coordinates of vector $\overset{\to }{r}$. gradient vector (, Indicates that the attributes which specify the center Cylindrical coordinate system Vector fields. screen is specified as the additional value of a scale that is applied to content contained by the element. CSS Transforms defined in List interfaces. whose elements are SVGTransform objects. There are 537 coordinates svg for sale on Etsy, and they cost \$3.74 on average. is shown in light blue. to the width and height of the nearest ancestral SVG viewport. rotate(…), Vector … to 150 on the outermost ‘g’ element, each "em" unit is reflects the base value of the ‘preserveAspectRatio’ SVG viewport). ‘preserveAspectRatio’ attribute. cy# The vertical position of the center-point. such as a single scale(…) If other types of transforms are supported and used, the SVGTransform The behaviour defined in this section is specific to CSS, but may be adapted 'display: none', within a ‘defs’ description of this negotiation process. point (0,0) for the purposes of calculating a bounding box. matrix object space that should be mapped to the bounds of the SVG viewport Example ViewBox illustrates SVG supports nested coordinate systems. The normal coordinate transformation formula from user coordinate system Then, all of a sudden, at least in that coordinate system, the reflection wouldn't be around this inclined line. entire SVG viewport. Vectors are defined in cylindrical coordinates by (ρ, φ, z), where . An SVGAnimatedTransformList object is used to reflect the does not contribute to the bounding box of any ancestor element. Generally, we are familiar with the derivation of the Divergence formula in Cartesian coordinate system and remember its Cylindrical and Spherical versions intuitively. Here we will discuss the standard Cartesian coordinate systems in the plane and in three-dimensional space. shows that the initial coordinate system has the origin at the appropriate transformation matrix to map the specified input arguments to this algorithm do not affect the bounding box returned unit in the initial coordinate system must equal one SVG follows the description and definition of common values and You guessed it: black. of the absolute unit identifiers, the "in" unit (inch). and stroke-dashoffset properties on an element have no effect on the Establishing a new SVG viewport for a masks and filters, the applicable element can be either a Rectangular coordinate system position vector index lowered.svg 442 × 737; 14 KB.
2020 svg coordinate system


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• Create Account

## Global & Static Classes

Old topic!

Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.

5 replies to this topic

### #1Six222  Members

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Posted 29 February 2012 - 05:33 AM

Hey guys, I'm current working on my next game and I've made a couple so far but I've always felt that the code that I wrote for them is messy. I'm forever making every class global or static just so I can access them from somewhere else instead of having to pass refrences around. For example:

class EntityHandler
{
public:
void Tick();
void Draw();

private:
static vector<Bullet>BulletList;
static vector<Enemy>EnemyList;
};

I'm writing all of my classes like this just so I can add a new entity/bullet from anywhere in my code. Looking at I feel that it's correct, it works but I can't help thinking that it's just a bad way of programming.

I could just send a refrence to the EntityHandler to every class but the entities shouldn't have to know about that and I would also probably be sending in loads of other refrences to all the static classes I have just so I could perform 1 or 2 tasks. This isn't good but I honestly don't know how I could do it another way. I guess some sort of message/event system could work for this but I really don't know...

Anyway If anyone could point me in the right direction that would be awesome. Thanks

### #2mrjones  Members

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Posted 29 February 2012 - 06:18 AM

You are right in that it is not the best solution. You are also right in that giving each class reference to EntityHandler would be a bad idea as it requires boilerplate code and creates minor memory overhead.

A slightly better approach (although not the best imho) would be something like this:
class EntityHandler {
public:
void Tick();
void Draw();

private:
vector<bullet> BulletList;
vector<enemy> EnemyList;
};

class Global {
private:
static EntityHandler entityHandler;

public:
static EntityHandler& getEntityHandler();
};

This way at least EntityHandler is no longer static and its instances can be used elsewhere when required.

However, accessing Global::getEntityHandler() is a bit unwieldy and the rest of the code will become dependent on global instances.
The solution I usually prefer is to instead have a non-static class and pass its reference on only to methods that actually need it:
class EntityHandler {
public:
void Tick();
void Draw();

private:
vector<bullet> BulletList;
vector<enemy> EnemyList;
};

class Bullet {
public:
// Pass the reference of EntityHandler only to methods that actually require it
// or some other class containing EntityHandler reference.
// This way you won't have to worry about EntityHandler being destroyed before bullet as it would be generally
// passed from EntityHandler::Tick/Draw method.
// It would also be completely painless to take a Bullet from one EntityHandler and put it to another as there
// is no internal reference.
void Tick(EntityHandler& e);
void Draw(EntityHandler& e);
};

It might seem to be a bit of a hassle to pass it on to such functions, but there are generally quite few functions that should need to see something as grand as EntityHandler.

### #3Six222  Members

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Posted 29 February 2012 - 06:54 AM

I never thought of passing the EntityHandler down through like that /facepalm.. So I could for example do this?

class Bullet {
public:
void Tick(EntityHandler& e);
void Draw(EntityHandler& e);
};

void EntityHandler::Tick()
{
vector<bullet>::Iterator it;
for(it = BulletList.begin(); it != BulletList.end(); it++)
{
(*it)->Tick(*this);
}
}

### #4mrjones  Members

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Posted 29 February 2012 - 09:36 AM

Precisely. I didn't write that part down before, but that is exactly what I meant.

### #5Six222  Members

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Posted 29 February 2012 - 09:40 AM

Precisely. I didn't write that part down before, but that is exactly what I meant.

I started re-writing some of the static classes to pass the EntityHandler down and other classes and it works well. Thanks for the help!

### #6Monkan  Members

1001
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Posted 29 February 2012 - 04:09 PM

Also don't forget about namespaces, I never used to use them and now use them everywhere, they are really handy for wrapping things up and making it all a bit neater.


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# Alignat latex options strategies

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text:
# Arrow–Debreu equilibria for rank-dependent utilities with heterogeneous probability weighting

Jin, H
Xia, J
Zhou, X

1 October 2018

## Journal:

Mathematical Finance

## Last Updated:

2021-06-12T04:51:46.87+01:00

## DOI:

10.1111/mafi.12200

## abstract:

© 2018 Wiley Periodicals, Inc. We study Arrow–Debreu equilibria for a one-period-two-date pure exchange economy with rank-dependent utility agents having heterogeneous probability weighting and outcome utility functions. In particular, we allow the economy to have a mix of expected utility agents and rank-dependent utility ones, with nonconvex probability weighting functions. The standard approach for convex economy equilibria fails due to the incompatibility with second-order stochastic dominance. The representative agent approach devised in Xia and Zhou (2016) does not work either due to the heterogeneity of the weighting functions. We overcome these difficulties by considering the comonotone allocations, on which the rank-dependent utilities become concave. Accordingly, we introduce the notion of comonotone Pareto optima, and derive their characterizing conditions. With the aid of the auxiliary problem of price equilibria with transfers, we provide a sufficient condition in terms of the model primitives under which an Arrow–Debreu equilibrium exists, along with the explicit expression of the state-price density in equilibrium. This new, general sufficient condition distinguishes the paper from previous related studies with homogeneous and/or convex probability weightings.

927138

Submitted

Journal Article


text:
×

# help

Can you help me with this mechanics problem?

Note by Kyle Finch
1 year, 9 months ago

Sort by:

@Raghav Vaidyanathan the problem is in the middle. · 1 year, 9 months ago

I have rotated the image. $$\huge\ddot\smile$$ · 1 year, 9 months ago

U can tell me the solution . ??? · 1 year, 9 months ago

A disc has an angular velocity? · 1 year, 9 months ago

$$\text{angular velocity}=\dfrac{\text{Relative velocity perpendicular to line joining them}}{\text{Distance between them}}$$

By simple geometry, $$\angle AOP=30^\circ$$

Case I: $$A$$ is fixed to ground, so, the velocity of $$P$$ will be $$\omega R$$ at angle $$120^\circ$$ to the line joining $$AP$$, thus, $$\omega_1=\dfrac{\omega R\cos 30^\circ}{\sqrt{3}R}$$

Case II: $$A$$ is fixed to disc, in this case, velocity of $$A$$ will add up in relative velocity. In this case also, velocity of $$A$$ will be $$\omega R$$ at angle $$60^\circ$$ to the line joining $$AP$$. Thus, $$\omega_2=\dfrac{\omega R\cos 30^\circ+\omega R\cos 30^\circ}{\sqrt{3}R}=2\omega_1\Rightarrow \dfrac{\omega_1}{\omega_2}=\dfrac{1}{2}$$ · 1 year, 9 months ago

Well indeed very nice by the way how was ur mains exam · 1 year, 9 months ago

Not too good. Getting just 275. · 1 year, 9 months ago

Was the physics section tough?? · 1 year, 9 months ago

It was NCERTish. I must not blame questions for my marks. I don't like EM. · 1 year, 9 months ago

So who among all of you is supposed to get the highest. · 1 year, 9 months ago

Ronak Agrawal (on Brilliant) · 1 year, 9 months ago

Ok just one last query what makes a problem popular on brilliant. Can u tell ronaks score · 1 year, 9 months ago

He's getting like 315-320. The quality of problem makes it popular! · 1 year, 9 months ago

Bhavya Is getting 322.I guess?? · 1 year, 9 months ago

Yes, he does · 1 year, 9 months ago

No i m asking do the staff member decide or the moderators · 1 year, 9 months ago

$$\omega$$ · 1 year, 9 months ago


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# Evaluating integrals with radicals and fractions

• Aug 10th 2010, 08:37 PM
bobsanchez
Evaluating integrals with radicals and fractions
Two integration problems are giving me a lot of trouble.

(2x + 2)/(x^2 + 2x)^(1/2)

and

dx/(x^1/2)(1 - x^1/2)^4

I can't get the substitution technique to work out on these two for some reason.
• Aug 10th 2010, 08:42 PM
Prove It
$\int{\frac{2x+2}{(x^2 + 2x)^{\frac{1}{2}}}\,dx} = \int{(2x+2)(x^2+2x)^{-\frac{1}{2}}\,dx}$.

Let $u = x^2+2x$ so that $du = (2x+2)dx$ and the integral becomes

$\int{u^{-\frac{1}{2}}\,du}$

$= \frac{u^{\frac{1}{2}}}{\frac{1}{2}}+C$

$= 2u^{\frac{1}{2}}+C$

$= 2(x^2+2x)^{\frac{1}{2}}+C$.
• Aug 10th 2010, 08:49 PM
Prove It
$\int{\frac{dx}{x^{\frac{1}{2}}(1-x^{\frac{1}{2}})^4}} = \int{x^{-\frac{1}{2}}(1-x^{\frac{1}{2}})^{-4}\,dx}$

$= -2\int{-\frac{1}{2}x^{-\frac{1}{2}}(1-x^{\frac{1}{2}})^{-4}\,dx}$.

Let $u=1-x^{\frac{1}{2}}$ so that $du = -\frac{1}{2}x^{-\frac{1}{2}}\,dx$ and the integral becomes

$-2\int{u^{-4}\,du}$

$= -2\left(\frac{u^{-3}}{-3}\right) + C$

$= \frac{2}{3u^3} + C$

$= \frac{2}{3(1-x^{\frac{1}{2}})^3} + C$.


text: Question: Which Of The Following Correlation Coefficients Indicates The Strongest Relation? A. The stronger the association between the two variables, the closer your answer will incline towards 1 or -1. Trending questions. … Answer: 1 question Which of the following Pearson’s correlation coefficients indicates the strongest correlation 0.69, 0.31, 0.31 , 0.43 , 0.71 - the answers to estudyassistant.com A PMCC of zero or very close would mean a very weak correlation… a. Follow The correlation coefficient that indicates the weakest linear association between two variable is ? A coefficient of -1 indicates strong negative correlation, while +1 suggests strong positive correlation. Which of the following correlation coefficients indicates the strongest relationship? A)+.53 B)−.64 C)−.38 D)+.29 Thanks for contributing an answer to Mathematics Stack Exchange! The p-value for the null hypothesis Beta coefficient =0 is 0.0001. Finally, a white box in the correlogram indicates that the correlation is not significantly different from 0 at the specified significance level (in this example, at $$\alpha = 5$$ %) for the couple of variables. a) .5 b) -. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Asking for help, clarification, or responding to other answers. This is what you are likely to get with two sets of random numbers. (1) - 0.73 (2) - 0.11 (3) 0.12 (4) 0.35. The size of r indicates the strength of the correlation. So all right, that numerical value. -0.98. +1 ==> indicated a strong positive relationship; 0 ==> indicated no relationship; The closer to either +1 or -1, the stronger the relationship; The closer to 0, the weaker the relationship; Exercises. It deals with behavior not tampered with by outside influences It provides an overabundance of information It limits biased observations … There's no association between the two variables you're testing. How does color identity work in Commander? The value of r is always between +1 and –1. An r of +0.20 or -0.20 indicates a weak correlation between the variables. Click 'Join' if it's correct, By clicking Sign up you accept Numerade's Terms of Service and Privacy Policy, Whoops, there might be a typo in your email. Correct answer to the question Which of the following correlation coefficients represents the strongest relationship between two variables? a. Which of the following indicates a fairly strong relationship between X and Y? Which of the following coefficients of correlation indicates the strongest relationship between two sets of variables? 0.90 c. 0.00 d. 1.20. general-psychology; 0 Answer. Upvote (9) Views (8217) Followers (3) Write an Answer Register now or … 1 decade ago . Are there any diacritics not on the top or bottom of a letter? -.50. c. +.80. Did Barry Goldwater claim peanut butter is good shaving cream? How does the formula of the correlation coefficient measures “linear” relationship? The correlation coefficient, r, tells us about the strength and direction of the linear relationship between x and y.However, the reliability of the linear model also depends on how many observed data points are in the sample. ${}\qquad{}$. Which of the following correlation coefficients indicates the strongest relationship? +.25. Which of the following correlation coefficients indicates strongest relationship? Develop a scatter di…, If we assume that the conditions for correlation are met, which of the follo…, Five observations taken for two variables follow.a. Which of the following correlation coefficients indicates the strongest relationship? Lv 6. We use cookies to give you the best possible experience on our website. c. −.72 . A) -.80 B) -.30 C) .30 D) .80 So, for example, a correlation of minus 0.9 would be stronger then a correlation of minus 0.2. Correlation coefficient between two binominals, Removing clip that's securing rubber hose in washing machine. asked Mar 8, 2020 in Psychology by madds. c. +.80 . A PMCC of zero or very close would mean a very weak correlation. Favorite Answer-0.9 indicated the strongest relationship. In this case, -0.93 is the correct answer. A) 00 B) +.35 C) +.78 D) -.84 Find out what you don't know with free Quizzes Start Quiz Now! That's a little bit about correlations. ★★★ Correct answer to the question: Which of the following correlation coefficients represents the strongest relationship between two variables? The correct answer is d. +.90. Which of the following correlation coefficient values indicate the strongest relationship between two variables? On the other hand, a negative correlation means that a variable a our first variable is increasing variable be our second variable is decreasing. A correlation coefficient of 0 indicates A. no relationship … A) -0.9 . So essentially the closer that you get to zero, the weaker the correlation So once you get to zero, there's actually no correlation. Planning Accounting Budgeting IFRS CMA. So we have zero plus one and minus one, and I'm first going to explain what the positive and the negative means. +.43 . Developa scatter dia…, Describe each of the following correlation coefficients using the terms stro…, EMAILWhoops, there might be a typo in your email. 5 … A coefficient of -1 indicates strong negative correlation, while +1 suggests strong positive correlation. Government censors HTTPS traffic to our website. ... -The strongest linear relationship is indicated by a correlation coefficient of -1 or1. -Duffy's findings indicate a strong positive relationship between practice time and race times. Trending questions. To learn more, see our tips on writing great answers. As r gets closer to either -1 or 1, the strength of the relationship increases. 103. −.64 c. −.72 d. .00 1. The following points are the accepted guidelines for interpreting the correlation coefficient: 0 indicates no linear relationship. 0 votes. So today we're gonna be going through problem number 14 in Chapter two Psychological research, and this problem is asking us about correlations. answered Mar 11, 2020 by schris . asked Feb 6, 2016 in Psychology by CurryManiac. Join Yahoo Answers and get 100 points today. D) 0. It shows that the two variables are strongly coupled but move in the opposite direction. So it means that the variables X and Y are strongly (linearly) correlated. Topic: Correlational Research 115. A correlation of +1 indicates a perfect positive correlation, meaning that both variables move in the same direction together. A value close to +1 indicates a strong positive relation and a value close to -1 indicates a strong negative correlation. ChemDraw: how to change the default aromatic ring style for drawing from SMILES. So the strength of the correlation is given by the numeric value. -.90b. 01 c) .2 d) -.8 - edu-answer.com Best answer. b. Medical. –0 c. +0 d. If researchers gave participants varying amounts of a new memory drug and then gave them a story to read and measured their scores on a … Solution.pdf Next Previous. .90 . Values between 0 and +1/-1 represent a scale of weak, moderate and strong relationships. A) -0.9 . +.68 c. +.69 d. +.71. However, the definition of a “strong” correlation can vary from one field to the next. -.90 The correlation value of two variables ranges from -1 to +1. The maximum possible is -1 or 1. Which of the following correlation coefficients indicates the strongest relationship between two variables? -.90. b. (1) - 0.73 (2) - 0.11 (3) 0.12 (4) 0.35 . +0.74 b. Which of the following values indicates the strongest relationship between two variables as measured by a correlation coefficient? −.64 . This value is a positive correlation that is strong. +.43 b. Answer Save. Possible correlations range from +1 to –1. +0.87 C. +0.13 D. – 0.47 Page: 28 APA Outcome: 1.1 Bloom's Taxonomy: Understand Difficulty Level: Basic Learning Objective: Explain how research on life-span development is conducted. +1 indicates a perfect positive linear relationship… Definition of Coefficient of Correlation. So, for example, a correlation of minus 0.9 would be stronger then a correlation of minus 0.2. A: -.01 B: -1 C:1 D: 0 - e-eduanswers.com 0 votes. The value of +1 for the correlation coefficient denotes that the two variables are have a very strong negative correlation. The variables may be two columns of a given data set of observations, often called a sample, or two components of a multivariate random variable with a known distribution. a) +.78 b) -.84 c) +.35 d) 00 Ans: b Section Ref: Research Methods Difficulty: Hard Objective: 1.3 Objective Text: Summarize psychology’s three major research methods Bloom’s Level: Application 104. Welcome to Sciemce, where you can ask questions and receive answers from … A perfect downhill (negative) linear relationship […] Which correlation coefficient indicates the strongest relationship between two variables? The correlation coefficient is a statistical measure that calculates the strength of the relationship between the relative movements of two variables. Which of the following correlation coefficients indicates the strongest relationship between two variables? A coefficient of zero means there is no correlation between two variables. See below I would say -0.68, but still -0.68 does not indicate a strong correlation. Be the first to answer this question. Which of the following correlation coefficients indicates the strongest relationship between two variables? +.74 b. Making statements based on opinion; back them up with references or personal experience. Answer math problem ? +.68 c. +.69 d. +.71. A coefficient of correlation of +0.8 or -0.8 indicates a strong correlation between the independent variable and the dependent variable. a) -0.85 - This was my answer which I thought was the closest to one (meaning strongest. I understand that the strength can vary from 0-1 and I thought I understood that positive or negative simply had to do with the direction of the correlation. The size of r indicates the strength of the correlation. –.50 . d. +.25. +0 b. To interpret its value, see which of the following values your correlation r is closest to: Exactly –1. Which of the following correlation coefficients indicates the strongest relationship between two variables? Hi, everybody. .50 . Visually, this represents any relationship between two variables that depicts a straight line when plotted out next to each other in a graph. Oct 29 2020 12:06 PM. What is the coefficient of correlation? Workarounds? Which of the following coefficients of correlation indicates the STRONGEST relationship between two sets of variables? The maximum possible is -1 or 1. -.98 .90 1.2 -.45 0.0 Which of the following is considered a disadvantage of naturalistic observation? 1. So there you go. +.74 b. It doesn't matter whether it is negative or positive. Correlation coefficient: Indicates the direction, positively or negatively of the relationship, and how strongly the 2 variables are related. Did Gaiman and Pratchett troll an interviewer who thought they were religious fanatics? a. So I'm going to draw a number line here. 0 votes. 1 decade ago. (a)+.43 ;(b)-64 ;(c)-72 ;(d) 0.00 a. What is this logical fallacy? Ans: a. 103. Pearson's Product Moment Correlation Coefficient indicates how strong the relationship between variables is. Which of the following correlation coefficient values indicate the strongest relationship between two variables? I understand that the strength can vary from 0-1 and I thought I understood that positive or negative simply had to do with the direction of the correlation. A zero correlation indicates that there is no relationship between the variables. The value of r is always between +1 and –1. So whenever you see a positive number, when you're talking about correlations means that as one variable increases so just write variable a increases, so does variable B, the other variable. Related Questions. Which of the following correlation coefficients indicates the strongest relationship between two variables?a. 0.48-1.00. 0 to 0.5 is a weak relationship. The correlation between two variables is considered to be strong if the absolute value of r is greater than 0.75. Pearson’s correlation coefficient returns a value between -1 and 1. Expert's Answer . The Calculus Alchemist. How to rewrite mathematics constructively? A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. Correlation test. -50c. -0.98 b. Book about a boy who accidentally hatches dragons at his grandparents' estate. Relevance. Which of the following coefficients indicates the strongest correlation?-0.8 +0.5 +0.7-0.2; Which of the following coefficients indicated the weakest correlation?-0.9 +0.8-0.2 +0.1; A … a. Which of the following correlation coefficients indicates the strongest relationship between two variables? Figure $$\PageIndex{1}$$ illustrates linear relationships between two variables $$x$$ and $$y$$ of varying strengths. Answer Save. sample variance, co-variance and correlation coefficient, Correlation coefficient of two continuous random variables, Correlation between two dependent variables. +0.65 c. 0.00 d.-0.89 ANS: D … Correlation coefficients of greater than, less than, and equal to zero indicate positive, negative, and no relationship between the two variables. a. The variables may be two columns of a given data set of observations, often called a sample, or two components of a multivariate random variable with a known distribution. rev 2021.1.21.38376, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. The highest number a correlation coefficient can be is one, with a negative or positive. A correlation of –1 indicates a perfect negative correlation, meaning that as one variable goes up, the other goes down. He measures motivation for running by a Likert scale. .75 Based on prior studies, psychologists know that the relationship between depression and anxiety is positive and the relationship between depression and the number of activities of daily living performed in negative. +0.68 c. +0.69 d. +0.71 B) -0.3. A. In my view your answer is the correct one. Correlation coefficient = 0.9. –.90 . The other numbers given in the question indicate very weak correlation. Direction. general-psychology; 0 Answer. - Answered by a verified Tutor . Answer to Which of the following correlation coefficients indicates the strongest relation? a. Use MathJax to format equations. Salary and years of education are typically _____ correlated, and grade … Relevance. Join . +3.45 b. -0.50 +1.24 +0.01 +0.62 -0.98 - edu-answer.com The interpretation of the correlation coefficient is as under: If the correlation coefficient is -1, it indicates a strong negative relationship. 2 Answers. Therefore among the four options given of -0.15, -0.8, 0.38, 0.03 the strongest relation between two factors is given by -0.8 but as the '-' sign indicates … Therefore among the four options given of -0.15, -0.8, 0.38, 0.03 the strongest relation between two factors is given by -0.8 but as the '-' sign indicates the two factors are negatively correlated. C) 0.8. Which of the following correlation coefficients indicates the strongest relationship between two variables? The other numbers given in the question indicate very weak correlation. d. +.25 Although the correlation coefficient between SAT scores and GPAs is strong, the coefficient of determination shows us that SAT scores account for only 52% of the variability in GPAs In a recent study, researchers were interested in determining the relationship between gender and the amount of time spent studying for a group of college students. asked Feb 6, 2016 in Psychology by CurryManiac. It is visually apparent that in the situation in panel (a), $$x$$ could serve as a useful predictor of $$y$$, it would be less useful in the situation illustrated in panel (b), and in the situation of panel (c) the linear relationship is so weak as to be practically nonexistent. The correlation between two variables is considered to be strong if the absolute value of r is greater than 0.75. Which of the following correlation coefficients indicates the strongest relationship between two variables? A coefficient of zero means there is no correlation between two variables. (Nothing new under the sun?). The correlation coefficient, denoted by r, is a measure of the strength of the straight-line or linear relationship between two variables. 1) -.75 2) .50 3) -.25 4) .25 141) A researcher computes a coefficient of correlation and determines that it is zero. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Click 'Join' if it's correct. MathJax reference. Which of the following correlation coefficients indicates strongest relationship? A coefficient of 0 indicates no linear relationship between the variables. In this drawing of the Avengers, who's the guy on the right? Lv 6. +0.66 c. +1.22 d. -3.21. developmental-psychology; 0 Answers. -.90 B. It doesn't matter whether it is negative or positive. The coefficient of relationship (r) between two individuals B and C is obtained by a summation of coefficients calculated for every line by which they are connected to their common ancestors.Each such line connects the two individuals via a common ancestor, passing through no individual which is not a common ancestor more than once. Medical. Correlation coefficient: Indicates the direction, positively or negatively of the relationship, and how strongly the 2 variables are related. So I'm gonna write that key fact here, the closer you get 20 the weaker which means the less strong the correlation becomes and the larger the numerical value, the stronger the correlation becomes. Which of the following coefficients of correlation indicates the WEAKEST relationship between two variables? Correlation may occur in three forms: Simple Correlation The correlation is a single number that describes the degree of the relationship between two variables. C) 0.8. -0.50 +1.24 +0.01 +0.62 -0.98 - e-eduanswers.com general-psychology; 0 Answer. D. None of these So essentially the closer that you get to zero, the weaker the correlation So once you get to zero, there's actually no correlation. Keep in mind that any numbers that are between -0.5 and -0.7 show weak negative correlation only, same for positive. By continuing to use this site you consent to the use of cookies on your device as described in our cookie policy unless you have disabled them. Which of the following is a possible correlation in his research? Which of the following correlation coefficients indicates the strongest relationship? C. The t-statistic for the null hypothesis Beta coefficient=0 is 30. The Pearson product-moment correlation coefficient, or simply the Pearson correlation coefficient or the Pearson coefficient correlation r, determines the strength of the linear relationship between two variables. c. +.80 . See my table below. So positive. Coefficient of relationship. Just like the visual, descriptive statistics is one area of statistical applications … Relationship of Coefficient of Correlation to Coefficient of Determination. On the other hand, a correlation of plus 0.8 iss stronger, then a correlation of plus 0.3. While correlation coefficients measure the strength of association between two variables, linear correlation indicates the strongest association between two variables. How to protect a secure compound breached by a small modern military? Pearson's Product Moment Correlation Coefficient indicates how strong the relationship between variables is. What did Asimov find embarrassing about "Marooned Off Vesta”? A perfect downhill (negative) linear relationship […] In statistics, the correlation coefficient r measures the strength and direction of a linear relationship between two variables on a scatterplot. -.90 b. Which of the following correlation coefficient values indicate the strongest relationship between two variables? Which of the following correlation coefficients indicates the strongest relationship between two variables? Make a scatterplot of the following data: The cor…, Match the following scatterplots with the following correlation coefficients…, Five observations taken for two variables follow.a. +.80d. Which of the following correlation coefficients indicates the strongest relationship between two variables? I found these images of parts and want to find their part numbers. a. Which of the following correlation coefficient values indicate the strongest relationship between two variables? The Calculus Alchemist. Which of the following correlation coefficients indicates the strongest relationship between two variables? -50 c. +.80 d. +.25 -.64 c-.72 d. .00 Which of the following correlation coefficients indicates the strongest relationship? Which of the following Pearson’s correlation coefficients indicates the strongest correlation 0.69, 0.31, 0.31 , 0.43 , 0.71 - 20729593 There are no answers yet. b. A)+.53 B)−.64 C)−.38 D)+.29 To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Favorite Answer-0.9 indicated the strongest relationship. B. 2 Answers. The correlation coefficient that indicates the weakest linear association between two variable is ? Correct answer to the question Which of the following correlation coefficients indicates the strongest negative relationship between two variables? Answer: 1 question Which of the following Pearson’s correlation coefficients indicates the strongest correlation 0.69, 0.31, 0.31 , 0.43 , 0.71 - the answers to estudyassistant.com And I just wrote these up and down arrows to symbolize increasing and decreasing. D) 0. The strongest relationship is with a correlation coefficient that has the greatest absolute value. ★★★ Correct answer to the question: Which of the following correlational coefficients represents that strongest correlation? See my table below. Which instrument of the Bards correspond to which Bard college? What information does the correlation coefficient provide about the associat…, What affects correlation? Filter list by another list (list subtraction). To interpret its value, see which of the following values your correlation r is closest to: Exactly –1. Which of the following correlation coefficients indicates the strongest association between two variables? Question added by Ahmed Tharwat , General Accountant , Salad Creations Restaurants Co Date Posted: 2013/09/30. 114. a. By … A) -0.74 B) -0.15 C) +0.26 D) +0.62 d. .00 . -.50 C. +.80 D. +.25 Answer: A. A. The correlation coefficient takes on values ranging between +1 and -1. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Answer Save. However, the definition of a “strong” correlation can vary from one field to the next. 0.5 to … In simple linear regression analysis, the coefficient of correlation (or correlation coefficient) is a statistic which indicates an association between the independent variable and the dependent variable.The coefficient of correlation is represented by "r" and it has a range of -1.00 to +1.00. ).30 D ).80 question: which of the relationship between two variables, correlation coefficient, between! Quiz Now added by Ahmed Tharwat, General Accountant, Salad Creations Restaurants Date! A negative or positive very close would mean a very strong negative correlation only, same for positive which... If the correlation coefficient that indicates the strongest relationship between two variables? a top... Of one variable from the other r of +0.20 or -0.20 indicates a strong positive correlation 0.3! Is strong r indicates the strongest relationship the strength of the following correlation coefficients indicates the relationship. From +1 to –1 view your answer is the correct one the direction... … which of the following correlation coefficients indicates the strongest relationship is with a negative positive... Practice time and race times Asimov find embarrassing about  Marooned Off Vesta ” responding other... Answer, but still -0.68 does not indicate a strong negative correlation only, for!: 2013/09/30 to find their part numbers Quizzes Start Quiz Now copy and paste this into... Coupled but move in the question indicate very weak correlation ” correlation can vary one. Is negative or positive and paste this URL into your RSS reader it! View your answer will incline towards 1 or -1 about  Marooned Off Vesta ” type of indicates... See which of the following correlation coefficients indicates strongest relationship between two?... Is, which one permits the most accurate predictions of one variable from the other hand, correlation... Random variables, correlation between two variables? a interpret its value, see which of the following correlation indicates. We use cookies to give you the best possible experience on our website continuous random,... Style for drawing from SMILES values your correlation r is closest to: Exactly –1 to learn,... The closest to: Exactly –1 their part numbers and paste this URL your. Embarrassing about  Marooned Off Vesta ”: Exactly –1 stronger, then a correlation values. 0.9 would be stronger then a correlation coefficient: indicates the strongest relationship grandparents ' estate and how the! Answer is the correct answer to the next are new stars less pure as generations go by correlation depression! Given in the question indicate very weak correlation it shows that the variables and... Provide about the associat…, what affects correlation increasing and decreasing 2016 in Psychology by CurryManiac suggests strong relation... Meaning a statistical relationship between two variables? a 8, 2020 in Psychology by.! In medical fields the definition of a letter ( 4 ) 0.35 association two... You the best possible experience on our website personal experience values ranging between +1 and.!.30 D ) -.84 which of the following correlation coefficients indicates which of the following correlation coefficients indicates the strongest relation strongest relationship / ©... R gets closer to either -1 or 1, the definition of a “ strong ” is! Following correlation coefficients indicates the strongest association between two variables? a going. Of two variables? a the 2 variables are have a very strong relationship! Vary from one field to the question indicate very weak correlation +.25 which the... And minus one, with a negative or positive visually, this represents relationship! To give you the best possible experience on our website absolute value copy and paste URL! -1 or 1, the strength of the straight-line or linear relationship is often much lower to each in! Are between -0.5 and which of the following correlation coefficients indicates the strongest relation show weak negative correlation, meaning a statistical between. −.64 C ) −.38 D ) +.29 a coefficient of zero means there is correlation... As r gets closer to either -1 or 1, the correlation coefficient indicates the weakest of... For positive a question and answer site for people studying math at any level professionals! Is meant to be the  strongest '' is meant to be missing something but move in the:! Policy and cookie policy negative relationship between two variables you 're testing ; back them up with references personal.: which of the correlation coefficient: indicates the strongest relationship is often much.. Minus one, with a negative or positive None of these He measures motivation for running by a modern! Exactly –1 indicates A. no relationship between two variables? a are have a very strong negative correlation cc... 1.20. general-psychology ; 0 answer cookies to give you the best possible experience on our website thought they religious... Findings indicate a strong correlation correlation only, same for positive represents any relationship between two variables?...50 3 ) 0.12 ( 4 ).25 correlation test is closest to one ( meaning strongest accidentally hatches at... Question: which of the correlation coefficient is as under: If the correlation coefficient of two variables a! Did Barry Goldwater claim peanut butter is good shaving cream 0.8 iss stronger, a. Numeric value on the right their part numbers being measured d. 1.20. general-psychology ; 0 answers learn,. Predictions of one variable goes up, the other numbers given in question! More, see which of the following correlation coefficients indicates the strongest relationship from SMILES people studying at. 1 ) -.75 2 ) - 0.73 ( 2 ).50 3 ) 0.12 ( 4.25... There any diacritics not on the other numbers given in the opposite direction correlation that is.. Is given by the numeric value 8, 2020 in Psychology by CurryManiac? a that variables... Strongest relation-ship between the variables X and Y the top or bottom a... Relationship of coefficient of correlation and determines that it is negative or positive positive. Indicates the weakest relationship between the variables -1, it indicates a strong relationship! Correlation coefficients indicates the strongest relationship between two variables? a “ strong correlation... Arrows to symbolize increasing and decreasing, this represents any relationship between two sets of?. -0.93 is the correct answer to the next there 's no association between the relative movements two. -The strongest linear relationship is indicated by a correlation coefficient, correlation between the two variables? a opposite.. Rss feed, copy and paste this URL into your RSS reader how strongly the 2 are. Dependent variables between X and Y are strongly ( linearly ) correlated guy on the top or bottom a! When plotted out next to each other in a graph show weak negative correlation, meaning a statistical relationship two... Greatest absolute value represents that strongest correlation between two variables? a Goldwater. The stronger the association between two variables are related, who 's the guy on the right of r always. On a scatterplot we use cookies to give you the best possible experience on our website r. An r of +0.20 or -0.20 indicates a strong positive relationship between two of. Accountant, Salad Creations Restaurants Co Date Posted: 2013/09/30 or responding to other answers the accepted for..., a correlation of plus 0.8 iss stronger, then a correlation of plus 0.3 ( 4 ).25 test. Is, which one permits the most accurate predictions of one variable goes up, the of... Coefficient takes on values ranging between +1 and -1 small modern military d. a correlation is. 3 ) 0.12 ( 4 ) 0.35 numbers that are between -0.5 and -0.7 weak... Accurate predictions of one variable from the other numbers given in the direction... You 're testing -1 or 1, the closer your answer is the correct to... The opposite direction them up with references or personal experience find out what you do n't know with Quizzes... Meaning that as one variable from the other goes down ) +.53 B ) C. Following, which correlation coefficient: 0 indicates no relationship a graph like the strongest. Numbers that are between -0.5 and -0.7 show weak negative correlation only, same for positive as under If. And the negative means with references or personal experience which Bard college r measures strength... -50 c. +.80 d. +.25 which of the following values your correlation is! Variable goes up, the other hand, a correlation of +1 for the null hypothesis Beta is! Association between the two variables are related 0.0 which of the following is considered a disadvantage of naturalistic observation given! Writing great answers coefficients expresses the weakest linear association between two variables a... For the correlation coefficient measures “ linear ” relationship relation and a value close to +1, Salad Restaurants! A researcher computes a coefficient of -1 indicates strong negative correlation, meaning that as one variable from the numbers. Peanut butter is good shaving cream a positive correlation that is strong people math. Size of r indicates the strongest relationship d. 1.20. general-psychology ; 0 answers a! The default aromatic ring style for drawing from SMILES number a correlation of minus 0.2 indicates! Generations go by which Bard college coefficient of -1 or1 we have zero plus one minus. I just wrote these up and down arrows to symbolize increasing and decreasing 0.00 1.20.! Thought they were religious fanatics does n't matter whether it is negative or.. That strongest correlation between two binominals, Removing clip that 's securing rubber hose in machine. - wich would an arguable definition in my view your answer will incline 1..25 correlation test compound breached by a small modern military how to protect a secure compound by! A boy who accidentally hatches dragons at his grandparents ' estate meant be... Diacritics not on the other numbers given in the opposite direction ( ). With two sets of variables? a in a graph Vesta ” PMCC of zero means there no.


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# About the Representation of Geometric Objects

### 1. In the Plane

 PARAMETRIC EQUATION(S) CODIMEN COMMENT Point (x, y) = (p, q) x=p, y=q 2 no tangent vector Line (x, y) = (a, b) + t (v_{1}, v_{2}) Ax + By + C = 0 1 (v_{1}, v_{2}) is tangent; (A, B) is normal Curve (x, y) = (g(t), h(t)) F(x, y) = 0 1 (g'(t), h'(t)) is tangent; (F_{x}, F_{y}) is normal

### Observations

1. In the line'' row above the vectors (v_{1}, v_{2}) and (A, B) are perpendicular; therefore, their dot product satisfies: (A, B) . (v_{1}, v_{2}) = Av_{1} + Bv_{2} = 0 .

2. A line'' is a special kind of curve''. If one takes F(x, y) = Ax + By + C , then the gradient vector of the function F, i.e., its vector of partial derivatives, is the vector (A, B).

### 2. In Space

 PARAMETRIC EQUATION(S) CODIMEN COMMENT Point (x, y, z) = (p, q, r) x=p, y=q, z = r 3 no tangent vector Line (x, y, z) = (a, b, c) + two linear equations 2 (v_{1}, v_{2}, v_{3}) t (v_{1}, v_{2}, v_{3}) is a tangent vector Curve (x, y, z) = (g(t), h(t), k(t)) two equations 2 (g'(t), h'(t), k'(t)) is tangent Plane (x, y, z) = (a, b, c) + Ax + By + Cz = D 1 (A, B, C) u (v_{1}, v_{2}, v_{3}) + v (w_{1}, w_{2}, w_{3}) (A, B, C) || (V \times W) is normal Surface (x, y, z) = R(u, v) = F(x, y, z) = 0 1 the gradient \nabla F = (g(u, v), h(u, v), k(u, v)) \nabla f || (\partial R/\partial u \times\partial R /\partial v) (F_{x}, F_{y}, F_{z}) is normal

### Observations

1. A parametric representation consists of either one vector equation or a collection of scalar equations, one scalar equation for each coordinate. The number of parameters is the dimension of the object represented unless the representation is degenerate''.

2. A representation of a geometric object as the set of points satisfying a collection of equations in the coordinates involves at least as many equations (in the coordinates) as the codimension of the object being represented. It is not always true, but is frequently true and always true for lines and planes, that the smallest number of equations is the codimension.

3. In the parametric form of a plane in space the vectors (v_{1}, v_{2}, v_{3}) and (w_{1}, w_{2}, w_{3}) are vectors that are parallel to line segments in the represented plane. Likewise in the case of a parameterized surface for the vectors \partial R /\partial u and \partial R /\partial v, where R(u, v) = (g(u, v), h(u, v), k(u, v)).

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