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Calculus 2 : Parametric, Polar, and Vector

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Example Questions

Example Question #62 : Parametric Form

Convert the following parametric equation to rectangular form:

$x=2t, y=\ln(t)$

Possible Answers:

x=2e^y

x=e^y

x=0

x=y

Correct answer:

x=2e^y

Explanation:

To convert from parametric to rectangular coordinates, we must eliminate the parameter by finding t in terms of x or y:

We will start by taking the exponential of both sides of the equation y=ln(t) . Recall that $e^{ln(t)}=t$ .

Therefore we get,

t=e^y .

Now, replace t with the above term in the equation for x:

x=2e^y

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Example Question #66 : Parametric Form

When $x=-\frac{1}{3}t+8$ and $y=\frac{2}{3}t+7$ , what is in terms of (rectangular form)?

Possible Answers:

y=-2x-9

y=-2x+23

y=2x-9

y=9x-2

y=-9x-2

Correct answer:

y=-2x+23

Explanation:

Given $x=-\frac{1}{3}t+8$ and $y=\frac{2}{3}t+7$ , wet's solve both equations for :

$x=-\frac{1}{3}t+8\rightarrow t=-3(x-8)$

$y=\frac{2}{3}t+7\rightarrow t=\frac{3}{2}(y-7)$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{3}{2}(y-7)=-3(x-8)$

$y-7=\frac{2}{3}(-3)(x-8)$

y-7=-2(x-8)

y-7=-2x+16

y=-2x+23

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Example Question #72 : Parametric

Given x=7-t and y=2t+5 , what is in terms of ?

Possible Answers:

y=19x+2

y=19+2x

y=19x-2

y=19-2x

None of the above

Correct answer:

y=19-2x

Explanation:

Given x=7-t and y=2t+5 , let's solve both equations for :

$x=7-t\rightarrow t=7-x$

$y=2t+5\rightarrow t=\frac{y-5}{2}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y-5}{2}=7-x$

y-5=2(7-x)

y=2(7-x)+5

y=14-2x+5

y=19-2x

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Example Question #68 : Parametric Form

Given $x=-\frac{3}{2}t$ and y=7+2t , what is in terms of ?

Possible Answers:

$y=7-\frac{4}{3}x$

None of the above

$y=7x-\frac{4}{3}$

$y=7x+\frac{4}{3}$

$y=7+\frac{4}{3}x$

Correct answer:

$y=7-\frac{4}{3}x$

Explanation:

Given $x=-\frac{3}{2}t$ and y=7+2t , let's solve both equations for :

$x=-\frac{3}{2}t\rightarrow t=-\frac{2}{3}x$

$y=7+2t\rightarrow t=\frac{y-7}{2}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y-7}{2}=-\frac{2}{3}x$

$y-7=-\frac{4}{3}x$

$y=7-\frac{4}{3}x$

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Example Question #63 : Parametric Form

Given $x=4+\frac{1}{2}t$ and y=-6t , what is in terms of ?

Possible Answers:

y=48-12x

y=48x-12

y=48x+12

y=48+12x

None of the above

Correct answer:

y=48-12x

Explanation:

Given $x=4+\frac{1}{2}t$ and y=-6t , wlet's solve both equations for :

$x=4+\frac{1}{2}t\rightarrow t=2(x-4)$

$y=-6t\rightarrow t=-\frac{1}{6}y$

Since both equations equal , let's set them equal to each other and solve for :

$-\frac{1}{6}y=2(x-4)$

y=-12(x-4)

y=48-12x

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Example Question #64 : Parametric Form

Given x=-3t+8 and y=4t+1 , what is in terms of (rectangular form)?

Possible Answers:

$y=-\frac{4}{3}(x+8)-1$

$y=\frac{4}{3}(x-8)-1$

$y=-\frac{4}{3}(x-8)+1$

$y=-\frac{4}{3}(x-8)-1$

$y=\frac{4}{3}(x-8)+1$

Correct answer:

$y=-\frac{4}{3}(x-8)+1$

Explanation:

Given x=-3t+8 and y=4t+1 , let's solve both equations for :

$x=-3t+8\rightarrow t=-\frac{x-8}{3}$

$y=4t+1\rightarrow t=\frac{y-1}{4}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y-1}{4}=-\frac{x-8}{3}$

$y-1=-4\left(\frac{x-8}{3}\right)$

$y=-\frac{4}{3}(x-8)+1$

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Example Question #71 : Parametric, Polar, And Vector

Given x=t+12 and y=6-t , what is in terms of (rectangular form)?

Possible Answers:

y=18+x

None of the above

y=1+18x

y=18-x

y=1-18x

Correct answer:

y=18-x

Explanation:

Given x=t+12 and y=6-t , let's solve both equations for :

$x=t+12\rightarrow t=x-12$

$y=6-t\rightarrow t=6-y$

Since both equations equal , let's set them equal to each other and solve for :

6-y=x-12

-y=x-18

y=18-x

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Example Question #72 : Parametric, Polar, And Vector

Given x=2t-19 and y=4-8t , what is in terms of (rectangular form)?

Possible Answers:

y=4x-72

y=-4x-72

None of the above

y=-72x-4

y=72x-4

Correct answer:

y=-4x-72

Explanation:

Given x=2t-19 and y=4-8t , let's solve both equations for :

$x=2t-19\rightarrow t=\frac{x+19}{2}$

$y=4-8t\rightarrow t=-\frac{y-4}{8}$

Since both equations equal , let's set them equal to each other and solve for :

$-\frac{y-4}{8}=\frac{x+19}{2}$

y-4=-4(x+19)

y=-4(x+19)+4

y=-4x-76+4

y=-4x-72

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Example Question #73 : Parametric, Polar, And Vector

Given $x=2t^{2}+3$ and y=5t+8 , what is in terms of (rectangular form)?

Possible Answers:

$y=5\left(\pm \sqrt{\frac{x-3}{2}}\right)+8$

$y=5\sqrt{\frac{x+3}{2}}-8$

$y=5\sqrt{\frac{x+3}{2}}+8$

$y=5\sqrt{\frac{x-3}{2}}-8$

None of the above

Correct answer:

$y=5\left(\pm \sqrt{\frac{x-3}{2}}\right)+8$

Explanation:

Given $x=2t^{2}+3$ and y=5t+8 , let's solve both equations for :

$x=2t^{2}+3\rightarrow t=\pm \sqrt{\frac{x-3}{2}}$

$y=5t+8\rightarrow t=\frac{y-8}{5}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y-8}{5}=\pm \sqrt{\frac{x-3}{2}}$

$y-8=5\left(\pm\sqrt{\frac{x-3}{2}}\right)$

$y=5\left(\pm \sqrt{\frac{x-3}{2}}\right)+8$

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Example Question #74 : Parametric, Polar, And Vector

Find dy/dx at the point corresponding to the given value of the parameter without eliminating the parameter:

x=3t^2-6, y=9t+5, t=3

Possible Answers:

$\frac{32}{21}$

$\frac{1}{2}$

$\frac{1}{9}$

$\frac{21}{32}$

Correct answer:

$\frac{1}{2}$

Explanation:

The formula for dy/dx for parametric equations is given as:

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ From the problem statement:

$\frac{dy}{dt}=9,\frac{dx}{dt}=6t$

If we plug in t=3, into the above equations:

$\frac{dy}{dx}=\frac{9}{6(3)}=\frac{9}{18}=\frac{1}{2}$

This is one of the answer choices.

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Calculus 2 : Parametric, Polar, and Vector

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #62 : Parametric Form

Example Question #66 : Parametric Form

Example Question #72 : Parametric

Example Question #68 : Parametric Form

Example Question #63 : Parametric Form

Example Question #64 : Parametric Form

Example Question #71 : Parametric, Polar, And Vector

Example Question #72 : Parametric, Polar, And Vector

Example Question #73 : Parametric, Polar, And Vector

Example Question #74 : Parametric, Polar, And Vector

Find dy/dx at the point corresponding to the given value of the parameter without eliminating the parameter:

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