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Calculus 2 : Parametric Form

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Example Questions

Example Question #61 : Parametric, Polar, And Vector

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

Possible Answers:

y=-6x

$y=-\frac{1}{6}x$

None of the above

$y=\frac{1}{6}x$

y=6x

Correct answer:

y=-6x

Explanation:

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

$x=-\frac{1}{2}t\rightarrow t=-2x$

$y=3t\rightarrow t=\frac{y}{3}$

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

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

y=-6x

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Example Question #62 : Parametric Form

Given x=6t-7 and y=8t+9 , what is in terms of (rectangular form)?

Possible Answers:

$y=\frac{4}{3}(x+7)+9$

$y=\frac{8}{6}(x-7)-9$

$y=\frac{8}{6}(x-7)+9$

None of the above

$y=\frac{8}{6}(x+7)-9$

Correct answer:

$y=\frac{4}{3}(x+7)+9$

Explanation:

Given x=6t-7 and y=8t+9 , let's solve both equations for :

$x=6t-7\rightarrow t=\frac{x+7}{6}$

$y=8t+9\rightarrow t=\frac{y-9}{8}$

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

$\frac{y-9}{8}=\frac{x+7}{6}$

$y-9=8\left(\frac{x+7}{6}\right)$

$y=\frac{4}{3}(x+7)+9$

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Example Question #65 : Parametric

When $x=\frac{3}{4}t-5$ and y=7t+4 , what is in terms of (rectangular form)?

Possible Answers:

$y=\frac{28}{3}(x-5)+4$

$y=\frac{28}{3}(x+5)-4$

$y=\frac{28}{3}(x-5)-4$

$y=\frac{28}{3}(x+5)+4$

None of the above

Correct answer:

$y=\frac{28}{3}(x+5)+4$

Explanation:

Given $x=\frac{3}{4}t-5$ and y=7t+4 , wet's solve both equations for :

$x=\frac{3}{4}t-5\rightarrow t=\frac{4}{3}(x+5)$

$y=7t+4\rightarrow t=\frac{y-4}{7}$

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

$y-4=(7)(\frac{4}{3})(x+5)$

$y=(7)(\frac{4}{3})(x+5)+4$

$y=\frac{28}{3}(x+5)+4$

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Example Question #64 : Parametric

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

Possible Answers:

$y=3x+\frac{1}{2}$

$y=-3x-\frac{1}{2}$

$y=3x-\frac{1}{2}$

$y=-3x+\frac{1}{2}$

None of the above

Correct answer:

$y=-3x-\frac{1}{2}$

Explanation:

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

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

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

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

$\frac{y+2}{3}=\frac{1}{2}-x$

$y+2=(3)(\frac{1}{2}-x)$

$y=(3)(\frac{1}{2}-x)-2$

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

$y=\frac{3}{2}-3x-\frac{4}{2}$

$y=-3x-\frac{1}{2}$

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Example Question #71 : Parametric, Polar, And Vector

Convert the following parametric equation to rectangular form:

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

Possible Answers:

x=e^y

x=2e^y

x=y

x=0

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 #71 : Parametric, Polar, And Vector

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

Possible Answers:

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

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

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

None of the above

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

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 #71 : Parametric, Polar, And Vector

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

Possible Answers:

y=48-12x

y=48x+12

None of the above

y=48x-12

y=48+12x

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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Calculus 2 : Parametric Form

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #61 : Parametric, Polar, And Vector

Example Question #62 : Parametric Form

Example Question #65 : Parametric

Example Question #64 : Parametric

Example Question #71 : Parametric, Polar, And Vector

Example Question #66 : Parametric Form

Example Question #72 : Parametric

Example Question #71 : Parametric, Polar, And Vector

Example Question #71 : Parametric, Polar, And Vector

Example Question #64 : Parametric Form

All Calculus 2 Resources

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