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Calculus 2 : Parametric Form

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Example Questions

Example Question #45 : Parametric

Convert the following equation from parametric to rectangular form:

$x=e^t, y=2\sin(t)$

Possible Answers:

$y=2\sin(\ln(x))$

$y=-2\sin(\ln(x))$

$y=2\sin(e^t)$

y=x

Correct answer:

$y=2\sin(\ln(x))$

Explanation:

To convert from parametric to rectangular form, eliminate the parameter (t) from one of the equations:

$t=\ln(x)$

Now plug this into the equation for y to get our final answer:

$y=2\sin(\ln(x))$

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Example Question #46 : Parametric

Given x=3t-6 and y=7t+9 , what is in terms of (rectangular form)?

Possible Answers:

$y=23x-\frac{7}{3}$

None of the above

$y=23x+\frac{7}{3}$

$y=\frac{7}{3}x+23$

$y=\frac{7}{3}x-23$

Correct answer:

$y=\frac{7}{3}x+23$

Explanation:

Given x=3t-6 and y=7t+9 , let's solve both equations for :

$x=3t-6\rightarrow t=\frac{x+6}{3}$

$y=7t+9\rightarrow t=\frac{y-9}{7}$

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

$\frac{y-9}{7}=\frac{x+6}{3}$

$y-9=\frac{7}{3}(x+6)$

$y-9=\frac{7}{3}x+14$

$y=\frac{7}{3}x+23$

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Example Question #47 : Parametric, Polar, And Vector

Given x=2t+11 and y=11t+2 , what is in terms of (rectangular form)?

Possible Answers:

$y=\frac{11}{2}(x-11)+2$

$y=\frac{11}{2}(x-11)-2$

$y=\frac{11}{2}(x+11)+2$

None of the above

$y=\frac{11}{2}(x+11)-2$

Correct answer:

$y=\frac{11}{2}(x-11)+2$

Explanation:

Given x=2t+11 and y=11t+2 , let's solve both equations for :

$x=2t+11\rightarrow t=\frac{x-11}{2}$

$y=11t+2\rightarrow t=\frac{y-2}{11}$

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

$\frac{y-2}{11}=\frac{x-11}{2}$

$y-2=\frac{11}{2}(x-11)$

$y=\frac{11}{2}(x-11)+2$

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Example Question #48 : Parametric, Polar, And Vector

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

Possible Answers:

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

None of the above

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

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

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

Correct answer:

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

Explanation:

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

$x=t+3\rightarrow t=x-3$

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

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

2(y+3)=x-3

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

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

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Example Question #51 : Parametric, Polar, And Vector

Given x=10-t and y=5t+3 , what is in terms of (rectangular form)?

Possible Answers:

y=5-53x

None of the above

y=5+53x

y=53+5x

y=53-5x

Correct answer:

y=53-5x

Explanation:

Given x=10-t and y=5t+3 , let's solve both equations for :

$x=10-t\rightarrow t=10-x$

$y=5t+3\rightarrow t=\frac{y-3}{5}$

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

$\frac{y-3}{5}=10-x$

y-3=5(10-x)

y=5(10-x)+3

y=50-5x+3

y=53-5x

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Example Question #52 : Parametric, Polar, And Vector

Given x=8-3t and y=7-4t , what is in terms of (rectangular form)?

Possible Answers:

$y=\frac{4}{3}(x-8)+7$

None of the above

$y=\frac{4}{3}(x+8)-7$

$y=\frac{4}{3}(x-8)-7$

$y=\frac{4}{3}(x+8)+7$

Correct answer:

$y=\frac{4}{3}(x-8)+7$

Explanation:

Given x=8-3t and y=7-4t , let's solve both equations for :

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

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

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

$-\frac{y-7}{4}=-\frac{x-8}{3}$

$-(y-7)=-4\left(\frac{x-8}{3}\right)$

$y-7=\frac{4}{3}(x-8)$

$y=\frac{4}{3}(x-8)+7$

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Example Question #53 : Parametric, Polar, And Vector

Given x=7t-4 and y=2t+9 , what is in terms of (rectangular form)?

Possible Answers:

$y=\frac{2}{7}(x-4})+9$

$y=\frac{2}{7}(x+4})-9$

None of the above

$y=\frac{2}{7}(x+4})+9$

$y=\frac{2}{7}(x-4})-9$

Correct answer:

$y=\frac{2}{7}(x+4})+9$

Explanation:

Given x=7t-4 and y=2t+9 , let's solve both equations for :

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

$y=2t+9\rightarrow t=\frac{y-9}{2}$

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

$\frac{y-9}{2}=\frac{x+4}{7}$

$y-9=2\left(\frac{x+4}{7}\right)$

$y=\frac{2}{7}(x+4})+9$

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Example Question #54 : Parametric, Polar, And Vector

Given x=2t-11 and y=-t+9 , what is in terms of (rectangular form)?

Possible Answers:

$y=9-\frac{x+11}{2}$

$y=9+\frac{x-11}{2}$

$y=9+\frac{x+11}{2}$

$y=9-\frac{x-11}{2}$

None of the above

Correct answer:

$y=9-\frac{x+11}{2}$

Explanation:

Given x=2t-11 and y=-t+9 , let's solve both equations for :

$x=2t-11\rightarrow t=\frac{x+11}{2}$

$y=-t+9\rightarrow t=9-y$

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

$9-y=\frac{x+11}{2}$

$y=9-\frac{x+11}{2}$

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Example Question #55 : Parametric, Polar, And Vector

Find $\frac{dy}{dx}$ when $x=t^{2}-5$ and $y=t^{3}-5t^{2}-10t-50$ .

Possible Answers:

$\frac{dy}{dx}=\frac{3t^{2}-10t-10}{t^{2}-5}$

$\frac{dy}{dx}=\frac{3t^{2}-10t-10}{2t}$

$\frac{dy}{dx}=2t$

$\frac{dy}{dx}=t^{3}-5t^{2}-10t-50$

$\frac{dy}{dx}=3t^{2}-10t-10$

Correct answer:

$\frac{dy}{dx}=\frac{3t^{2}-10t-10}{2t}$

Explanation:

If x=x(t) and y=y(t) , then we can use the chain rule to define $\frac{dy}{dx}$ as

$\frac{dy}{dx}=\frac{dy}{dt}/\frac{dx}{dt}$ .

We use the power rules

$\frac{d}{dt}(t^{c})=ct^{c-1}$ and $\frac{d}{dt}(t)=1$ where is a constant, the constant rule

$\frac{d}{dt}(c)=0$ where is a constant, and the additive property of derviatives

$\frac{d}{dt}(a+b)=\frac{d}{dt}(a)+\frac{d}{dt}(b)$ .

In this case

$\frac{dy}{dt}=\frac{d}{dt}(t^{3}-5t^{2}-10t-50)=3t^{2}-10t-10$

and

$\frac{dx}{dt}=\frac{d}{dt}(t^{2}-5)=2t$ ,

therefore

$\frac{dy}{dx}=\frac{dy}{dt}/\frac{dx}{dt}=\frac{3t^{2}-10t-10}{2t}$

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Example Question #56 : Parametric, Polar, And Vector

Given x=12-t and y=2t+5 , what is in terms of (rectangular form)?

Possible Answers:

y=29x+2

y=29+2x

None of the above

y=29x-2

y=29-2x

Correct answer:

y=29-2x

Explanation:

Given x=12-t and y=2t+5 , let's solve both equations for :

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

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

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

$\frac{y-5}{2}=12-x$

y-5=2(12-x)

y=2(12-x)+5

y=24-2x+5

y=29-2x

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Calculus 2 : Parametric Form

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #45 : Parametric

Example Question #46 : Parametric

Example Question #47 : Parametric, Polar, And Vector

Example Question #48 : Parametric, Polar, And Vector

Example Question #51 : Parametric, Polar, And Vector

Example Question #52 : Parametric, Polar, And Vector

Example Question #53 : Parametric, Polar, And Vector

Example Question #54 : Parametric, Polar, And Vector

Example Question #55 : Parametric, Polar, And Vector

Example Question #56 : Parametric, Polar, And Vector

All Calculus 2 Resources

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