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Calculus 2 : Parametric Form

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Example Question #81 :Parametric Form

Which of the following set of parametric equations parametrizes a section of parabola

y=1-x^2 ? (Assume $0\leq t\leq 2\pi$ )

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

We can see that the parametric equations

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

describe a section of the parabola

y=1-x^2

because if do some manipulations of the parametric equations, we get

$x(t)^2+y(t)=\cos^2 t+\sin^2 t=1$

$\Rightarrow x(t)^2+y(t)=1$

So then we get

y(t)=1-x(t)^2

which describes part of the parabola

y=1-x^2 .

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Example Question #82 :Parametric Form

Which of the following parametric equations parametrizes an ellipse? (Assume that $0\leq t\leq 2\pi$ )

Possible Answers:

$x(t)=\sqrt{2}\cos t, y(t)=\sqrt{2}\cos t$

$x(t)=2\cos t, y(t)=3\sin t$

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

$x(t)=\sqrt{2}\cos t, y(t)=\sqrt{3}\cos t$

Correct answer:

$x(t)=2\cos t, y(t)=3\sin t$

Explanation:

The parametric equations

$x(t)=2\cos t, y(t)=3\sin t$

describe an ellipse because we have

$\frac{x(t)^2}{4}+\frac{y(t)^2}{9}=\cos^2 t+\sin^2 t=1$

which means

$\frac{x(t)^2}{4}+\frac{y(t)^2}{9}=1$

which is the equation for an ellipse.

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Example Question #83 :Parametric Form

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

Possible Answers:

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

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

$y=3(\pm\sqrt{x+5})+8$

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

None of the above

Correct answer:

$y=3(\pm\sqrt{x+5})+8$

Explanation:

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

$x=t^{2}-5 \rightarrow t=\pm\sqrt{x+5}$

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

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

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

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

$y=3(\pm\sqrt{x+5})+8$

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Example Question #84 :Parametric Form

Given x=4+10t and y=2-t , what isin terms of(rectangular form)?

Possible Answers:

$y=\frac{x-4}{10}-2$

$y=\frac{x+4}{10}-2$

None of the above

$y=2-\frac{x-4}{10}$

$y=\frac{x+4}{10}+2$

Correct answer:

$y=2-\frac{x-4}{10}$

Explanation:

Given x=4+10t and y=2-t , let's solve both equations for:

$x=4+10t\rightarrow t=\frac{x-4}{10}$

$y=2-t\rightarrow t=2-y$

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

$2-y=\frac{x-4}{10}$

$y=2-\frac{x-4}{10}$

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Example Question #85 :Parametric Form

Given x=7-9t and $y=3t^{2}$ , what isin terms of(rectangular form)?

Possible Answers:

$y=-3\left(\frac{7+x}{9}\right)^{2}$

None of the above

$y=-3\left(\frac{7-x}{9}\right)^{2}$

$y=3\left(\frac{7-x}{9}\right)^{2}$

$y=3\left(\frac{7+x}{9}\right)^{2}$

Correct answer:

$y=3\left(\frac{7-x}{9}\right)^{2}$

Explanation:

Given x=7-9t and $y=3t^{2}$ , let's solve both equations for:

$x=7-9t\rightarrow t=\frac{7-x}{9}$

$y=3t^{2}\rightarrow t=\sqrt{\frac{y}{3}}$

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

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

$\frac{y}{3}=\left(\frac{7-x}{9}\right)^{2}$

$y=3\left(\frac{7-x}{9}\right)^{2}$

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Example Question #86 :Parametric Form

Given $x=6t^{2}-14$ and y=5t+1 , what isin terms of?

Possible Answers:

$y=5\sqrt{\frac{x-14}{6}}-1$

None of the above

$y=5\sqrt{\frac{x+14}{6}}-1$

$y=5\left(\pm\sqrt{\frac{x+14}{6}}\right)+1$

$y=5\sqrt{\frac{x-14}{6}}+1$

Correct answer:

$y=5\left(\pm\sqrt{\frac{x+14}{6}}\right)+1$

Explanation:

Given $x=6t^{2}-14$ and y=5t+1 , let's solve both equations for:

$x=6t^{2}-14\rightarrow t=\pm\sqrt{\frac{x+14}{6}}$

$y=5t+1\rightarrow t=\frac{y-1}{5}$

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

$\frac{y-1}{5}=\pm\sqrt{\frac{x+14}{6}}$

$y-1=5\left(\pm\sqrt{\frac{x+14}{6}}\right)$

$y=5\left(\pm\sqrt{\frac{x+14}{6}}\right)+1$

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Example Question #87 :Parametric Form

Given x=2t-7 and $y=8-t^{2}$ , what isin terms of?

Possible Answers:

$y=8-\left(\frac{x-7}{2}\right)^2$

$y=8-\left(\frac{x+7}{2}\right)^2$

$y=8+\left(\frac{x+7}{2}\right)^2$

None of the above

$y=8+\left(\frac{x-7}{2}\right)^2$

Correct answer:

$y=8-\left(\frac{x+7}{2}\right)^2$

Explanation:

Given x=2t-7 and $y=8-t^{2}$ , let's solve both equations for:

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

$y=8-t^{2}\rightarrow t=\sqrt{8-y}$

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

$\sqrt{8-y}=\frac{x+7}{2}$

$8-y=\left(\frac{x+7}{2}\right)^2$

$y=8-\left(\frac{x+7}{2}\right)^2$

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Example Question #88 :Parametric Form

Given the parametric equations

x(t)=t+t^2, y(t)=t^3

what is $\frac{dy}{dx}$ ?

Possible Answers:

$\frac{dy}{dx}=\frac{3t^2}{2t+1}$

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

$\frac{dy}{dx}=\frac{2t+1}{3t^2}$

$\frac{dy}{dx}=2t+1$

Correct answer:

$\frac{dy}{dx}=\frac{3t^2}{2t+1}$

Explanation:

It is known that we can derive $\frac{dy}{dx}$ with the formula

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

So we just find $\frac{dy}{dt}, \frac{dx}{dt}$ :

In order to find these derivatives we will need to use the power rule which states,

$x^n \ dx =nx^{n-1}$ .

Applying the power rule we get the following.

$\frac{dy}{dt}=3t^2$

$\frac{dx}{dt}=2t+1$

so we have

$\frac{dy}{dx}=\frac{3t^2}{2t+1}$ .

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Example Question #89 :Parametric Form

Given the parametric equations

$x(t)=\tan t, y(t)=e^t$

what is $\frac{dy}{dx}$ ?

Possible Answers:

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

$\frac{dy}{dx}=\frac{e^t}{\sec^2 t}$

$\frac{dy}{dx}=\frac{\sec^2 t}{e^t}$

$\frac{dy}{dx}=e^t$

Correct answer:

$\frac{dy}{dx}=\frac{e^t}{\sec^2 t}$

Explanation:

It is known that we can derive $\frac{dy}{dx}$ with the formula

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

So we just find $\frac{dy}{dt}, \frac{dx}{dt}$ :

To find the derivatives we will need to use trigonometric and exponential rules.

Trigonometric Rule for tangent: tan(t)dt=sec^2(t)

Rules of Exponentials: $e^u dx=e^u\cdot \frac{du}{dx}$

Thus, applying the above rules we get the following derivatives.

$\frac{dy}{dt}=e^t$

$\frac{dx}{dt}=\sec^2 t$

so we have

$\frac{dy}{dx}=\frac{e^t}{\sec^2 t}$ .

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Example Question #90 :Parametric Form

Find the arc length of the curve:

$x=5t+3, y=4t-5, 0\leq t\leq2$

Possible Answers:

$2\sqrt{41}$

$\sqrt{41}$

Correct answer:

$2\sqrt{41}$

Explanation:

Finding the length of the curve requires simply applying the formula:

$L=\int_{a}^{b}\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dt$

Where:

a=0, b =2

Since we are also givenand, we can easily compute the derivatives of each:

$\frac{dx}{dt}=\frac{d}{dt}[5t+3]=5$

$\frac{dy}{dt}=\frac{d}{dt}[4t-5]=4$

Applying these into the above formula results in:

$L=\int_{0}^{2}\sqrt{(5)^2+(4)^2}\,dt=\int_{0}^{2}\sqrt{25+16}\,dt=\int_{0}^{2}\sqrt{41}\,dt$

$L=\sqrt{41}\int_{0}^{2}dt=\sqrt{41}\,(2-0)=2\sqrt{41}$

This is one of the answer choices.

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Calculus 2 : Parametric Form

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #81 :Parametric Form

Example Question #82 :Parametric Form

Example Question #83 :Parametric Form

Example Question #84 :Parametric Form

Example Question #85 :Parametric Form

Example Question #86 :Parametric Form

Example Question #87 :Parametric Form

Example Question #88 :Parametric Form

Example Question #89 :Parametric Form

Example Question #90 :Parametric Form

Find the arc length of the curve:

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