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Calculus 2 : Polar

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Example Questions

Example Question #1 : Derivatives Of Polar Form

Evaluate the area given the polar curve: $r=\frac{\theta}{2}$ from $[0,2\pi]$ .

Possible Answers:

$\frac{\pi^3}{3}$

$8\pi$

$\frac{\pi^3}{4}$

$\pi$

$\frac{\pi^3}{6}$

Correct answer:

$\frac{\pi^3}{3}$

Explanation:

Write the formula to find the area in between two polar equations.

$A=\frac{1}{2}\int_{a}^{b}[(\textup{Outer Radius}^2)-(\textup{Inner Radius})^2]\: d\theta$

The outer radius is $r=\frac{\theta}{2}$ .

The inner radius is r=0 .

Substitute the givens and evaluate the integral.

$A=\frac{1}{2}\int_{0}^{2\pi}\left(\frac{\theta}{2}\right)^2\: d\theta=\frac{1}{2}\int_{0}^{2\pi}\frac{\theta^2}{4}\: d\theta=\frac{1}{8}\left[\frac{\theta^3}{3}\right]_{0}^{2\pi}$

$\frac{1}{8}\left[\frac{\theta^3}{3}\right]_{0}^{2\pi}= \frac{1}{8}\left[\frac{(2\pi)^3}{3}\right] = \frac{\pi^3}{3}$

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Example Question #1 : Derivatives Of Polar Form

Find the derivative $\frac{dy}{dx}$ of the polar function $r(\theta)=3-4sin(\theta)$ .

Possible Answers:

$\frac{dy}{dx}=-4cos(\theta)$

$\frac{dy}{dx}=\frac{-4cos(\theta)sin(\theta)+(3-4sin(\theta))cos(\theta)}{-4cos^2(\theta)-(3-4sin(\theta))sin(\theta)}$

$\frac{dy}{dx}=4cos(\theta)$

$\frac{dy}{dx}=\frac{-4cos(\theta)cos(\theta)+(3-4sin(\theta))sin(\theta)}{-4cos(\theta)sin(\theta)-(3-4sin(\theta))cos(\theta)}$

Correct answer:

$\frac{dy}{dx}=\frac{-4cos(\theta)sin(\theta)+(3-4sin(\theta))cos(\theta)}{-4cos^2(\theta)-(3-4sin(\theta))sin(\theta)}$

Explanation:

The derivative of a polar function is found using the formula

$\frac{dy}{dx}=\frac{r'(\theta)sin(\theta)+r(\theta)cos(\theta)}{r'(\theta)cos(\theta)-r(\theta)sin(\theta)}$

The only unknown piece is $r'(\theta)$ . Recall that the derivative of a constant is zero, and that

$\frac{d}{d\theta}sin(\theta)=cos(\theta)$ , so

$r'(\theta)=-4cos(\theta)$

Substiting $r'(\theta)\ and\ r(\theta)$ this into the derivative formula, we find

$\frac{dy}{dx}=\frac{-4cos(\theta)sin(\theta)+(3-4sin(\theta))cos(\theta)}{-4cos(\theta)cos(\theta)-(3-4sin(\theta))sin(\theta)}$

$\frac{dy}{dx}=\frac{-4cos(\theta)sin(\theta)+(3-4sin(\theta))cos(\theta)}{-4cos^2(\theta)-(3-4sin(\theta))sin(\theta)}$

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Example Question #803 : Calculus Ii

Find the first derivative of the polar function

$r(\theta)=sin(3\theta)$ .

Possible Answers:

$\frac{dy}{dx}=3cos(3\theta)$

$\frac{dy}{dx}=\frac{3cos(3\theta)sin(\theta)+sin(3\theta)cos(\theta)}{3cos(3\theta)cos(\theta)-sin(3\theta)sin(\theta)}$

$\frac{dy}{dx}=\frac{3sin(3\theta)cos(\theta)+cos(3\theta)sin(\theta)}{3sin(3\theta)sin(\theta)-cos(3\theta)cos(\theta)}$

$\frac{dy}{dx}=\frac{sin(3\theta)sin(\theta)+3cos(3\theta)cos(\theta)}{sin(3\theta)cos(\theta)-3cos(3\theta)sin(\theta)}$

Correct answer:

$\frac{dy}{dx}=\frac{3cos(3\theta)sin(\theta)+sin(3\theta)cos(\theta)}{3cos(3\theta)cos(\theta)-sin(3\theta)sin(\theta)}$

Explanation:

In general, the dervative of a function in polar coordinates can be written as

$\frac{dy}{dx}=\frac{r'(\theta)sin(\theta)+r(\theta)cos(\theta)}{r'(\theta)cos(\theta)-r(\theta)sin(\theta)}$ .

Therefore, we need to find $r'(\theta)$ , and then substitute $r(\theta)\ and\ r'(\theta)$ into the derivative formula.

To find $r'(\theta)$ , the chain rule,

$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$ , is necessary.

We also need to know that

$\frac{d}{dx}sin(x)=cos(x)$ .

Therefore,

$r'(\theta)=3cos(3\theta)$ .

Substituting $r(\theta)\ and\ r'(\theta)$ into the derivative formula yields

$\frac{dy}{dx}=\frac{3cos(3\theta)sin(\theta)+sin(3\theta)cos(\theta)}{3cos(3\theta)cos(\theta)-sin(3\theta)sin(\theta)}$

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Example Question #3 : Derivatives Of Polar Form

Find the derivative of the following function:

$r=2\cos(\theta)-1$

Possible Answers:

$\frac{-2sin^2(\theta)+2\cos^2(\theta)-\cos(\theta)}{4\cos(\theta)\sin(\theta)-\sin(\theta)}$

$\frac{-2sin^2(\theta)+2\cos^2(\theta)-\cos(\theta)}{-4\cos(\theta)\sin(\theta)+\sin(\theta)}$

$\cos^2(\theta)-\sin^2(\theta)$

Correct answer:

$\frac{-2sin^2(\theta)+2\cos^2(\theta)-\cos(\theta)}{-4\cos(\theta)\sin(\theta)+\sin(\theta)}$

Explanation:

The formula for the derivative of a polar function is

$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{\frac{\mathrm{dr} }{\mathrm{d} \theta}\sin(\theta)+r\cos(\theta)}{\frac{\mathrm{dr} }{\mathrm{d} \theta}\cos(\theta)-r\sin(\theta)}$

First, we must find the derivative of the function given:

$\frac{\mathrm{dr} }{\mathrm{d} \theta}=-2\sin(\theta)$

Now, we plug in the derivative, as well as the original function, into the above formula to get

$\frac{-2sin^2(\theta)+2\cos^2(\theta)-\cos(\theta)}{-4\cos(\theta)\sin(\theta)+\sin(\theta)}$

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Example Question #1 : Derivatives Of Polar Form

Find the derivative of the following function:

$r=\theta^2+1$

Possible Answers:

$\frac{2\theta\sin(\theta)+(\theta^2+1)(\cos(\theta))}{2\theta\cos(\theta)+(\theta^2+1)(\sin(\theta))}$

$\frac{2\theta\sin(\theta)+(\theta^2+1)(\cos(\theta))}{2\theta\cos(\theta)-(\theta^2+1)(\sin(\theta))}$

$2\theta$

$\frac{2\theta\sin(\theta)-(\theta^2+1)(\cos(\theta))}{2\theta\cos(\theta)+(\theta^2+1)(\sin(\theta))}$

Correct answer:

$\frac{2\theta\sin(\theta)+(\theta^2+1)(\cos(\theta))}{2\theta\cos(\theta)-(\theta^2+1)(\sin(\theta))}$

Explanation:

The derivative of a polar function is given by the following:

$\frac{dy}{dx}=\frac{\frac{dr}{d\theta}\sin(\theta)+r\cos(\theta)}{\frac{dr}{d\theta}\cos(\theta)-r\sin(\theta)}$

First, we must find $\frac{dr}{d\theta}=2\theta$

The derivative was found using the following rule:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Finally, plug in the derivative we just found along with r, the function given, into the above formula:

$\frac{2\theta\sin(\theta)+(\theta^2+1)(\cos(\theta))}{2\theta\cos(\theta)-(\theta^2+1)(\sin(\theta))}$

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Example Question #2 : Derivatives Of Polar Form

Find the derivative of the following function:

$r=\cos^2(\theta)$

Possible Answers:

$-2\cos(\theta)\sin(\theta)$

$\frac{-2\cos(\theta)\sin^2(\theta)-\cos^3(\theta)}{-2\cos^2(\theta)\sin(\theta)+\sin(\theta)\cos^2(\theta)}$

$\frac{-2\cos(\theta)\sin(\theta)+\cos^3(\theta)}{-2\cos^2(\theta)\sin(\theta)-\sin(\theta)\cos^2(\theta)}$

$\frac{-2\cos(\theta)\sin^2(\theta)+\cos^3(\theta)}{-2\cos^2(\theta)\sin(\theta)-\sin(\theta)\cos^2(\theta)}$

Correct answer:

$\frac{-2\cos(\theta)\sin^2(\theta)+\cos^3(\theta)}{-2\cos^2(\theta)\sin(\theta)-\sin(\theta)\cos^2(\theta)}$

Explanation:

The derivative of a polar function is given by the following:

$\frac{dy}{dx}=\frac{\frac{dr}{d\theta}\sin(\theta)+r\cos(\theta)}{\frac{dr}{d\theta}\cos(\theta)-r\sin(\theta)}$

First, we must find $\frac{dr}{d\theta}=-2\cos(\theta)\sin(\theta)$

The derivative was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x)) \cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$

Finally, plug in the above derivative and our original function into the above formula:

$\frac{-2\cos(\theta)\sin^2(\theta)+\cos^3(\theta)}{-2\cos^2(\theta)\sin(\theta)-\sin(\theta)\cos^2(\theta)}$

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Example Question #2 : Derivatives Of Polar Form

Find the derivative of the following function:

$r=3\theta+e^\theta$

Possible Answers:

$\frac{(3+e^\theta)\sin(\theta)-(3\theta+e^\theta)\cos(\theta)}{(3+e^\theta)\cos(\theta)+(3\theta+e^\theta)\sin(\theta)}$

$\frac{(3+e^\theta)\sin(\theta)+(3\theta+e^\theta)\cos(\theta)}{(3+e^\theta)\cos(\theta)-(3\theta+e^\theta)\sin(\theta)}$

$3+e^\theta$

$\frac{(3+e^\theta)\sin(\theta)+(3\theta+e^\theta)\cos(\theta)}{(3+e^\theta)\cos(\theta)+(3\theta+e^\theta)\sin(\theta)}$

Correct answer:

$\frac{(3+e^\theta)\sin(\theta)+(3\theta+e^\theta)\cos(\theta)}{(3+e^\theta)\cos(\theta)-(3\theta+e^\theta)\sin(\theta)}$

Explanation:

The derivative of a polar function is given by the following:

$\frac{dy}{dx}=\frac{\frac{dr}{d\theta}\sin(\theta)+r\cos(\theta)}{\frac{dr}{d\theta}\cos(\theta)-r\sin(\theta)}$

First, we must find $\frac{dr}{d\theta}=3+e^\theta$

We found the derivative using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x}e^u=e^u\frac{\mathrm{du} }{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x}x^n=nx^{n-1}$

Finally, we plug in the above derivative and the original function into the above formula:

$\frac{(3+e^\theta)\sin(\theta)+(3\theta+e^\theta)\cos(\theta)}{(3+e^\theta)\cos(\theta)-(3\theta+e^\theta)\sin(\theta)}$

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Example Question #7 : Derivatives Of Polar Form

Find the derivative of the function:

$r=5\cos(2\theta)$

Possible Answers:

$\frac{-10\sin^2(2\theta)+5\cos(2\theta)^2}{-10\sin(2\theta)\cos(\theta)-5\cos(2\theta)\sin(\theta)}$

$\frac{-10\sin(2\theta)\sin(\theta)+5\cos(2\theta)\cos(\theta)}{-10\sin(2\theta)\cos(\theta)+5\cos(2\theta)\sin(\theta)}$

$\frac{-10\sin(2\theta)\sin(\theta)-5\cos(2\theta)\cos(\theta)}{-10\sin(2\theta)\cos(\theta)+5\cos(2\theta)\sin(\theta)}$

$\frac{-10\sin(2\theta)\sin(\theta)+5\cos(2\theta)\cos(\theta)}{-10\sin(2\theta)\cos(\theta)-5\cos(2\theta)\sin(\theta)}$

Correct answer:

$\frac{-10\sin(2\theta)\sin(\theta)+5\cos(2\theta)\cos(\theta)}{-10\sin(2\theta)\cos(\theta)-5\cos(2\theta)\sin(\theta)}$

Explanation:

The derivative of a polar function is given by

$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{\frac{dr}{d\theta}\sin(\theta)+r\cos(\theta)}{\frac{dr}{d\theta}\cos(\theta)-r\sin(\theta)}$

First, we must find the derivative of the function, r:

$\frac{dr}{d\theta}=-10\sin(2\theta)$

which was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$

Now, using the derivative we just found and our original function in the above formula, we can write out the derivative of the function in terms of x and y:

$\frac{-10\sin(2\theta)\sin(\theta)+5\cos(2\theta)\cos(\theta)}{-10\sin(2\theta)\cos(\theta)-5\cos(2\theta)\sin(\theta)}$

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Example Question #301 : Parametric, Polar, And Vector

Find the derivative of the following function:

$r=3\cos(\theta)+\theta^2$

Possible Answers:

$\frac{(-3\sin(\theta)+2\theta)\sin(\theta)-(3\cos(\theta)+\theta^2)(\cos(\theta))}{(-3\sin(\theta)+2\theta)(\cos(\theta))-(3\cos(\theta)+\theta^2)(\sin(\theta))}$

$\frac{(-3\sin(\theta)+2\theta)\sin(\theta)-(3\cos(\theta)+\theta^2)(\cos(\theta))}{(-3\sin(\theta)+2\theta)(\cos(\theta))+(3\cos(\theta)-\theta^2)(\sin(\theta))}$

$\frac{(-3\sin(\theta)+2\theta)\sin(\theta)+(3\cos(\theta)+\theta^2)(\cos(\theta))}{(-3\sin(\theta)+2\theta)(\cos(\theta))-(3\cos(\theta)+\theta^2)(\sin(\theta))}$

$\frac{(-3\sin(\theta)+2\theta)\cos(\theta)+(3\cos(\theta)+\theta^2)(\cos(\theta))}{(-3\sin(\theta)+2\theta)(\sin(\theta))-(3\cos(\theta)+\theta^2)(\sin(\theta))}$

Correct answer:

$\frac{(-3\sin(\theta)+2\theta)\sin(\theta)+(3\cos(\theta)+\theta^2)(\cos(\theta))}{(-3\sin(\theta)+2\theta)(\cos(\theta))-(3\cos(\theta)+\theta^2)(\sin(\theta))}$

Explanation:

The derivative of a polar function is given by

$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{\frac{dr}{d\theta}\sin(\theta)+r\cos(\theta)}{\frac{dr}{d\theta}\cos(\theta)-r\sin(\theta)}$

First, we must find the derivative of the function, r:

$\frac{dr}{d\theta}=-3\sin(\theta)+2\theta$

We used the following rules to find the derivative:

$\frac{\mathrm{d} }{\mathrm{d} x} \cos(\theta)=-\sin(\theta)$ , $\frac{\mathrm{d} }{\mathrm{d} x}x^n=nx^{n-1}$

Now, plug in the derivative and the original function r into the above formula:

$\frac{(-3\sin(\theta)+2\theta)\sin(\theta)+(3\cos(\theta)+\theta^2)(\cos(\theta))}{(-3\sin(\theta)+2\theta)(\cos(\theta))-(3\cos(\theta)+\theta^2)(\sin(\theta))}$

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Example Question #302 : Parametric, Polar, And Vector

Find the derivative of the following function:

$r=\theta+\tan(\theta)$

Possible Answers:

$\frac{(1+\sec^2(\theta))\sin(\theta)+(\theta+\tan(\theta))\cos(\theta)}{(1+\sec^2(\theta))\cos(\theta)-(\theta+\tan(\theta))\sin(\theta)}$

$\frac{(1+\sec^2(\theta))\sin(\theta)-(\theta+\tan(\theta))\cos(\theta)}{(1+\sec^2(\theta))\cos(\theta)-(\theta+\tan(\theta))\sin(\theta)}$

$\frac{(1+\sec^2(\theta))\sin(\theta)+(\theta+\tan(\theta))\cos(\theta)}{(1+\sec^2(\theta))\cos(\theta)+(\theta+\tan(\theta))\sin(\theta)}$

$\frac{(1+\sec^2(\theta))\cos(\theta)+(\theta+\tan(\theta))\sin(\theta)}{(1+\sec^2(\theta))\cos(\theta)-(\theta+\tan(\theta))\sin(\theta)}$

Correct answer:

$\frac{(1+\sec^2(\theta))\sin(\theta)+(\theta+\tan(\theta))\cos(\theta)}{(1+\sec^2(\theta))\cos(\theta)-(\theta+\tan(\theta))\sin(\theta)}$

Explanation:

The derivative (slope of the tangent line) of a polar function is given by the following formula:

$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{\frac{dr}{d\theta}\sin(\theta)+r\cos(\theta)}{\frac{dr}{d\theta}\cos(\theta)-r\sin(\theta)}$

So, we must simply find $\frac{dr}{d\theta}$ and plug it into the above formula:

$\frac{dr}{d\theta}=1+\sec^2(\theta)$

The following rules were used to find the derivative:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{^{n-1}}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \tan(x)=\sec^2(x)$

Now, plug the given function and its derivative into the above formula to get our answer:

$\frac{(1+\sec^2(\theta))\sin(\theta)+(\theta+\tan(\theta))\cos(\theta)}{(1+\sec^2(\theta))\cos(\theta)-(\theta+\tan(\theta))\sin(\theta)}$

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Calculus 2 : Polar

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #1 : Derivatives Of Polar Form

Example Question #1 : Derivatives Of Polar Form

Example Question #803 : Calculus Ii

Example Question #3 : Derivatives Of Polar Form

Example Question #1 : Derivatives Of Polar Form

Example Question #2 : Derivatives Of Polar Form

Example Question #2 : Derivatives Of Polar Form

Example Question #7 : Derivatives Of Polar Form

Example Question #301 : Parametric, Polar, And Vector

Example Question #302 : Parametric, Polar, And Vector

All Calculus 2 Resources

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