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For the following exercises, find the slope of a tangent line to a polar curve $r=f(\theta) .$ Let $x=r \cos \theta=f(\theta) \cos \theta$ and $y=r \sin \theta=f(\theta) \sin \theta, \quad$ so the polar equation$r=f(\theta)$ is now written in parametric form.$$r=4 \cos \theta ; \quad\left(2, \frac{\pi}{3}\right)$$

$$\mathrm{Slop}=\frac{1}{\sqrt{3}}$$

Calculus 2 / BC

Chapter 7

Parametric Equations and Polar Coordinates

Section 4

Area and Arc Length in Polar Coordinates

Parametric Equations

Polar Coordinates

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so in this problem were given that we have a polar curve are that is equal to four times coasts on data. And we are asked, what is the slope or D y d x of our at polar coordinates to comma pi over three? And to answer that, we used the definition of the slope for a polar at some angle, which is this, and you can see we already have our And so the only thing we don't know in this equation is drd Keita. So what we'll do is we'll say we'll take our our formula and take d d data of both sides find the derivative of our respective data. So Sam on this side and four times co signed data and so the derivative of co sign is negative signs. We get negative for times. Signed data equals D R. De data. So now evaluating d r d theta and are and signing co signs evaluating. Do you Why d x at the angle pi over three, Because do I d actually going to see it takes data and its input, and that gives us the answer, which is Route three over three, which is a little more than 1/2

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