Question

Find the slope of the tangent line to the given polar r = cos(2theta) curve at theta=pi/4

Answer 1

The answer is  slope of the tangent tothe given polar curve $r=\cos (2 \theta)$ at $\theta=\frac{\pi}{4}$ is $m=1$

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

As we know that the slope m of the polar curve$r=f(\theta)$ at a point P is given by

$m=\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{\theta=\theta_{0}}=\left.\frac{f^{\prime}(\theta) \sin (\theta)+f(\theta) \cos (\theta)}{f^{\prime}(\theta) \cos (\theta)-f(\theta) \sin (\theta)}\right|_{\theta=\theta_{0}}$..........(1)

Here $f(\theta)=\cos (2 \theta) \quad$ and $\quad \theta_{0}=\frac{\pi}{4}$

$f^{\prime}(\theta)=-2 \sin (2 \theta) \quad$ [ by using chain rule]

Substituting in eq. (1) we get

$m=\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{\theta=\frac{\pi}{4}}=\left.\frac{-2 \sin (2 \theta) \sin (\theta)+\cos (2 \theta) \cos (\theta)}{-2 \sin (2 \theta) \cos (\theta)-\cos (2 \theta) \sin (\theta)}\right|_{\theta=\frac{\pi}{4}}$

$m=\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{\theta=\frac{\pi}{4}}=\frac{-2 \sin \left(2 \frac{\pi}{4}\right) \sin \left(\frac{\pi}{4}\right)+\cos \left(2 \frac{\pi}{4}\right) \cos \left(\frac{\pi}{4}\right)}{-2 \sin \left(2 \frac{\pi}{4}\right) \cos \left(\frac{\pi}{4}\right)-\cos \left(2 \frac{\pi}{4}\right) \sin \left(\frac{\pi}{4}\right)}$

$m=\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{\theta=\frac{\pi}{4}}=\frac{-2 \sin \left(\frac{\pi}{2}\right) \sin \left(\frac{\pi}{4}\right)+\cos \left(\frac{\pi}{2}\right) \cos \left(\frac{\pi}{4}\right)}{-2 \sin \left(\frac{\pi}{2}\right) \cos \left(\frac{\pi}{4}\right)-\cos \left(\frac{\pi}{2}\right) \sin \left(\frac{\pi}{4}\right)}$

$\begin{aligned}&m=\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{\theta=\frac{\pi}{4}}=\frac{-2 \frac{1}{\sqrt{2}}+0}{-2 \frac{1}{\sqrt{2}}-0} \\&m=\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{\theta=\frac{\pi}{4}}=\frac{-\frac{2}{\sqrt{2}}}{-\frac{2}{\sqrt{2}}} \\&m=\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{\theta=\frac{\pi}{4}}=1\end{aligned}$

Hence slope of the tangent to the given polar curve $r=\cos (2 \theta)$ at $\theta=\frac{\pi}{4}$ is $m=1$

What is chain rule ?

• The chain rule is a method for calculating the derivative of composite functions, which are functions created by joining one or more functions.
• The chain rule is used to differentiate a function that contains another function. The product rule is used to distinguish a function that is the product of two functions. The quotient rule is used to distinguish a function that is the sum of two functions.

To learn more about chain rule visit:

https://brainly.com/question/25854839

#SPJ4