find the equation of the line that is tangent to the curve y = 4x cos x at the point (π, -4π). the equation…

Explanation:

Step1: Find the derivative of $y$

Use the product - rule $(uv)^\prime = u^\prime v+uv^\prime$, where $u = 4x$ and $v=\cos x$. $u^\prime=4$ and $v^\prime=-\sin x$. So $y^\prime=4\cos x-4x\sin x$.

Step2: Calculate the slope $m$

Substitute $x = \pi$ into $y^\prime$. $m=y^\prime|_{x = \pi}=4\cos\pi-4\pi\sin\pi$. Since $\cos\pi=- 1$ and $\sin\pi = 0$, then $m=4\times(-1)-4\pi\times0=-4$.

Step3: Find the y - intercept $b$

We know the line equation is $y = mx + b$, and the line passes through the point $(\pi,-4\pi)$. Substitute $x=\pi$, $y = - 4\pi$ and $m=-4$ into $y=mx + b$. We get $-4\pi=-4\pi + b$, so $b = 0$.

Answer:

$m=-4$ $b = 0$