assignment 5: problem 10\n(1 point)\nfind the equation of the line that is tangent to the curve\n\nat the…

Explanation:

Step1: Differentiate the function

Use the product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = 6x) and (v=\cos x). (u^\prime=6), (v^\prime=-\sin x). So (y^\prime=6\cos x-6x\sin x).

Step2: Find the slope (m)

Substitute (x = \pi) into (y^\prime). (m=y^\prime|_{x = \pi}=6\cos\pi-6\pi\sin\pi). Since (\cos\pi=- 1) and (\sin\pi = 0), then (m=6\times(-1)-6\pi\times0=-6).

Step3: Find the (y -)intercept (b)

Use the point - slope form (y - y_0=m(x - x_0)), with ((x_0,y_0)=(\pi,-6\pi)) and (m=-6). (y+6\pi=-6(x - \pi)). Expand: (y+6\pi=-6x + 6\pi). Subtract (6\pi) from both sides: (y=-6x). So (b = 0).

Answer:

(m=-6), (b = 0)