find the equation of the tangent line to the curve ( y = 5xcos x ) at the point ( (pi,-5pi) ).\nthe equation…

find the equation of the tangent line to the curve ( y = 5xcos x ) at the point ( (pi,-5pi) ).\nthe equation of this tangent line can be written in the form ( y = mx + b ) where\n( m=)\nand ( b=)\nquestion help: video message instructor\nsubmit question jump to answer

find the equation of the tangent line to the curve ( y = 5xcos x ) at the point ( (pi,-5pi) ).\nthe equation of this tangent line can be written in the form ( y = mx + b ) where\n( m=)\nand ( b=)\nquestion help: video message instructor\nsubmit question jump to answer

Answer

Explanation:

Step1: Find the derivative of the function

Use the product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = 5x) and (v=\cos x). (u^\prime=5), (v^\prime=-\sin x) (y^\prime=(5x\cos x)^\prime=5\cos x-5x\sin x)

Step2: Calculate the slope (m)

Substitute (x = \pi) into (y^\prime) (m=y^\prime|_{x = \pi}=5\cos\pi-5\pi\sin\pi) Since (\cos\pi=- 1) and (\sin\pi = 0) (m=5\times(-1)-5\pi\times0=-5)

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

Use the point - slope form (y - y_1=m(x - x_1)), where ((x_1,y_1)=(\pi,-5\pi)) and (m=-5) (y+5\pi=-5(x - \pi)) Expand: (y+5\pi=-5x + 5\pi) (y=-5x), so (b = 0)

Answer:

(m=-5), (b = 0)