What is the equation of the line tangent to #f(x)=x^2 + sin(x) # at #x=pi#?

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Answer 1 · 2016-01-17T17:57:22

#y-pi^2=(2pi-1)(x-pi)#

Find the point the tangent line will intercept.

#f(pi)=pi^2+sin(pi)=pi^2#

The tangent line will intercept the point #(pi,pi^2)#.

To find the slope of the tangent line, find the derivative of the function.

#f'(x)=2x+cos(x)#

The slope of the tangent line is

#f'(pi)=2pi+cos(pi)=2pi-1#

Write the equation of the tangent line in point slope form knowing it passes through the point #(pi,pi^2)# and has a slope of #2pi-1#.

#y-pi^2=(2pi-1)(x-pi)#

Graphed are the function and tangent line:

graph{(x^2+sin(x)-y)(y-pi^2-(2pi-1)(x-pi))=0 [-29.44, 52.78, -5.24, 35.85]}