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

##### 1 Answer

#### Explanation:

Find the point the normal line will intercept.

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

The normal line will pass through the point

To find the slope of the normal line, first find the slope of the tangent line. Since the tangent line and normal line are perpendicular, their slopes will be opposite reciprocals of one another.

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

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

The derivative of

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

The slope of the tangent line is

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

Thus, the slope of the normal line will be the opposite reciprocal of

The normal line passes through the point

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

Graphed are the function and the normal line:

graph{(x^2+sinx-y)(y-pi^2-1/(1-2pi)(x-pi))=0 [-15.51, 20.53, -0.57, 17.46]}