A function is defined for all real numbers by

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

A function

f

is defined for all real numbers by

f(t)=\sqrt{1+t^{2}}

. Let

F(x)=\int_{x^{2}}^{3} f(t)\,dt.

Answer the following. (a) Calculate

F'(x)

for all real

x

. (b) Evaluate

F'(2)

. Express your answer in simplest exact form. (c) Interpret the meaning of

F'(2)

in the context of the area accumulated by the integral definition of

F

. Your interpretation should refer to the geometric or graphical significance of the sign and magnitude of

F'(2)

. (No sketch is required.)