Mathematics

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.)

Intuitive explanation

The key here is recognizing that $F(x)$ is a composite function—an integral with a variable limit that itself depends on $x$ . To find $F'(x)$ , you'll need the Fundamental Theorem of Calculus combined with the chain rule: when the upper limit is constant and the lower limit is a function of $x$ , the derivative "flips the sign" and multiplies by the derivative of that lower limit. Think of it as tracking how the accumulated area changes as the input $x^2$ shifts.

Intuitive explanation

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