## The Fundamental Theorem of Calculus

Read this section to see the connection between derivatives and integrals. Work through practice problems 1-5.

### Part 1: Antiderivatives

Every continuous function has an antiderivative, even those nondifferentiable functions with "corners" such as absolute value.

**The Fundamental Theorem of Calculus** (Part 1)

then . is an antiderivative of .

Proof: Assume is a continuous function and let . By the definition of derivative of ,

By Property 6 of definite integrals (Section 4.3), for

Dividing each part of the inequality by , we have that is between the minimum and the maximum of on the interval . The function is continuous (by the hypothesis) and the interval is shrinking (since h approaches 0), so and . Therefore, is stuck between two quantities (Fig. 2) which both approach .

Then must also approach , and .

**Example 1: ** for in Fig. 3. Evaluate and for .

**Practice 1:** for in Fig. 4. Evaluate and for and .

**Example 2: ** for the function shown in Fig. 5.

For which value of is maximum?

For which is the rate of change of maximum?

Solution: Since is differentiable, the only critical points are where or at endpoints. and has a maximum at . Notice that the values of as goes from 0 to 3 and then the values decrease. The rate of change of is , and appears to have a maximum at so the rate of change of is maximum when . Near , a slight increase in the value of yields the maximum increase in the value of .