The Fundamental Theorem of Calculus Part 1 & 2 Chapter 5.3 February 1, 2007

Let , where g is graphed below Find: On what interval(s) is f increasing? What are the Max/Min values of f on [-4,4]?

If what is g(1)?

Improper Integrals (We’ll evaluate them in chapt. 7) An integral having at least one nonfinite limit or an integrand that becomes infinite between the limits of integration. Interval is infinite (easiest to identify) Function “Blows” up! (down)

Which of the following integrals are improper?

Fundamental Theorem of Calculus (Part 1) (Chain Rule) If f is continuous on [a, b], then the function defined by is continuous on [a, b] and differentiable on (a, b) and

Examples:

Fundamental Theorem of Calculus (Part 1)

Fundamental Theorem of Calculus (Part 2) If f is continuous on [a, b], then : Where F is any antiderivative of f. ( ) Helps us to more easily evaluate Definite Integrals in the same way we calculate the Indefinite!

Evaluate:

Evaluate: Improper (@x = 0)

In-class Assignment Estimate (by counting the squares) the total area between f(x) and the x-axis. Using the given graph, estimate Why are your answers in parts (a) and (b) different?

The Total Change Theorem The integral of a rate of change is the total change from a to b. (displacement)

The Total Change Theorem Ex: Given Find the displacement and total distance traveled from time 1 to time 6. Displacement: Total Distance:

Practice Test 1 In-class version