1. Clicker Question 1 What is an antiderivative of f(x) = (5x – 3)  ? – A. (5/(  + 1))(5x – 3)  +1 – B. (1/(  + 1))(5x – 3)  +1 – C. 5  (5x – 3)  - 1 – D. (1/(5(  + 1)))(5x – 3)  +1 – E. (5(  + 1))(5x – 3)  +1

2. Clicker Question 2 What is an antiderivative of g(x) = x / (x 2 + 1)? – A. x ln(x 2 + 1) – B. (1/2) ln(x 2 + 1) – C. 1 / (x 2 + 1) 2 – D. (-1/4) / (x 2 + 1) 2 – E. 2 ln(x 2 + 1)

3. Definite Integrals, Substitution and Average Value (1/27/14) If you use substitution and the Fundamental Theorem to evaluate a definite integral, there are two possible approaches: – Go back to the original variable and evaluate at the endpoints as usual, or – Never return to the original variable! Instead, change the endpoints to correspond to your new variable, and then stay with that variable.

4. Using the Definite Integral This semester we shall study numerous applications of the definite integral to geometry, physics, economics, probability, and so on. Remember that whenever you want to “add up” the values of a function over some interval, the definite integral may well be the ticket! We start with an easy application: – Average value of a function on an interval

5. Average Value of a Function on an Interval To find the average value of a list of numbers, you add them up and divide by how much is there. It’s the exact same for functions: add up the values of the function on the interval in question and then divide by how much is there (i.e., the length of the interval). Thus the average value of f on [a, b] is

6. Example of Average Value What is the average value of sin(t) on the interval [0,  ] ? Look at the picture and make a guess. The answer is = 2 /  .637 Check that this answer makes sense. (The average value on a graph is the average height, i.e. the height whose rectangle has the same area as the area under the curve.)

7. Clicker Question 3 What is the average value of f(x) = x 2 on the interval [0, 4]? – A. 8 – B. 21 1/3 – C. 5 1/3 – D. 6 2/3 – E. 7 2/3

8. Assignment for Wednesday On page 414, do Exercises 53, 55, 57, and 59. Read Section 6.5. On page 453-4, do Exercises 1,3,5,10 and 17.