1. Clicker Question 1 – A. converges to 1 – B. converges to 1/5 – C. converges to -1/5 – D. converges to 5 – E. diverges

2. Clicker Question 2 – A. converges to ln(3) – B. converges to 3 ln(3) – C. converges to ln(ln(3)) – D. converges to ln(1/3) – E. diverges

3. More on Improper Integrals: The p-Test (2/21/14) converges for p > 1 and diverges for p  1 (a is any positive constant). Why? converges for p < 1 and diverges for p  1. Why? (These two are called The p-Test) converges for all a and all positive c. Why?

4. Global Behavior (Comparison) With improper integrals of the Type 1, a quotient of algebraic functions will behave like the ratio of its highest degree terms. Example: Does converge or diverge? Example: What about ? If one of these converges, we still need to work to find out to what number. If one diverges, we’re done.

5. Clicker Question 3 The improper integral – A. converges to 1 – B. converges to 2/  7 – C. converges, but we don’t know to what – D. diverges – E. I’m clueless

6. Indeterminate Forms If limits end up having the form 0 / 0,  / , 0 *  or   , they are called indeterminate. This means they can actually have any number value or be +  or - . Their value can’t be determined, hence the name! Note, for example, that every derivative is of the form 0 / 0. When you have such a form, you may be able to manipulate it algebraically to find its value.

7. L’Hopital’s Rule If an indeterminate form is in the form 0 / 0 or  / , you may be able to evaluate it by L’Hopital’s Rule, which says that the limit of the ratio will be the same as the limit of the ratio of the (separate) derivatives. Example: What is lim x->  x 2 / e x ?

8. Assignment for Monday Check over your Hand-in #2. On pages 527-8 (in Section 7.8), try Exercises 14 (answer 2/e), 31 (why is it improper?), 39 (will need L’Hopital’s Rule – write as a quotient), 49 and 51. Monday’s class is an optional Q & A (no clicker questions). Test #1 is on Wed Feb 26. It covers everything we’ve done to this point. You may bring one reference sheet. Start at 8:30.