Math 161 Calculus I

Reading Outlines

You should answer these questions as you read the text.

Section 4.10

1. What is the intermediate value theorem?
2. Explain in everyday language what the intermediate value theorem (IVT) tells us.
3. Suppose that f (1) < 0 and f (4) > 0, and f is continuous. What can you say about the roots of f?
4. What is the extreme value theorem?
5. Explain in everyday language what the extreme value theorem (EVT) tells us.
6. Why is the EVT important in finding solutions to optimization problems?
7. If you wanted to find a solution to 5^x = 32, what function would you use in the bisection method? What interval would you start with?
8. In your own words, describe the process of the bisection method. What does it do for us?
9. What other method have we learned for approximating the roots of an function?
10. Let f (x) = 1/x
    1. What does the EVT say about f on the interval [1, 2]?
    2. Is f continuous on [1,$\infty$)? Does it have a minimum value on [1,$\infty$)? Does this contradict the EVT? Why or why not?
    3. Is f continuous on (1, 2]? Does it have a maximum value on (1, 2]? Does this contradict the EVT? Why or why not?