Math 161 Calculus I

Reading Outlines

You should answer these questions as you read the text.

Section 4.11 Read up until the subsection titled ``Proving the MVT'' on page 335.

1. Suppose that a function f is differentiable at x = 1. What does Theorem 6 tell us about f?
2. Is it also true that if f is continuous at x = 1, then f is differentiable at x = 1?
3. What are the hypotheses of the mean value theorem (MVT)? What is the conclusion?
4. Why is it called the mean value theorem?
5. Explain in everyday language what the MVT says.
6. In words, what does the left side of the MVT equation represent (be specific)? What does the right side represent?
7. In the picture at the bottom of page 333, which line is the secant line between x = a and x = b? Where are the tangent lines with the same slope? Use the picture to explain graphically what the MVT tells us.
8. Let f (x) = | x|. What is the slope of the secant line between x = - 1 and x = 1? Is there a point x = c inside the interval [- 1, 1] where f'(c) equals this slope? Look at the graph! Does this contradict the MVT?