MATH 131, section 1 – Calculus for Physical and Life Sciences I Discussion 3 – What do f′ and f′′ tell us about f? October 12, 2007

Background

When f′(x) > 0 at all points on an interval, then from our geometric interpretation of f′(x) as the slope of the tangent line to y = f(x), it is natural to expect that f is an increasing function on the interval. Similarly, when f′(x) < 0 at all points on an interval, then we expect that f is an decreasing function on the interval. We will use accept these observations as facts today; we will be able to give precise proofs later in the semester.

Discussion Questions

A.

1. What is true about the first derivative f′(x) if f′′(x) > 0 on some interval? What if f′′(x) < 0 on an interval?

2. One of the following graphs has f′′(x) < 0 for all x shown and the other has f′′(x) > 0 for all x shown. Which is which? Explain how you can tell.

“concave up” “concave down”

B.

1. Draw a graph y = f(x) such that f′(x) > 0 and f′′(x) > 0 for all x.

2. Draw a graph y = f(x) such that f′(x) > 0 and f′′(x) < 0 for all x.

3. Draw a graph y = f(x) such that f′(x) < 0 and f′′(x) > 0 for all x.

4. Draw a graph y = f(x) such that f′(x) < 0 and f′′(x) < 0 for all x.

Please turn over for question C.

C. The following graph shows y = f′(x) for a continuous function f(x).

1. Over which interval(s) is f(x) increasing?

2. Over which interval(s) is f(x) decreasing?

3. Over which interval(s) is y = f(x) concave up?

4. Over which interval(s) is y = f(x) concave down?

5. What happens at x = 2 on the graph y = f(x)?

6. What happens at x = 4 on the graph y = f(x)?

7. What happens at x = 6 on the graph y = f(x)?

8. What happens at x = 3 on the graph y = f(x)? (The technical name is “inflection point” but what does this mean and how can you tell?)

9. Using the information in the plot y = f′(x) and your answers to parts 1 - 8, sketch y = f(x), given that f(0) = 0. Note: It is possible to get at least reasonable estimates for y-coordinates on the graph of f(x) in this case. Do you see how?