Mathematics 36, section 1 -- AP Analysis

Discussion 2 -- Using Derivatives

September 22, 1997

Background

Our discussion of derivatives for functions f(x) has given us three related but distinct ways to think about the meaning of f'(a) at a particular x = a:

1. f'(a) is the instantaneous rate of change of f with respect to x -- the limit of the average rate of change (f(a+h) - f(a))/h as h -> 0.
2. f'(a) is the slope of the tangent line to the graph y = f(x) at x = a.
3. f'(a) and the function value f(a) are the information needed to write down the linear approximation: f(x) ~ f(a) + f'(a)(x - a) which is valid near x = a. In Lab 2, the "zooming-in" process you used produced an interval where this approximation was so good that you couldn't distinguish the tangent line from the actual graph y = f(x). So for instance the slope f'(a) can also be thought of as ``the slope of the graph'' y = f(x) at x = a.

Discussion Questions

A) Suppose you know that f is a function that is differentiable at all x in some interval I.

1. If f'(x) > 0 for all x in I, what can you say about f on I? Explain, relating to the linear approximation in point 3 in the Background material above.
2. If f'(x) < 0 for all x in I, what can you say about f on I? Explain, relating to the linear approximation in point 3 in the Background material above.
3. What if f'(x) = 0 for all x in I?

B) Suppose f is a differentiable function on some interval.

1. Suppose that f'(x) > 0 for x < a while f'(x) < 0 for x > a. What happens at x = a on the graph of f? Explain your reasoning.
2. Suppose that f'(x) < 0 for x < a while f'(x) > 0 for x > a. What happens at x = a on the graph of f? Again, explain your reasoning.

C) To give a patient an antibiotic slowly, the drug is injected into a muscle (rather than directly into a vein). The quantity of the drug in the bloodstream of the patient starts at 0 at t = 0 when the injection is given, increases to a maximum at t = 3 hours, then decays asymptotically to zero for t > 3. The rate of increase of the quantity of the drug is greatest at the time of the injection.

1. Sketch a possible graph for the function Q(t) giving the quantity of the drug in the bloodstream as a function of time.
2. Using that graph, make a qualitative sketch of the graph of the rate of change of the amount of the drug, Q'(t) -- the rate at which the drug is entering or leaving the bloodstream. How are the values of Q' different for t < 3 and t > 3? Explain.
3. Now repeat the process and construct a third sketch showing the graph of Q'' -- the rate of change of the rate of change of the quantity of the drug. What is the meaning of the t-axis intercept of this graph?