Mathematics 36, section 5 -- AP Calculus
Discussion 1 -- Using Derivatives
September 22, 1999
Background
Our techniques for analyzing critical points of functions
can be applied to many problems, especially optimization problems,
where we seek a maximum or minimum value of some function.
Discussion Questions
- A) Sketch the graphs y = f(x) and y = f '' (x),
given the graph of y = f '(x).
- B) 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.
- Sketch a possible graph for the function Q(t)
giving the quantity of the drug in the bloodstream as a function of time.
- 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.
- 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?
- C) You have a 10 inch length of flexible metal wire. If
you cut the wire into two pieces, bend one to enclose a square, and
the other to enclose a circle, what is the largest total
area you can enclose? Give a full justification of your answer.
- D) The efficiency of a screw is given by
E = (theta - mu theta2) / (mu + theta)
where theta is the pitch angle of the threads of the screw,
and mu is a positive constant. What value of theta
maximizes E?
Assignment
One write-up per group
of solutions for these problems, due Monday, September 27.