[Note: These problems appeared on
my M106 exams in the fall of 1995.]
Instructions: Show all of your work and clearly
explain your answers.
This is particularly important on problems
with a numerical answer, to allow the possibility
of partial credit. No books are allowed
during the exam, but you may use your calculator.
Problem  Suppose the graph shown
is the graph of y = f(x):
(a) For what values of x is f '(x) < 0?
(b) For what values of x is f ''(x) > 0?
(c) Sketch the graph of f '(x).
Problem  Suppose now that
the graph above gives the graph of g '(x),
where g(x) is some differentiable function. Although what you are given
is the graph of the derivative, pay attention to the fact that
the questions below refer to g(x) itself.
(a) For what values of x is g(x) decreasing?
(b) For what values of x is g(x) concave up?
(c) For what value of x is g(x) biggest? Justify your answer.
Problem  Suppose that f(t)
is a differentiable function such that f(3.99)=251.1,
f(4)=251, and whose graph is concave up.
(a) What is the average rate of change of
f(t) over the interval 3.99 <= t <= 4?
(b) Is the answer to (a) bigger or smaller than f '(4)? Explain.
(c) Suppose that f(5) = 245 and f '(5) = -2. Use this
information to estimate f(5.001);
indicate whether the estimate should be bigger or smaller
than the actual value of
f(5.001), and explain why.
Problem  The temperature one winter
day in Lincoln was 42 degrees Fahrenheit at noon
and decreased all day until at 8:00 pm it was only 10 degrees Fahrenheit.
Let T = f(t) be the temperature at time t,
where t is the number of hours past noon and T is in degrees Fahrenheit.
(a) Express the average temperature between noon and 8:00 pm in terms of
a definite integral.
(b) Suppose you took the temperature every two hours,
obtaining the following table of data:
t (hrs) 0 2 4 6 8
f(t) (deg F) 42 26 18 13 10
Using this table and appropriate Riemann sums, give upper and
lower estimates for the average
temperature between noon and 8:00 pm:
Would taking the temperature every 5 minutes
be often enough to ensure that the Left Hand Riemann Sum estimate
for the average temperature between noon and 8:00 pm
is within 0.1 degrees Fahrenheit of the actual value? Why or why not.
Problem  This problem refers to the
function T = f(t) discussed in the preceding problem.
(a) Use the table in the preceding problem to estimate the derivative
of f(t) at t = 5 (show your work), and specify the units of your answer.
(b) Which is a possible value of the derivative of f -1(T) at T = 18, 0.32
or -0.32? Justify your answer.
(c) What are the units to your answer to (b), and
what is its practical meaning?
Problem  The graph of the
derivative F '(t) of a differentiable function F(t) is given below:
(a) Given that F(1) = 0, determine the values of F(-1) and F(3).
(b) Evaluate the definite integral of F '(t) from t = 0 to t = 3,
and indicate how you obtain your answer.
Problem  Suppose that f(x) is a
differentiable function such that f(2) = 3, f '(2) = 5 and f '(3) = 7.
(a) Find the derivative of (f(x))2 at x = 2.
(b) Find the derivative of f(f(x)) at x = 2.
(c) Find the derivative of x/f(x) at x = 2.
(d) Find the line tangent to the graph of
g(x) = x7 - 8x2 + 35x - 40 at x = 0.
Problem  Consider the limit
This limit is actually the definition of the derivative of some function
f(x) at some value x = a. Determine f(x) and a.