November 19, 2001
(1) Consider the function f(x) = .
(a) Find the inverse of f.
(b) Find the range of .
(c) Find the domain of .
(d) Find the range of f.
(2) Simplify the following expressions. Your answers should contain neither an e nor an ln.
(b) ln( ln(e) ) =
(4) Differentiate the following functions.
(5) This problem involves the function .
(a) Find all critical points of f.
(b) Find the interval(s) on which f increases and on which it decreases.
(c) Identify the locations of any extrema of f. Classify them as relative maxima or minima.
Using the first derivative test,
(6) The graph of the derivative f ' of
a function f is sketched.Supply the following information about the function
(a) List the critical points of f.
(b) State the interval(s) on which f is decreasing.
(c) State the intervals(s) on which f is concave up.
(d) At which value of x does f have a relative minimum?
(e) Using the same coordinate axis, sketch a possible graph of f.
(7) The graph of the second derivative g''
of a function g is sketched. Suppose you also know that the first
derivative g ' has x-intercepts at -3, 0, and 4. Supply
the following information about the function g.