M106 Practice Exam 2

[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 [1] Suppose the graph shown is the graph of y = f(x): Graph goes here.
Problem [2] 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.
Problem [3] Suppose that f(t) is a differentiable function such that f(3.99)=251.1, f(4)=251, and whose graph is concave up.
Problem [4] 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.
Problem [5] This problem refers to the function T = f(t) discussed in the preceding problem.

Problem [6] The graph of the derivative F '(t) of a differentiable function F(t) is given below: Graph goes here.

Problem [7] Suppose that f(x) is a differentiable function such that f(2) = 3, f '(2) = 5 and f '(3) = 7.

Problem [8] Consider the limit limh->0 (ln(e+h)-1)/h. This limit is actually the definition of the derivative of some function f(x) at some value x = a. Determine f(x) and a.