# 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):
• (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 [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.
• (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 [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.
• (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 [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.
• (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:
• Upper estimate:
• Lower estimate:
• 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 [5] 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 [6] 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 [7] 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 [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.