## MAT 131 Calculus I~~~~Final Examination

May 11, 1994

This is an 3 hour test. Work all questions. You may use a programmable, graphing calculator (make sure it is in radian mode). As usual, no credit for unexplained work!

1. A function f(x) is monotonic increasing on [0,1]. Some of its values are given in the following table:


x        0.0     0.2      0.4     0.6      0.8      1.0
f(x)    0.00   0.040    0.155   0.339    0.573    0.841

a. Estimate the derivative of f at x=0.4.

b. Estimate the second derivative of f at x=0.4.

2. Give the derivatives of the following functions:

a. f(x) = x + e^x.

b. h(t) = \sin(2t).

c. g(s) = e^{-s}\cos s.

d. k(z) = sin(z)/ (cos(z) + 2).

e. p(w) = \sqrt[1+sin^2(w)].

3. A can in the shape of a rectangular solid with a square top and bottom is made to hold 1000 cm^3 of liquid. The top and the bottom are made of a special material that costs .02 per cm^2, while the material used for the sides costs only .01 per cm^2. Calculate the dimensions of the can that minimize the total cost of material.

4. a. Estimate \int_0^{1.2} cos(x^2)dx

          /1.2
|
|cos(x^2)dx
|
/0

by a left-hand sum with three subintervals. Do not use a program; show all your steps.

b. How many subintervals would it take to get an estimate for this integral with error less than 10^{-5} ? Note that cos(x^2) is monotonic decreasing on the interval of integration.

5. Find ANTI-derivatives for the following functions:

a. f(x) = 2x + 2e^x.

b. h(t) = cos(t).

c. g(s) = 2 sin(3s).

d. k(z) = 1/z.

e. p(w) = w^{-4}

6. The function f(x)= e^x + x is invertible; let the function g be its inverse.

a. Use your calculator to determine the value of g(3) to 2 decimal places.

b. Calculate the derivative g'(3). You may use the formula g'(x) = 1/[f'(g(x))] for the derivative of an inverse function. Explain your work clearly.

7. Figure 1 shows the graph of the derivative f' of a certain function f. Given that f(0)=0, give a rough sketch of the graph of f, paying attention to having critical points and inflection points at the correct x-values, and to showing the proper concavity. Do not worry about getting the y-values exactly.

(height=3in,width=6in) Figure 1: The graph of f'.

8. Figure 2 shows the graph of a function g(x). Estimate \int_0^4g(x)dx. Explain your work.

(height=3in,width=6in) Figure 2: The graph of g.

Questions left off the exam:

A. A water gate at the bottom of a dam is opened at t=0; the flow (in cubic feet/second) is measured initially and every hour during the first five hours:

t in hours                     0      1      2      3     4      5
rate of flow in ft^3/sec      550    400    250    200   150    125


Since as the water level falls the pressure at the gate decreases, the rate of flow decreases with time. Use the data in the table to give upper and lower bounds on the quantity of water that has passed through the gate during the first five hours. Watch your units!

B. The height of a roller-coaster track is given as a function of horizontal distance from the start by the equation

h(x) = (10-x)sin^2(x), 0 <= x <=10.

For what x-value is the downward slope greatest? Hint: This problem requires a combination of symbolic calculation and use of the graphing calculator. Explain carefully how you obtain your answer.

C. The graph of the function f(x) = x^3+x^2-2x intersects the x-axis in three places: x = -2,0,1. Calculate the total area enclosed by the graph and the x-axis. Draw a picture and explain your work carefully.