Math 4111
Test 2
Spring 1996

No books, notes etc. are permitted.
Show all your work! Box in your answers!
The test has 7 problems on 4 pages.
Read the problems carefully!

Problem 1 (15 points)   Sketch the graph of the derivative of f(x), where f(x) has the following graph:

Problem 2 (15 points)   Let f(x)=x³-2x. Compute f '(-2) algebraically by using the definition of the derivative.

Problem 3 (15 points)   The following table gives the population of the City of El Paso during the last 90 years:

YEAR	POPULATION	YEAR	POPULATION
1890	10,338	1940	96,810
1900	15,906	1950	130,485
1910	39,279	1960	276,687
1920	77,560	1970	322,261
1930	102,421	1980	425,259

(a) Estimate the rate of change of the population for 1970. Explain your reasoning!

(b) Approximately when was the rate of change of the population the greatest?

(c) Estimate the population for the year 1957.

(d) Estimate the rate of change of the population for 1957. Explain your reasoning!

Problem 4 (10 points)   Sketch the graph of a positive function on [-2,2], whose first and second derivatives are everywhere negative!

Problem 5 (15 points)   Consider the following functions, defined on [0,1]:

$$h_1(x)=\left\{\begin{array}{rl}1,&\mbox{ if } 0\leq x\leq \fr... ...frac{1}{2} \mbox{ or if } \frac{3}{4}<x\leq 1\end{array}\right.$$

Sketch both functions, then sketch their product $h_1(x) \cdot h_2(x)$. (It is best to use three separate graphs.) Then find $$\int_0^1 h_1(x)\,dx$$, $$\int_0^1 \vert h_2(x)\vert\,dx$$, and $$\int_0^1 h_1(x) \cdot h_2(x)\,dx$$. (You should give exact answers!)

Problem 6 (15 points)   Compute a ``good" approximation for the area enclosed by the graph of $f(x)=\cos(x^2)$, the x-axis, and the lines x=0 and $$x=\sqrt{\frac{3 \pi}{2}}$$. Explain your procedure! (Hint: Graph f(x) first!)

Problem 7 (15 points)   For the following problem you may use that the function
F(x)=6x-e^2x has the derivative F '(x)=6-2 e^2x.

Find the area enclosed by the graph of f(x)=6-2e^2x, the x-axis and the lines x=0 and x=1. Give an exact answer, not an approximation! (Hint: Graph f(x) first!)