Math 75

December 14, 1999

1. Evaluate $$\lim_{x \rightarrow 3} {x^2 - 9 \over x - 3}$$

(a) 0 (b) $\infty$ (c) 6 (d) does not exist

2. Evaluate $$\lim_{x \rightarrow \infty} {7x^5 - 6x^3 + 9 \over x + 2x^5}$$

(a) 3 (b) $\infty$ (c) 9 (d) ${7 \over 2}$

3. The tangent line to the curve $y \,=\,x^2$ at the point $P(1,1)$ intersects the y-axis at $y \,=\,$

(a) -1 (b) 0 (c) 1 (d) 2

4. Find the equation of the tangent line to the curve $y \,=\,x^2 + x$ at the point $x \,=\,-1$.

(a) $y \,=\,2x + 1$ (b) $y \,=\,x + 1$ (c) $y \,=\,-x - 1$ (d) $y \,=\,-x + 1$

5. Determine whether the following function is continuous and/or differentiable at $x \,=\,1$.

         $f(x) = \left\{ \begin{array} {r@{\quad :\quad}l} x + 1 & 0 \le x \le 1 \\ 2x & 1 < x \le 5 \end{array}\right.$

(a) continuous (b) differentiable (c) both (d) neither

6. Find $f'(1)$, where $$f(x) \,=\,{-2x \over 1 + x}$$.

(a) $-2$ (b) $-{1 \over 2}$ (c) $0$ (d) $2$

7. Consider the following function defined on the interval $0 \le x \le 12$.

Fill in the blanks of the table with either $+$, $-$ or 0 to indicate the sign of $f(x)$, $f'(x)$ and $f''(x)$ at the corresponding point.

	A	B	C	D
$f(x)$
$f'(x)$
$f''(x)$

8. Determine the interval for which the function $f(x) \,=\,x e^{-2x}$ is increasing.

(a) $x \,>\, 2$ (b) $x \,>\, {1 \over 2}$ (c) $x \,<\, 2$ (d) $x \,<\, {1 \over 2}$

9. Let $$f(x) = {1 \over e^{5x}}$$. Which of the following statements are true?

(i) $f$ is decreasing for $x > 0$.

true     false

(ii) $f(x) \,<\,0$ for $x <0$.

true     false

(iii) $f(0) \,=\,1$.

true     false

(1v) The graph of $f$ has a vertical asymptote at $x = 0$.

true     false

10. A stock portfolio increased in value from $100,000 dollars to $150,000 dollars in ten years. What rate of interest, compounded continously, did the portfolio earn?

(a) $ln(1.5)$ (b) $1.5\,ln(10)$ (c) $${ln(1.5) \over 10}$$ (d) $0.15$

11. Differentiate.

(a) $$y \,=\,x^2 - \sqrt{x}$$

(b) $$y \,=\,\bigg(x - {1 \over x} \bigg)^5$$

12. Differentiate $$g(x) \,=\,ln\bigg[ {(x^2 - 2)(x^2 + 4) \over x} \bigg]$$.

13. Find the absolute minimum value and the absolute maximum value of the function

$f(x) \,=\,x^3 - 6x$ on the interval $0 \le x \le 2$.

14. Find the first and second derivative of function $g(x) \,=\,ln(x^2 - 2)$.

15. Determine the intervals for which the function $f(x) \,=\,{1 \over 3}x^2 - {3 \over 2}x^2 + 2x + 3$ is concave up.

16. A rectangular playground is fenced off and divided in three by two additional fences parallel to one side of the playground. Twelve hundred feet of fencing is used. Find the values of W and L that will result in the greatest total area.

17. Assume that $x^3 + 2xy - y^3 \,=\,ln\,x$, find the slope of the graph at the point $(1, -1)$

18. The average ticket price for a concert at the Spectrum was $30. The average attendance was 20000. For each increase of $1 in the price of the ticket, attendance declined by 200. What should the ticket price be in order to maximize concert revenues?

19. The size of a certain insect population is given by $P(t) \,=\,300e^{0.2t}$, where $t$ is measured in days.

(a) How many insects were present initially?

(b) When will the the population to triple?

20. If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Toricelli's Law gives the volume $V$ remaining in the tank after $t$ minutes as

$$V \,=\,5000\bigg(1 \,-\, {t \over 40}\bigg)^2 .$$

(a) Find the amount of water left in the tank after 30 min.

(b) What is the rate the water is draining from the tank after 30 min?

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