MATH 1680

Final | Fall '15
1. Find the derivative of $y = \ln (x^2 + x)$.

A) $\frac{1}{x^2 + x}$
B) $\frac{2x + 1}{x^2 + x}$
C) $\frac{1}{2x + 1}$
D) $\frac{1}{x^2} + \frac{1}{x}$
E) $1$

2. Evaluate $\lim_{x \to 3} \frac{x^2 + 2x - 15}{x^2 - 2x - 3}$.

A) $1$
B) DNE
C) $2$
D) $4$
E) $\infty$
F) $-\infty$
G) $\frac{0}{0}$

3. Find the marginal average cost of $C(x) = 10x^2 + 20x + 40$ when $x = 2$.

A) $-2$
B) $-1$
C) $0$
D) $1$
E) $2$

4. Find the average value of $f(x) = \frac{6}{\sqrt{x}}$ over the interval $1 \le x \le 4$.

A) $0$
B) $2$
C) $4$
D) $6$
E) $8$

5. Find the derivative of $x^2 + \frac{x}{y} + y^2 = 3$ at $(1, 1)$.

A) $-3$
B) $-1$
C) $0$
D) $1$
E) $3$

6. $f(x) = \frac{1}{\sqrt{x}}$. Evaluate $\lim_{h \to 0} \frac{f(4 + h) - f(4)}{h}$.

A) $\frac{0}{0}$
B) DNE
C) $\frac{-1}{16}$
D) $\frac{-1}{15}$
E) $\frac{-1}{14}$
F) $\frac{-1}{13}$

7. Solve $f'(x) = 3x^2 + 2x$ for $f(x)$ when $f(1) = 1$.

A) $f(x) = 6x + 2$
B) $f(x) = x^3 + x^2 - 1$
C) $f(x) = x^3 + x^2 + 1$
D) $f(x) = x^3 + x^2$
E) $f(x) = x^3 + x^2 + C$

8. Evaluate $\int_{-2}^{1} 3x^2 + 2x + 1 \; dx$

A) $4$
B) $5$
C) $6$
D) $7$
E) $8$
F) $9$

9. Find the vertical asymptote(s) for $y = \frac{x^2 + 2x + 5}{x^2 - x}$.

A) $x = 0$
B) $x = 0, x = 1$
C) $x = -1$
D) $x = -1$
E) $x = -1, x = 0$
F) $x = 1$

10. Find the absolute maximum of the function $f(x) = x^3 - 12x$ on the interval $-1 \le x \le 3$.

A) $-16$
B) $-9$
C) $0$
D) $11$
E) $16$

11. $f(x) = x^3 + 3x^2$ and $f'(x) = 3x^2 + 6x$. Where is $f(x)$ concave up?

A) $x < -1$
B) $x > -1$
C) $x < 1$
D) $x > 1$
E) $x < 0$

12. Find the area bounded by the curves $f(x) = x$ and $g(x) = x^2 - 2x$.

A) $\frac{1}{2}$
B) $\frac{3}{2}$
C) $\frac{5}{2}$
D) $\frac{7}{2}$
E) $\frac{9}{2}$

13. $y = xe^x$. Find $y''$ at $x = 1$

A) $43$
B) $2e$
C) $3e$
D) $4e$
E) $5e$

14. Solve $e^{2t} - 4e^t + 4 = 0$ for $t$.

A) $\ln 1$
B) $\ln 2$
C) $\ln 3$
D) $\ln 4$
E) $\ln 5$

15. Find the accumulated amount after $5$ years if \$ $40,000$ is invested at $12%$ per year compounded quarterly.

A) $40000(1.03)^{20}$
B) $40000(1.12)^{20}$
C) $40000(1.12)^{5}$
D) $40000(1.03)^{5}$
E) $40000(1.03)^{-20}$

16. Solve $2^{x^2 + 1} = \frac{1}{4^x}$ for $x$.

A) $-2$
B) $-1$
C) $0$
D) $1$
E) $2$

17. Find the derivative of $y = x \ln x - x$.

A) $\frac{1}{x} - 1$
B) $0$
C) $x \ln x - 1$
D) $\ln x$
E) $\ln x - 1$

18. Evaluate $\int \frac{x^2 - x}{x} dx$

A) $\frac{\frac{x^3}{3} - x}{\frac{x^2}{2}} + C$
B) $x^{-2} + x + C$
C) $\frac{x^2 - 1}{x} + C$
D) $\frac{x^2}{2} - \ln x + C$
E) $\frac{x^2}{2} - x + C$

19. Evaluate $\int \frac{x}{\sqrt{1 + x^2}} dx$.

A) $\frac{x^2}{2\sqrt{1 + x^2}} + C$
B) $\frac{(1 + x^2)^{2/3}}{3} + C$
C) $\frac{2(1 + x^2)^{3/2}}{3} + C$
D) $2\sqrt{1 + x^2} + C$
E) $\sqrt{1 + x^2} + C$

20. Evaluate $\lim_{x \to 1^-} \frac{x^2 - 3}{x^2 - x}$.

A) $+\infty$
B) $-\infty$
C) $-\frac{2}{0}$
D) DNE
E) $1$

21. $f(x, y) = 3x^2 - 4xy + y^2 + 4y + 5$. Determine the relative extrema.

A) $(-4, 6)$ saddle.
B) $(-4, 6)$ min.
C) $(-4, 6)$ max.
D) $(4, 6)$ saddle.
E) $(4, 6)$ min.
F) $(4, 6)$ max.

22. Which of the following approximates the level curve set for $f(x, y) = x + y$ for $z = -2, -1, 0, 1, 2$.

A) !!q22-1.png!!
B) !!q22-2.png!!
C) !!q22-3.png!!
D) !!q22-4.png!!
E) !!q22-5.png!!

23. Find the minimum of the function $f(x, y) = 2xy$ subject to the constraint $2x - 3y - 6 = 0$

A) $-4$
B) $-3$
C) $-2$
D) $-1$
E) $0$

24. Find the critical point(s) for $f(x, y) = 2y^3 - 3y^2 - 12y + 2x^2 - 8x + 2$.

A) $(-2, 2), (-2, 1)$
B) $(-2, 2), (2, 1)$
C) $(2, 2), (2, 1)$
D) $(-2, -2), (2, 1)$
E) $(2, 2), (2, -1)$

25. Evaluate the double integral where $f(x, y) = xy$ over the region $0 \le x \le 2$ and $0 \le y \le 3$.

A) $0$
B) $3$
C) $6$
D) $9$
E) $12$