1. What positive number exceeds its cube by the greatest amount? (A) $\frac{1}{3}$ (B) $\frac{1}{2}$ (C) $\frac{\sqrt{3}}{3}$ (D) $\sqrt{3}$ (E) None of the above 2. Evaluate: $\lim _{x \rightarrow 0} \frac{\csc x}{\csc 2 x}$ (A) 0 (B) $\frac{1}{2}$ (C) 1 (D) 2 (E) None of the above 3. Which of the following is an odd function? (A) $\frac{x^3+x}{\sin x}$ (B) $\sqrt{x^2}$ (C) $\cos x$ (D) $3 x+1$ (E) None of the above 4. Find the positive number $A$, such that the logarithmic function $y=\log _A x$ intersects its inverse function at only one point. (A) $\frac{1}{\mathrm{e}}$ (B) $\mathrm{e}^{1 / e}$ (C) $\mathrm{e}$ (D) $e^e$ (E) None of the above 5. A point moves around the circle $x^2+y^2=36$. When the point is at $(2 \sqrt{6},-2 \sqrt{3})$, its abscissa is increasing at a rate of 12 units per second. At what rate, in units per second, is the ordinate changing at that instant? (A) $-12 \sqrt{2}$ (B) $-6 \sqrt{2}$ (C) $6 \sqrt{2}$ (D) $12 \sqrt{2}$ (E) None of the above 6. For what value of $k$ will the graph of the function $g(x)=x^3-6 x^2+k x-9$ be symmetric about the point $(2,1)$ ? (A) $-2$ (B) 4 (C) 12 (D) there is no such $\mathbf{k}$ (E) None of the above 7. If $f(x)=\left(-x^4+4 x^3-2 x^2-4 x+3\right)^{-1 / 2}$, find the domain of $f(x)$. (A) $\{x:-1<x<1$ or $x>3\}$ (B) $\{\mathrm{x}: \mathrm{x}<-1$ or $1<x<3\}$ (C) $\{\mathrm{x}:-1<\mathrm{x}<3\}$ (D) $\{x: x<-1$ or $x>3\}$ (E) None of the above 8. Evaluate (where a and $b$ are positive real numbers): $\lim _{x \rightarrow 0} e^{\left(\frac{a^x-b^x}{x}\right)}$ (A) 0 (B) 1 (C) $\ln a-\ln b$ (D) $\frac{a}{b}$ (E) None of the above 9. Find the slope of the line tangent to the graph of $(x+2)^2+(y-3)^2=25$ at the point $(1,7)$. (A) $-\frac{4}{3}$ (B) $-\frac{3}{4}$ (C) $\frac{3}{4}$ (D) $\frac{4}{3}$ (E) None of the above 10. Find the sum of $a$ and $b$ given that $f(x)$ is differentiable at $x=3$. $f(x)=\left\{\begin{array}{l}5-a x \text { if } x \leq 3 \\ b x^2-4 \text { if } x>3\end{array}\right.$ (A) $-3$ (B) $\frac{5}{3}$ (C) 5 (D) 6 (E) None of the above 11. What is the $n$th derivative, $f^n(x)$, of the function $y=x^n$ ? (A) $n$ (B) $n^2$ (C) $(n-1)$ ! (D) $n$ ! (E) None of the above 12. Using differentials, approximate $\sqrt[5]{240}$ as a rational number. (A) $2 \frac{99}{100}$ (B) $2 \frac{134}{135}$ (C) $2 \frac{404}{405}$ (D) $3 \frac{1}{135}$ (E) None of the above 13. What is the amplitude of the function $f(x)=A \cos 2 x+B \sin 2 x$ where $A$ and $B$ are non-zero real numbers? (A) $A+B$ (B) $|A-B|$ (C) $\sqrt{A^2+B^2}$ (D) $2 \sqrt{A^2+B^2}$ (E) None of the above 14. Which of the following is equal to $\mathrm{e}$ ? I. $\sum_{n=1}^{\infty} \frac{1}{n !}$ II. $2.718281828$ III. $\lim _{x \rightarrow 0}(\cos x+x)^{1 / x}$ (A) I only (B) I and II only (C) I and III only (D) I, II and III (E) None of the above 15. If $f(x)=4 x, g(x)=\cos x, h(x)=x^3$ and $k(x)=h(g(f(x)))$, find $k^{\prime}(x)$. (A) $-192 x^2 \sin 64 x^3$ (B) $-3 \cos ^2 4 x \sin 4 x$ (C) $-6 \sin 8 x \cos 4 x$ (D) $12 \cos ^2 4 x \sin 4 x$ (E) None of the above 16. Find the $y$-intercept of the line normal to the graph of $y=\sqrt{1+x^3}+2$ at the point $(2,5)$. (A) 3 (B) 4 (C) 6 (D) 9 (E) None of the above 17. Find $h^{\prime}(3)$ if $h(x)=3^{x^2-2 x}$. (A) $81 \ln 3$ (B) $108 \ln 3$ (C) 108 (D) $\frac{108}{\ln 3}$ (E) None of the above 18. Given $x^2+y^2=1$, find $\frac{d^2 x}{d y^2}$ in simplest form. (A) $\frac{1}{x^3}$ (B) $-\frac{1}{x^3}$ (C) $\frac{1}{y^3}$ (D) $-\frac{1}{y^3}$ (E) None of the above 19. If $f^{\prime\prime}(x)$, the second derivative of $f(x)$, equals $(x+1)(x-1)^2(x-3)$, then which of the following is true? (A) $f(x)$ has 2 inflection points and is concave up on the intervals $(-1,1)$ and $(3,+\infty)$. (B) $f(x)$ has 2 inflection points and is concave up on the intervals $(-\infty,-1)$ and $(3,+\infty)$. (C) $f(x)$ has 3 inflection points and is concave up on the intervals $(-1,1)$ and $(3,+\infty)$. (D) $f(x)$ has 3 inflection points and is concave up on the intervals $(-\infty,-1)$ and $(3,+\infty)$. (E) None of the above 20. Evaluate: $\int\left(3 t^2-12 t+12\right) d t$ (A) $(\mathrm{t}-2)^3+\mathrm{C}$ (B) $6 \mathrm{t}-12+\mathrm{C}$ (C) $\mathrm{t}^3-6 \mathrm{t}^2+6 \mathrm{t}+\mathrm{C}$ (D) $-t^3+6 t^2-12 t+C$ (E) None of the above 21. A particle is moving along the $\mathrm{X}$-axis according to the equation $x=3 t^2-t^3$ for $t \geq 0$ where $t$ is the time in seconds. When is the speed of the particle increasing? $\begin{array}{ll}\text { (A) }\{\mathrm{t}: 0 \leq t<1 \text { or } t \geq 2\} & \text { (B) }\{t: 0 \leq t<1\}\end{array}$ (C) $\{t: 0<t<1\}$ (D) $\{t: 1<t<2\}$ (E) None of the above 22. Using Rolle's Theorem, it can be demonstrated that the equation $x^3+12 x+C=0$, where $C$ is any real number, has at most how many real roots? (A) 0 (B) 1 (C) 2 (D) 3 (E) None of the above 23. Find the area of the region bounded by the curve $y=\sin 2 x$, the lines $x=\frac{\pi}{3}$ and $x=\frac{\pi}{2}$ and the $X$ - axis. (A) $\frac{3}{4}$ (B) $\frac{\sqrt{3}}{2}$ (C) $\frac{1}{4}$ (D) $\frac{3}{2}$ (E) None of the above 24. Find the coordinates of the point of intersection of the asymptotes of the function $\mathrm{y}=\frac{3 \mathrm{x}^2-13 \mathrm{x}+16}{\mathrm{x}-2}$. (A) $(2,-1)$ (B) $(2,0)$ (C) $(2,2)$ (D) no such point exists (E) None of the above 25. Find $\frac{d y}{d x}$ if $y=\ln \left|\frac{5 x^2}{x+1}\right|$ for $x \neq 0,-1$. (A) $\frac{x+2}{x^2+x}$ (B) $\frac{3 x+2}{x^2+x}$ (C) $\frac{5 x+10}{x^2+x}$ (D) $\frac{5 x^2+10 x}{x+1}$ (E) None of the above 26. Find the height, in inches, of the right circular cylinder of greatest lateral surface area that can be inscribed in a sphere of radius 12 inches. (A) $6 \sqrt{2}$ (B) $12 \sqrt{2}$ (C) $18 \sqrt{2}$ (D) $24 \sqrt{2}$ (E) None of the above 27. The edges of a regular tetrahedron are shrinking at a constant rate of $1 \mathrm{in} / \mathrm{sec}$. How fast, in in $^3 / \mathrm{sec}$, is the volume decreasing when the surface area of the tetrahedron is $4 \sqrt{3} \mathrm{in}^2$ ? (The volume of a regular tetrahedron with an edge of length $s$ is $\frac{s^3 \sqrt{2}}{12}$.) (A) $\sqrt{2}$ (B) $\sqrt{3}$ (C) $4 \sqrt{2}$ (D) $4 \sqrt{3}$ (E) None of the above 28. Evaluate: $\lim _{x \rightarrow 0} \frac{1+2^{1 / x}}{3+4^{1 / x}}$ (A) 0 (B) $\frac{1}{2}$ (C) $\frac{1}{3}$ (D) $\frac{1}{\ln 2}$ (E) None of the above 29. Find $f^{\prime\prime}(\sqrt{3})$ if $f(x)=x \operatorname{Arccot} \frac{1}{x}-\ln \sqrt{1+x^2}$ for $x>0$. (A) $-\frac{3}{4}$ (B) $\frac{1}{4}$ (C) $\frac{\pi}{6}$ (D) $\frac{\pi}{3}$ (E) None of the above 30. If $f(x)=\frac{1}{1+e^{2 x}}$, find the average value of $f(x)$ on the interval $[0,1]$. (A) $1+\ln \sqrt{\frac{2}{1+e^2}}$ (B) $\ln \sqrt{\frac{1+\mathrm{e}^2}{2 \mathrm{e}^2}}$ (C) $\frac{1-\mathrm{e}^2}{2+2 \mathrm{e}^2}$ (D) $\arctan e-\frac{\pi}{4}$ (E) None of the above