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Which of the following statements about the function $f(x) = 4\sin(3x - \pi) + 2$ is true?

Select one option

The amplitude is 3.

The period is $2\pi/3$ .

The phase shift is $\pi$ to the right.

The vertical shift is 4 units up.

Simplify the expression: $\frac{\cos^2\theta - \sin^2\theta}{\tan\theta}$ .

Select one option

$\cot\theta$

$\frac{1 - 2\sin^2\theta}{\sin\theta\cos\theta}$

$\frac{\cos(2\theta)}{\tan\theta}$

$\frac{\cos(2\theta)\cos\theta}{\sin\theta}$

Solve the equation $2\cos x + 1 = 0$ for $x$ in the interval $[0, 2\pi)$ .

Select one option

$x = \frac{\pi}{3}, \frac{5\pi}{3}$

$x = \frac{2\pi}{3}, \frac{4\pi}{3}$

$x = \frac{4\pi}{3}, \frac{5\pi}{3}$

$x = \frac{\pi}{6}, \frac{11\pi}{6}$

Which of the following is the inverse of the function $f(x) = 3^{x-1} - 2$ ?

Select one option

$y = \log_3(x+2)+1$

$y = \log_3(x-2)+1$

$y = \log_3(x+2)-1$

$y = \log_3(x-2)-1$

What is the domain of the function $g(x) = \sqrt{x^2 - 9}$ ?

Select one option

$(-3, 3)$

$(-\infty, -3] \cup [3, \infty)$

$[-3, 3]$

$[-3, \infty)$

What is the horizontal asymptote of the rational function $h(x) = \frac{2x^2 + 3x - 1}{5x^3 - 7x + 2}$ ?

Select one option

$y = 2/5$

$y = 0$

There is no horizontal asymptote.

$x = 2/5$

Given vectors $\vec{u} = \langle 1, 2, 3 \rangle$ and $\vec{v} = \langle 3, -1, 4 \rangle$ , what is $\vec{u} + \vec{v}$ ?

Select one option

$\langle 4, 1, 7 \rangle$

$\langle -2, 3, -1 \rangle$

$\langle 3, -2, 12 \rangle$

$4\vec{i} + \vec{j} + 7\vec{k}$

If $\vec{a} = \langle 3, -4 \rangle$ and $\vec{b} = \langle 2, 1 \rangle$ , what is the dot product $\vec{a} \cdot \vec{b}$ ?

Select one option

$6$

$-4$

$5$

$\langle 5, -3 \rangle$

Evaluate $\lim_{x \to 0} \frac{\sin(5x)}{x}$ .

Select one option

$0$

$1$

$5$

Does not exist.

10.

For what value of $x$ is the function $f(x) = \frac{x^2 - 9}{x - 3}$ discontinuous?

Select one option

$x = -3$

$x = 0$

$x = 3$

The function is continuous everywhere.

11.

Evaluate $\lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^x$ .

Select one option

$1$

$2$

$e$

$e^2$

12.

Find the derivative of $f(x) = x^4 - 3x^2 + 5$ .

Select one option

$4x^3 - 6x + 5$

$4x^3 - 6x$

$\frac{x^5}{5} - x^3 + 5x$

$x^3 - 3x$

13.

If $g(x) = \cos(x^2)$ , what is $g'(x)$ ?

Select one option

$\sin(x^2) \cdot 2x$

$-2x\sin(x^2)$

$-\sin(x^2)$

$2x\cos(x^2)$

14.

Consider a function $f(x)$ where $f'(x) = (x-1)(x-3)$ . Which of the following describes the local extrema of $f(x)$ ?

Select one option

Local maximum at $x=1$ , local minimum at $x=3$ .

Local minimum at $x=1$ , local maximum at $x=3$ .

Local maximum at $x=1$ only.

Local minimum at $x=3$ only.

15.

Evaluate $\int e^{2x} dx$ .

Select one option

$e^{2x} + C$

$\frac{1}{2}e^{2x} + C$

$2e^{2x} + C$

$\ln|2x| + C$

16.

Evaluate $\int_0^1 (4x^3 - 2x) dx$ .

Select one option

$0$

$1$

$2$

$-1$

17.

What does the definite integral $\int_a^b f(x) dx$ represent?

Select one option

The slope of the tangent line to $f(x)$ at $x=a$ .

The rate of change of $f(x)$ at $x=b$ .

The total antiderivative of $f(x)$ .

The area under the curve $f(x)$ from $x=a$ to $x=b$ .

18.

Solve for $x$ : $e^{2x} = 9$ .

Select one option

$x = \frac{1}{2}\ln 9$

$x = \ln 3$

$x = 3$

$x = \frac{9}{2e}$

19.

For the function $f(x) = \frac{x}{x^2 - 4}$ , determine its domain, equations of all asymptotes, intercepts, and identify any holes in the graph. Do not sketch the graph.

20.

Solve the trigonometric equation $2\sin^2 x - 3\sin x + 1 = 0$ for $x$ in the interval $[0, 2\pi)$ . Show all steps.

21.

Given vectors $\vec{u} = \langle 1, 2, 3 \rangle$ and $\vec{v} = \langle 2, 1, 2 \rangle$ . (a) Find the angle between $\vec{u}$ and $\vec{v}$ (in radians or degrees). (b) Find the projection of $\vec{u}$ onto $\vec{v}$ , i.e., $\text{proj}_{\vec{v}}\vec{u}$ .

22.

A rectangular piece of cardboard is $24 \text{ cm}$ long and $16 \text{ cm}$ wide. Equal squares are cut from each corner and the sides are folded up to form an open box. Find the dimensions of the box that will maximize its volume. What is the maximum volume?

23.

A $10 \text{ m}$ ladder is leaning against a vertical wall. The base of the ladder is pulled away from the wall at a rate of $0.5 \text{ m/s}$ . How fast is the angle between the ladder and the ground changing when the base of the ladder is $6 \text{ m}$ from the wall?

24.

Find the area of the region bounded by the curves $y = x^2$ and $y = x + 2$ . Show all steps, including finding intersection points.

25.

Prove the trigonometric identity: $\frac{1 + \cos(2x)}{\sin(2x)} = \cot x$ . Show all steps and justify each one.

26.

Using the epsilon-delta definition of a limit, prove that $\lim_{x \to 2} x^2 = 4$ .