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What is the domain of the function $f(x) = \frac{\sqrt{x-2}}{x-5}$ ?

Select one option

$[2, \infty)$

$[2, 5) \cup (5, \infty)$

$(2, 5) \cup (5, \infty)$

$(-\infty, 5) \cup (5, \infty)$

Find the inverse function $f^{-1}(x)$ for $f(x) = 3x - 2$ .

Select one option

$\frac{x+2}{3}$

$\frac{x-2}{3}$

$3x+2$

$2x-3$

If the graph of $y = f(x)$ is stretched vertically by a factor of 2 and shifted 3 units left, what is the equation of the transformed graph?

Select one option

$y = 2f(x+3)$

$y = 2f(x-3)$

$y = \frac{1}{2}f(x+3)$

$y = f(2x+3)$

What is the exact value of $\tan\left(\frac{5\pi}{6}\right)$ ?

Select one option

$\frac{\sqrt{3}}{3}$

$-\frac{\sqrt{3}}{3}$

$\sqrt{3}$

$-\sqrt{3}$

Simplify the expression $\sin^2\theta + \cos^2\theta + \tan^2\theta$ .

Select one option

$1$

$\sec^2\theta$

$\csc^2\theta$

$\cot^2\theta$

Given vector $\mathbf{v} = \langle 3, -4 \rangle$ , what is its magnitude, $|\mathbf{v}|$ ?

Select one option

$1$

$5$

$7$

$25$

For vectors $\mathbf{u} = \langle 1, 2 \rangle$ and $\mathbf{v} = \langle -3, 4 \rangle$ , calculate the dot product $\mathbf{u} \cdot \mathbf{v}$ .

Select one option

$-11$

$5$

$11$

$10$

Evaluate $\lim_{x \to 2} (3x^2 - 5x + 1)$ .

Select one option

$1$

$3$

$5$

$7$

Evaluate $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$ .

Select one option

$0$

$3$

$6$

Does not exist

10.

At which point is the function $f(x) = \frac{x+1}{x-2}$ discontinuous?

Select one option

$x = -1$

$x = 0$

$x = 1$

$x = 2$

11.

For the function $f(x) = \frac{x-1}{x^2-4}$ : a) Determine the domain and range. b) Find all x-intercepts and y-intercepts. c) Identify all vertical and horizontal asymptotes. d) Discuss its continuity. e) Sketch the graph of $f(x)$ . (Sketch not graded on perfect artistic merit, but on correctly representing features)

12.

Solve the equation $2\cos^2 x + \sin x - 1 = 0$ for $x \in [0, 2\pi)$ . Show all your work.

13.

A boat wants to travel due North across a river. The river flows East at $3 \text{ m/s}$ . The boat can travel at $5 \text{ m/s}$ in still water. a) In what direction should the boat head relative to the shore to travel directly North? b) What is the boat's resultant speed relative to the shore? Show your vector diagrams and calculations.

14.

For the function $f(x) = x^3 - 6x^2 + 9x - 2$ : a) Find the equation of the tangent line to the curve at $x=0$ . b) Find the critical points and the intervals where the function is increasing or decreasing. Show all your work.

15.

Prove the trigonometric identity: $\sin(2x) = 2\sin x \cos x$ . Show all steps clearly.

16.

Using the limit definition of the derivative, prove that if $f(x) = \sin x$ , then $f'(x) = \cos x$ . Show all steps clearly.