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

Select one option

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

$[2, \infty)$

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

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

If $f(x) = 3x - 7$ , what is $f^{-1}(x)$ ?

Select one option

$\frac{x+7}{3}$

$\frac{x-7}{3}$

$7-3x$

$\frac{1}{3x-7}$

The graph of $y = f(x)$ is stretched vertically by a factor of 2, then shifted 3 units to the right, and 1 unit up. Which equation represents this transformation?

Select one option

$y = 2f(x-3) + 1$

$y = 2f(x+3) + 1$

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

$y = 2f(x-3) - 1$

Given the piecewise function $g(x) = \begin{cases} x^2+1 & \text{if } x < 0 \\ 2x-1 & \text{if } x \ge 0 \end{cases}$ , what is $g(-2) + g(3)$ ?

Select one option

-2

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

Select one option

$\frac{7\pi}{6}, \frac{11\pi}{6}$

$\frac{\pi}{6}, \frac{5\pi}{6}$

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

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

Given vectors $\mathbf{u} = \langle 3, -1 \rangle$ and $\mathbf{v} = \langle 2, 4 \rangle$ , what is the value of $\mathbf{u} \cdot \mathbf{v} + \|\mathbf{u}\|^2$ ?

Select one option

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

Select one option

Does not exist

Using the graph provided, what is $\lim_{x \to 2} f(x)$ ?

Select one option

Does not exist

For a function $f(x)$ to be continuous at $x=c$ , which of the following conditions MUST be met?

Select one option

$\lim_{x \to c} f(x)$ exists

$f(c)$ is defined

$\lim_{x \to c} f(x) = f(c)$

All of the above

10.

The derivative of a function $f(x)$ at a point $x=a$ is defined as:

Select one option

$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

$\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

$\lim_{x \to a} (f(x) - f(a))$

$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

11.

Analyze and sketch the graph of the rational function $f(x) = \frac{x-2}{x^2 - x - 6}$ . Include all intercepts, asymptotes (vertical and horizontal), and holes. Show your work.

12.

A surveyor is measuring the distance across a small lake. She stands at point A and observes two points B and C on the opposite side of the lake. The angle BAC is $60^\circ$ . She measures the distance from A to B as 120 meters and the distance from A to C as 150 meters. Calculate the distance between points B and C across the lake, rounded to two decimal places.

13.

An object is pulled by two forces. Force $\mathbf{F_1}$ has a magnitude of 50 N at an angle of $30^\circ$ to the positive x-axis. Force $\mathbf{F_2}$ has a magnitude of 70 N at an angle of $120^\circ$ to the positive x-axis. Find the magnitude and direction (angle with the positive x-axis) of the resultant force. Round your answers to one decimal place.

14.

Find the values of $a$ and $b$ that make the function $f(x)$ continuous everywhere: $f(x) = \begin{cases} ax+3 & \text{if } x < 1 \\ 5 & \text{if } x = 1 \\ x^2 - bx + 2 & \text{if } x > 1 \end{cases}$ .

15.

a) Use the limit definition of the derivative to find $f'(x)$ for $f(x) = x^2 - 3x$ . (5 points)\nb) Find the derivative of $g(x) = 4x^5 - \frac{2}{x^3} + \sqrt{x}$ using basic differentiation rules. Simplify your answer. (5 points)

16.

Estimate the area under the curve of $f(x) = x^2$ from $x=0$ to $x=4$ using four subintervals of equal width and right endpoints (R_4). Sketch the function and the rectangles used for the approximation.

17.

Prove the trigonometric identity: $\frac{\sin x}{1 + \cos x} + \frac{1 + \cos x}{\sin x} = 2\csc x$ .

18.

Using the limit definition of the derivative, prove that if $f(x) = c$ (where $c$ is a constant), then $f'(x) = 0$ .