Exam Center

AI-Generated Content

This exam was created by a large language model. Please use it with caution, as inaccuracies may be present. Always verify critical information.

What does $\lim_{x \to c} f(x) = L$ mean?

Select one option

The value of $f(x)$ is $L$ when $x = c$ .

The value of $f(x)$ approaches $L$ as $x$ approaches $c$ .

The function $f(x)$ is undefined at $x = c$ .

The derivative of $f(x)$ at $x = c$ is $L$ .

What is the domain of the function $f(x) = \sqrt{x-1}$ ?

Select one option

$x > 1$

$x < 1$

$x \le 1$

$x \ge 1$

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

Select one option

$x = 1$

$x = -1$

$y = 0$

$y = 2$

Which of the following is equivalent to $\ln x + 2\ln y$ ?

Select one option

$\ln(x+2y)$

$\ln(xy^2)$

$(\ln x)(\ln y)^2$

$\ln(x/y^2)$

Find the derivative of $y = \ln(2x)$ .

Select one option

$\frac{1}{x}$

$\frac{2}{x}$

$\frac{1}{2x}$

$2x$

If $\mathbf{a} = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ -2 \\ 2 \end{pmatrix}$ , what is $\mathbf{a} + \mathbf{b}$ in component form?

Select one option

$\begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix}$

$\begin{pmatrix} -1 \\ 2 \\ -3 \end{pmatrix}$

$\begin{pmatrix} 2 \\ 0 \\ -2 \end{pmatrix}$

$\begin{pmatrix} 3 \\ 0 \\ -1 \end{pmatrix}$

Evaluate $\lim_{x \to 0} \frac{\sin x}{x} - 2$ .

Select one option

$1$

$0$

$-1$

Undefined

Which condition is NOT required for a function $f(x)$ to be continuous at $x=c$ ?

Select one option

$f(c)$ must be defined.

$\lim_{x \to c} f(x)$ must exist.

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

$f'(c)$ must exist.

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

Select one option

$12x^2 - 2x + 1$

$4x^2 - x + 1$

$12x^3 - 2x^2 + x$

$x^4 - x^3/3 + x^2/2 - 5x$

10.

Find the derivative of $f(x) = \cos(x^2)$ .

Select one option

$\sin(x^2)$

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

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

$2x \sin(x^2)$

11.

What is the equation of the tangent line to $y = x^2 - 4x + 5$ at $x = 1$ ?

Select one option

$y = 2x + 1$

$y = -2x + 1$

$y = -2x + 5$

$y = 2x - 3$

12.

The function $f(x) = x^3 - 3x^2 - 9x + 1$ has critical points at $x = -1$ and $x = 3$ . Which of the following represents a local maximum and local minimum?

Select one option

Local max at $x = -1$ , local min at $x = 3$ .

Local min at $x = -1$ , local max at $x = 3$ .

Both are local maxima.

Both are local minima.

13.

Evaluate $\int x^2 dx$ .

Select one option

$2x + C$

$x^3 + C$

$\frac{1}{3}x^3 + C$

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

14.

If $\int_1^5 f(x) dx = 7$ and $\int_1^3 f(x) dx = 3$ , what is $\int_3^5 f(x) dx$ ?

Select one option

$4$

$10$

$-4$

$7$

15.

What does the definite integral $\int_a^b f(x) dx$ represent graphically, assuming $f(x) \ge 0$ on $[a,b]$ ?

Select one option

The slope of the tangent line at $x=a$ .

The total volume under the curve.

The area between the curve $f(x)$ and the x-axis from $x=a$ to $x=b$ .

The perimeter of the region bounded by $f(x)$ .

16.

Find two positive numbers whose sum is $20$ and whose product is a maximum.

17.

Water is poured into a conical tank at a rate of $2 \text{ cm}^3/s$ . The tank's height is always equal to its diameter. How fast is the water level rising when the water is $4 \text{ cm}$ deep? (The volume of a cone is $V = \frac{1}{3}\pi r^2 h$ ).

18.

Sketch the graph of the function $f(x) = x^3 - 6x^2 + 9x$ . Clearly label all intercepts, local extrema, and inflection points.

19.

Given two vectors $\mathbf{u} = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix}$ .

a) Calculate the dot product $\mathbf{u} \cdot \mathbf{v}$ . b) Find the magnitude of vector $\mathbf{u}$ . c) Determine the angle $\theta$ between $\mathbf{u}$ and $\mathbf{v}$ (to the nearest degree).

20.

Solve the following equations for $x$ :

a) $2^{3x-1} = 16^x$ b) $\log_2(x-2) + \log_2(x) = 3$

21.

Find the area of the region bounded by the curves $y = x^2$ and $y = x+2$ .

22.

Prove the Product Rule for differentiation, which states that if $P(x) = f(x)g(x)$ , then $P'(x) = f'(x)g(x) + f(x)g'(x)$ . You must use the limit definition of the derivative.

23.

Prove the Pythagorean Identity $\sin^2\theta + \cos^2\theta = 1$ . You may use a right-angled triangle and its definition of trigonometric ratios.