Math 201 Sample Final Exam 1

Instructions:

This exam contains 10 questions. Students must solve all 10 questions. Total: 100 points. No calculators or electronic devices allowed. Turn off all sound-producing devices. You must show all your work to receive credit. Good luck!

Question 1 (15 points)

Compute $\frac{dy}{dx}$ for each of the functions below. Simplify your answer.

(a) (5 points) $y = \left(x + \frac{1}{x}\right)^9$

(b) (5 points) $y = \frac{x^5}{x}$ — Your final answer should be in terms of $x$ only.

(c) (5 points) $x^4 y^3 - x^5 = y^6 - 10$

Question 2 (12 points)

Evaluate each integral and simplify your answer.

(a) (4 points)
$$\int \frac{3 + x^2}{\sqrt{x}} \, dx$$

(b) (4 points)
$$\int_{1}^{e} \frac{\ln(x)}{3x} \, dx$$

(c) (4 points)
$$\int \frac{\sin(x)}{\sqrt{1 + \cos(x)}} \, dx$$

Question 3 (12 points)

Find the limit or state that the limit does not exist. Justify your answer.

(a) (4 points)
$$\lim_{x \to 6} \frac{\sqrt{10 - x} - 2}{x - 6}$$

(b) (4 points)
$$\lim_{x \to 0} \frac{x e^x}{e^x - 1}$$

(c) (4 points)
$$\lim_{x \to \infty} \frac{x + 5}{\sqrt{9x^2 - 1}}$$

Question 4 (10 points)

Let $f(x) = \sqrt{x}$.

(a) (6 points) Use the limit definition of derivative to find $f'(x)$. (No credit for other methods.)

(b) (4 points) Use differentials (or linear approximations) to estimate $f(4.2)$. Show your work and simplify.

Question 5 (10 points)

(a) (5 points)
Let
$$F(x) = \int_{1}^{2x} \frac{\ln(t+1)}{t} \, dt$$
Using the Fundamental Theorem of Calculus, Part I, find $F'(x)$.

(b) (5 points)
Find the value of $c$ that makes $f$ continuous if:
$$f(x) = \begin{cases} cx^2 + 2x & \text{if } x < 2 \\ x^3 - cx & \text{if } x \geq 2 \end{cases}$$

Question 6 (6 points)

If the surface area of a cube increases at a rate of 100 ft²/min, at what rate is the cube's volume changing when the edge length is 3 ft? Include correct units in your answer.

Question 7 (9 points)

(a) (5 points)
Determine the point $(x, y)$ on the curve
$$y = 2 + \sqrt{x}$$
that is closest to the point $\left(\frac{5}{2}, 2\right)$.

(b) (4 points)
Find the absolute extrema of
$$f(x) = x^3 - 3x^2 + 1$$
on the interval $[-1, 3]$.

Question 8 (10 points)

Let
$$f(x) = \frac{x^2 - 2}{x^2 - 4}, \quad f'(x) = \frac{-4x}{(x^2 - 4)^2}, \quad f''(x) = \frac{4(3x^2 + 4)}{(x^2 - 4)^3}$$

Find all significant features of $f(x)$; that is, find domain, $x$ and $y$ intercepts,limits and equations of all asymptotes, coordinates of all local maxima and minima, ntervals where $f$ is increasing/decreasing, coordinates of all inflection points, intervals of concavity. Then sketch the graph of $f(x)$.

Question 9 (6 points)

(a) (3 points)
Does there exist a function $f$ such that $f(0) = -1$, $f(2) = 4$, and $f'(x) \leq 2$ for all $x$?
(Hint: Use the Mean Value Theorem.)

(b) (3 points)
Use the Intermediate Value Theorem to show that there is a root of the equation
$$-3x = x^4 + 1$$
in the interval $(-2, -1)$.

Question 10 (10 points)

(a) (5 points)
Let $f$ be a continuous function. Give the definition of
$$\int_a^b f(x) \, dx$$
in terms of Riemann sums with $n$ equal subintervals.

(b) (5 points)
Use part (a) with 4 equal subintervals and right endpoints to estimate:
$$\int_0^1 x^4 \, dx$$
Simplify your answer. (No credit for other methods.)