Math 201 Sample Final Exam 3

Instructions

Please read each question carefully, show all work, and check afterwards that you have answered all of each question correctly.

Important: No books, calculators, cell phones, computers, laptops, or notes are allowed. You must show all your work to receive credit. Any crossed-out work will be disregarded (even if correct). Write one clear answer with a coherent derivation for each question.

You have 135 minutes to complete this exam. Good luck!

1. (16 points) Compute the derivative $f(x)$ for each of the functions below. You do not need to simplify your answer.

(a) (4 points)
$f(x) = \cos\left( \frac{x + 1}{x} \right)$

(b) (4 points)
$f(x) = \left[ \ln(2x) \right]^{\frac{\sqrt[3]{x} - 1}{x^2}}$

(c) (4 points)
$f(x) = \frac{x^2 - 5x^5 + x^7}{e^{2x}}$

(d) (4 points)
$f(x) = x \left( x^2 - 4 \right)$

2. (16 points) Find each integral and simplify your answer.

(a) (4 points)
$$\int \frac{e}{1 - \ln(2x)} \cdot \frac{1}{x} \, dx$$

(b) (4 points)
$$\int \left[ \cos^3(3x) \cos(3x) \right] dx$$

(c) (4 points)
$$\int x^{3x^2} \, dx$$

(d) (4 points)
$$\int \frac{1}{\sqrt{1 - x^2}} \, dx \quad \text{for} \quad x \in (-1,1)$$

3. (9 points) Find the limits, or state that the limit does not exist (justify your answer).

(a) (3 points)
$$\lim_{x \to +\infty} \left( \sqrt{x^2 + 3x} - \sqrt{x^2 - 2x} \right)$$

(b) (3 points)
$$\lim_{x \to 0^+} \left( \ln(x) - \ln(\sin x) \right)$$

(c) (3 points)
$$\lim_{x \to +\infty} \left( 1 + 2x \right)^{\frac{1}{2\ln x}}$$

4. (6 points) Let $f(x) = x^2 - 2x$.

(a) (3 points) Using the limit definition of the derivative, compute $f'(x)$ (no credit will be given for any other method).

(b) (3 points) Find an equation of the tangent line to the graph $y = x^2 - 2x$ at the point $(1, 1)$.

5. (8 points)

(a) (4 points)
Let
$$F(x) = \int_{\sin(x)}^{x^2} \sqrt{1 - t^3} \, dt$$
Find $F'(x)$.

(b) (4 points) Find an equation for the tangent line to the curve
$$e^{x^2 y} = 2x + 2y$$
at the point $(0, \frac{1}{2})$.

6. (4 points)

An object is moving along a hyperbola $xy = 12$. As it reaches the point $(6,2)$, the $y$-coordinate is decreasing at a rate of $12 \ \text{cm/sec}$. How fast is the $x$-coordinate of the point changing at that instant? Be sure to include units in your answer.

7. (6 points)

(a) (3 points) Let $f$ be a differentiable function at $x = a$. Write down the expression for the linearization $L(x)$ of the function $f$ at $x = a$.

(b) (3 points) Find an approximation for $\tan(0.01)$ using calculus.

8. (8 points)

(a) (4 points) State the Mean Value Theorem, including the hypotheses.

(b) (4 points) Suppose $f(x)$ is a differentiable function such that $f(1) = 2$ and $f'(x) \leq 5$ for all $x$. What is the largest possible value for $f(4)$? Justify your answer.

9. (8 points) Let

$$f(x) = \begin{cases} x, & x < 1 \\ 2, & x = 1 \\ x^2, & x > 1 \end{cases}$$

(a) (2 points) Sketch the graph of $y = f(x)$ for $x \in [0, 4]$.

(b) (2 points) Is $f(x)$ continuous at $x = 1$? Justify your answer.

(c) (4 points) For the function $f$ above, use a Riemann Sum to estimate
$$\int_{0}^{4} f(x) \, dx$$
using the Midpoint Rule with 4 subdivisions. You may leave your answer as a sum of unsimplified fractions.

10. (9 points)

A cylindrical can is made of two different materials: the side is made of a material that costs 1 dollar per square foot, while the top and bottom are made of a material that costs 2 dollars per square foot. If the total volume of the can must be $1 \ \text{ft}^3$, find the dimensions of the can that minimize cost. Justify your answer using calculus.