Math 201 Sample Final Exam 3

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

1. (4 points)
   $$f(x) = \cos\left( \frac{x+1}{x} \right) \tag{1.1}$$

2. (4 points)
   $$f(x) = \left[ \ln(2x) \right]^{\frac{\sqrt[3]{x} - 1}{x^2}} \tag{1.2}$$

3. (4 points)
   $$f(x) = \frac{x^2 - 5x^5 + x^7}{e^{2x}} \tag{1.3}$$

4. (4 points)
   $$f(x) = x\left(x^2 - 4\right) \tag{1.4}$$

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

1. (4 points)
   $$\int \frac{e}{1 - \ln(2x)} \cdot \frac{1}{x} \, dx \tag{2.1}$$

2. (4 points)
   $$\int \left[ \cos^3(3x) \cos(3x) \right] dx \tag{2.2}$$

3. (4 points)
   $$\int x~2^{3x^2} \, dx \tag{2.3}$$

4. (4 points)
   $$\int \frac{1}{\sqrt{1 - x^2}} \, dx, \quad x \in (-1, 1) \tag{2.4}$$

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

1. (3 points)
   $$\lim_{x \to +\infty} \left( \sqrt{x^2 + 3x} - \sqrt{x^2 - 2x} \right) \tag{3.1}$$

2. (3 points)
   $$\lim_{x \to 0^+} \left( \ln(x) - \ln(\sin x) \right) \tag{3.2}$$

3. (3 points)
   $$\lim_{x \to +\infty} \left( 1 + 2x \right)^{\frac{1}{2\ln(x)}} \tag{3.3}$$

4. (6 points) Let

$$f(x) = x^2 - 2x \tag{4.0}$$

1. (3 points) Using the limit definition of the derivative, compute $f'(x)$. (No credit for other methods)

2. (3 points) Find an equation of the tangent line to the graph $y = x^2 - 2x$ at the point
   $$(1, 1) \tag{4.1}$$

5. (8 points)

1. (4 points) Let
   $$F(x) = \int_{\sin(x)}^{x^2} \sqrt{1 - t^3} \, dt \tag{5.1}$$
   Find $F'(x)$.

2. (4 points) Find an equation for the tangent line to the curve
   $$e^{x^2 y} = 2x + 2y \tag{5.2}$$
   at the point
   $$\left( 0, \frac{1}{2} \right) \tag{5.3}$$

6. (4 points)

An object is moving along a hyperbola
$$xy = 12 \tag{6.1}$$
As it reaches the point $(6, 2)$, the $y$-coordinate is decreasing at a rate of
$$\frac{dy}{dt} = 12 \, \text{cm/sec} \tag{6.2}$$
How fast is the $x$-coordinate changing at that instant? Include units in your answer.

7. (6 points)

1. (3 points) Let $f$ be a differentiable function at $x = a$. Write the expression for the linearization $L(x)$ of $f$ at $x = a$.

2. (3 points) Find an approximation for
   $$\tan(0.01) \tag{7.1}$$
   using calculus.

8. (8 points)

1. (4 points) State the Mean Value Theorem, including the hypotheses.

2. (4 points) Suppose $f(x)$ is differentiable and
   $$f(1) = 2, \quad f'(x) \le 5 \quad \text{for all } x \tag{8.1}$$
   What is the largest possible value for $f(4)$? Justify your answer.

9. (8 points) Let

$$f(x) = \begin{cases} x, & x < 1 \\ 2, & x = 1 \\ x^2, & x > 1 \end{cases} \tag{9.0}$$

1. (2 points) Sketch the graph of $y = f(x)$ for $x \in [0, 4]$.

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

3. (4 points) Use a Riemann Sum to estimate
   $$\int_{0}^{4} f(x) \, dx \tag{9.1}$$
   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 materials: the side costs $1/ft² and the top & bottom cost $2/ft².
If the total volume is
$$V = 1 \ \text{ft}^3 \tag{10.1}$$
find the dimensions of the can that minimize cost.
Note: Justify your answer using calculus.