#### MATH 16B

##### Final | Winter '17
1. Exponential and Logarithmic Functions
(a) Find the limit> Show ALL work. (Hint: To use L'Hospital's Rule, be sure to verify "something" first.) $\lim_{x \to 1} \frac{\ln x}{1 - x^2}$

(b) Find the exponential function $y = Ce^{kt}$ that passes through the points $\left( 0, \frac{1}{2} \right)$ and $\left( \frac{1}{5}, \frac{e}{2} \right)$. Show ALL work.

2. Trigonometric Functions
(a) Use $u$-substitution to find the indefinite integral. Show ALL work. (Hint: Remember to use absolute values where appropriate.) $\int \frac{e^{-x} \sin e^{-x}}{\cos e^{-x}} dx$

3. Partial Fractions
Find the indefinite integral. Show ALL work. (Hint: 10 pts to find the partial fraction, 5 pts to compute the integral. Remember to use absolute values where appropriate.) $\int \frac{x + 8}{x^3 - 4x^2 + 4x}dx$

4. Numerical Integration

Use Simpson's Rule with $n = 4$ to find the approximate value of the definite integral. Show ALL work. (Hint: Pay attention to the limits!) $\int_1^5 f(x) dx$

5. Areas and Volumes

(a) Find the definite integral which represents the area of the shaded region. Do not evaluate the integral. Show ALL work.
(Hint: 4pts for finding all 4 vertices of the parallelogram, 6pts for finding the correct definite integral.)

(b) Find the definite integral which represents the volume of the solid obtained by rotating the shaded region about the $y$-axis. Do NOT evaluate the integral. Show ALL work.

6. Discrete Random Variables
The following table is the probability distributionf for a discrete random variable $X$.

(a) Determine the probability that $X$ takes on a value greater than or equal to $3$. Show ALL work.

(b) Find the center, or expected value $\mu = \mathbb{E}[X]$, of the data set. You do not need to simplify. Show ALL work.

(c) Assuming that the expected value is $\mu$, find the variance of $X$. Show all work.

7. Continuous Random Variables
(a) Find the expected value of a continuous random variable $X$ with probability density function $f(x) = e^{-x}$ over the interval $[0, \infty)$. Show ALL work.
(Hint: You may find a useful formula on the "Formulas" sheet. Try using $u = -x$. To use L'Hospital's Rule, be sure to verify "something" first.)

(b) Find the standard deviation of a continuous random variable $X$ with probability density function $f(x) = e^{-x}$ over the interval $[0, \infty)$. Show ALL work.