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.) $\lim_{x \to 1} \frac{\ln x}{1 - x^2}$

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(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.

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

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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.) $\int \frac{x + 8}{x^3 - 4x^2 + 4x}dx$

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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!) $\int_1^5 f(x) dx$

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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.)

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(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.

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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.

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

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

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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.)

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(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.

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