MATH 2B

Final - Practice 2 | General

1. (6 points) Suppose that $\int_{-1}^{1} f(x) \; dx = 6, \int_1^4 f(x) \;dx = -2$ and $\int_{-1}^{1} h(x) \; dx = 9$. Use this information to compute the following.
a. $\int_4^1 6f(x) \; dx$

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b. $\int_{-1}^{1} [2f(x) + 3h(x)] dx$

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c. $\int_{-1}^{4} f(x) \; dx$

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2. (6 points)
a. Evaluate the following derivative $\frac{d}{dx} \int_{\sin(x)}^{x^2} t^3 \tan(t) \; dt$

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b. Let $r(t)$ be the rate at which the world's oil is consumed, where $t$ is measured in years starting at $t = 0$ representing January 1, 2000, and $r(t)$ is measured in barrels per year. What does $\int_0^{13} r(t) \; dt$ represent and what are its units?

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3. (6 points) Evaluate $\int x^2 \tan^{-1} x \; dx$

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4. (6 points) Evaluate $\int \frac{1}{x \ln (3x)} \; dx$

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5. (6 points) Evaluate $\int \sin^{5}(x) \cos^2(x) \; dx$

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6. (6 points) Evalute $\int \frac{\sqrt{x^2 - 25}}{x} dx$, where $x > 5$.

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7. (6 points) Determine whether each of the following improper integrals are convergent or divergent. Evaluate the integral if it is convergent.
a. $\int_0^2 \frac{1}{(x - 2)^2}dx$

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b. $\int_{-\infty}^{\infty} \frac{1}{1 + x^2}dx$

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8. (6 points)
a. Find the average value of the function $f(x) = \sec^2(x)$ on the interval $[0, \frac{\pi}{4}]$.

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b. Find the arc length of the curve given by $y = 2x^{3/2}$ from $x = 0$ to $x = 1$.

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9. (6 points) Find the first $5$ non-zero terms in the Maclaurin series for $f(x) = (1 - x)^{-2}$. Find the associated radius of convergence of this power series.

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10. (6 points) Determine whether each of the following sequences converges or diverges. If it converges, find the limit.
a. $a_n = (\frac{2}{3})^n + 3$

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b. $b_n = n^3e^{-n}$

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c. $c_n = \tan^{-1}(\ln(n))$

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11. (10 points) Find the area of the region(s) bounded by the curves $y = x^3$ and $y = 4x$.

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12. (10 points)
a. The region bounded by the curve $y = x^2 + 1$ and the line $y = -x + 3$ is revolved around the line $y = 5$ to generate a solid. Find the volume of that solid.

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b. Let $R$ be the region bounded by the curve $y = x^2 + 1$ and the line $y = -x + 3$. Find the volume of the solid with base $R$ and cross-sections perpendicular to the $x$-axis are squares.

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13. (10 points) Answer True or False to each of the following and briefly explain your answers.
a. True/False: We have $\int_0^5 |x^2 - 3x - 4| dx \ge 0$

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b. True/False: We have $\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!} \pi^{2k} = -1$.

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c. True/False: We have $\frac{d}{dx} \left( \int_0^{\pi/4} \cos(x) \; dx \right) = \frac{\sqrt{2} - 2}{2}$

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d. True/False: There is a positive integer $m$ such that $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{m-1} + \frac{1}{m} > 20$.

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14. (10 points) Determine whether each of the following series is convergent or divergent. Indicate test used.
a. $\sum_{n=1}^{\infty} \frac{n}{n^3 + 1}$

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b. $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n+1}}$

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c. $\sum_{n=1}^{\infty} \frac{n^2}{2^n}$

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d. $\sum_{n=1}^{\infty} n^2e^{-n^3}$

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e. $\sum_{n=1}^{\infty} \frac{1}{3^n - 1}$

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