MATH 2B

Final - Practice 1 | General

1. (8 points) Consider continuous functions $f$ and $f'$ (where $f'$ denotes the derivative of $f$) with values given by the following table:
x 0 1 2 3 4 5
$f(x)$ 3 4 6 9 13 18
$f'(x)$ 1 2 4 6 7 5

a. Find $\int_0^4 f'(x) \; dx$

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b. Estimate $\int_1^4 f(x) \; dx$ using a left-hand Riemann sum with $3$ equal subintervals.

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c. Evaluate the following derivative at the point $x = 3$ $\frac{d}{dx} \left( \int_2^x f(t) \; dt \right)$

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d. Suppose $f(x)$ gives the height of a rocket, measured in yards, $x$ minutes after its launch. What are the units of $\int_0^4 f'(x) \; dx$ and what does this quantity represent.

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2. (7 points) Evaluate $\int \frac{x}{1 + x^4} \; dx$

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3. (7 points) Evaluate $\int \frac{x^2}{e^{2x}} \; dx$

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4. (7 points) Evaluate $\int \sin^3(4t) \; dt$

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5. (7 points) Evaluate the following integral by making an appropriate trigonometric substitution. $\int \frac{dx}{x^2\sqrt{x^2 - 9}}$

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(8 points) Determine whether the following integral is converget or divergent. Evaluate the integral if it is convergent. If it is divergent, explain why. $\int_0^{\infty} \frac{dz}{z^2 + 3z + 2}$

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7. (10 points) Find the area of the region bounded by the curves $y = \frac{3}{2} - \frac{x^2}{2}$ and $y = |x|$.

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8. (10 points) Find the volume of the solid obtained by rotating about the $x$-axis the region bounded by the curves $y = \sqrt{4 - x^2}$ and $y = 2 - x$.

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9. (6 points) Determine whether each of the following sequences is convergent or divergent. Find the limit of the convergent sequences.
a. $a_n = \frac{e^{2n}}{\sqrt{n}}$

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b. $a_n = \frac{(-1)^n}{n!}$

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

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10. (6 points) Compute the arc length of the curve $y = \ln(\cos(x))$ over the interval $[0, \frac{\pi}{4}]$. (Hint: $\int \sec(x) \; dx = \ln |\sec(x) + \tan(x)| + C$.)

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11. (12 points) Use the indicated test to determine whether the given series is convergent or divergent.
a. $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n + 4}}$ (integral test)

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b. $\sum_{n=1}^{\infty} \frac{100^n}{n!}$ (ratio test)

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c. $\sum_{n=1}^{\infty} \frac{(-1)^n \sqrt{n}}{2n + 5}$ (alternating series test)

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d. $\sum_{n=2}^{\infty} \frac{n^2}{n^3 - 1}$ (comparison test or limit comparison test)

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12. (6 points) Find the sum of the following convergent series.
a. $\sum_{n=1}^{\infty} \frac{3^n}{5^{n+1}}$

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b. $\sum_{n=1}^{\infty} \frac{1}{n(n+3)}$

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13. (6 points) Find a power series representation for the function $f(x) = \frac{2}{3 - x}$ and determine the interval of convergence.

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