1. (8 points) Consider continuous functions $f$ and $f'$ (where $f'$ denotes the derivative of $f$) with values given by the following table:

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

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

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

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

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.

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

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

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

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

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

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

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 $y = 2 - x$.

9. (6 points) Determine whether each of the following sequences is convergent or divergent. Find the limit of the convergent sequences.

a. $a_n = \frac{e^{2n}}{\sqrt{n}}$

a. $a_n = \frac{e^{2n}}{\sqrt{n}}$

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

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

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)

a. $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n + 4}}$ (integral test)

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

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

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

12. (6 points) Find the sum of the following convergent series.

a. $\sum_{n=1}^{\infty} \frac{3^n}{5^{n+1}}$

a. $\sum_{n=1}^{\infty} \frac{3^n}{5^{n+1}}$

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

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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