Quiz 3 Review

Problem 1

3 points

$$\int_2^\infty e^{-3x} dx$$$$= \lim_{t\to \infty} \int_2^t e^{-3x} dx$$$$= \lim_{t\to \infty} \left[ \frac{e^{-3x}}{-3} \right]_{2}^{t}$$$$= -\frac{1}{3} \lim_{t\to \infty} \left( e^{-3t} - e^{-6} \right)$$$$= -\frac{1}{3} \left( 0 - \frac{1}{e^6} \right)$$$$\text{Solution }= \frac{1}{3e^6}$$

Converges

Problem 2

3 points

$$\int_0^1 \frac{3}{x^-5} dx$$$$= \lim_{t\to 0^{+}} \int_t^1 3x^5 dx$$$$= \lim_{t\to 0^{+}} \left[ \frac{3x^{-4}}{-4} \right]_{t}^{1}$$$$= \lim_{t\to 0^{+}} \left( \frac{-3}{4} + \frac{3}{4t^4} \right)$$$$\text{Solution }= \infty$$

Diverges



In [3]:

    
%matplotlib inline

import numpy as np
import sympy as sp
from matplotlib import pyplot as plt

# f(x) = 1/x --> Diverges
f = lambda x: 3/(x**5)

sp.mpmath.plot([f], xlim=[0,5], ylim=[0,5], points=500)

Problem 3

4 points

$$\{a_n\} = \frac{n^4}{5n^4+3}$$$$= \lim_{n\to \infty} \frac{n^4}{5n^4+3}$$$$= \lim_{n\to \infty} \frac{\frac{n^4}{n^4}}{5\left(\frac{n^4}{n^4}\right) + 3\left(\frac{1}{n^4}\right)}$$$$= \lim_{n\to \infty} \frac{1}{5 + 3(0)}$$$$= \frac{1}{5}$$

Converges



In [ ]: