Homework, due 2/23

1.

Find a function $f:(-\infty,\infty)\rightarrow(-\infty,\infty)$ so that f'(x) = 3f(x) and f(0) = 4.

2.

The rate at which a population of bacteria increases in proportional to the current population. If the initial population is 5 grams and the population after 3 days is 22 grams, what was the population after 1 day?

3.

A spring with mass 1 gram and spring constant K = 2 is stretched 5 cm and given an initial velocity of 1/100 cm per second. Find an equation for the spring's position.

4.

Use bisection to solve $\ln(x) + e^x - 4 = 0$. That is, find a value X so that $\ln(X) + e^X - 4 \in [-10^{-8}, 10^{-8}]$. How many iterations were needed?

5.

Use Newton's method to solve $\ln(x) + e^x - 4 = 0$. That is, find a value X so that $\ln(X) + e^X - 4 \in [-10^{-8}, 10^{-8}]$. How many iterations were needed?

6.

A spring with mass 1 gram and unknown spring constant K is stretched 5 cm and given an initial velocity of 1/100 cm per second. After 1 second it is stretched 5 cm. What is the value of K?

7.

A rubber ball is losing air at the rate of 6 cubic centimeters per second. When the volume is 1000 cubic centimeters, what is the rate of change of the radius with respect to time?

8.

Find the equations of the vertical and horizontal tangent lines to the ellipse x² + 2xy + 10y² + 4x + 2y = 19.