# Calculus homework due 3-20-14

1. When an electric current passes through two resistors with resistance $r_1$ and $r_2$, connected in parallel, the combined resistance, $R$, can be calculated from the equation $\frac{1}{R}=\frac{1}{r_1}+\frac{1}{r_2}$. Find the rate at which the combined resistance chances with respect to changes in $r_1$. Assume that $r_2$ is constant.

2. A museum has decided to sell one of its paintings and invest the proceeds. If the picture is sold between the years 2000 and 2020 and the money from the sale is invested in a bank account earning 5% interest per year compounded annually, then $B(t)$, the balance in the year 2020, depends on the year, $t$, in which the painting is sold and the sale price $P(t)$. If $t$ is measured from the year 2000 so that $0 then $B(t)=P(t)1.05^{20-t}$.

a. Explain why $B(t)$ is given by this formula.

b. Show that the formula for $B(t)$ is equivalent to $B(t)=1.05^{20} \frac{P(t)}{1.05^t}$.

c. Find $B'(10)$, given that $P(10)=150,000$ and $P'(10)=5000$.

3. Let $f(v)$ be the gas consumption (in liters/km) of a car going at velocity $v$ (in km/hr). In other words, $f(v)$ tell you how many liters of gas the car uses to go one kilometer, if it is going at velocity $v$. You are told that $f(80)=0.05$ and $f'(80)=0.0005$.

a. Let $g(v)$ be the distance the same car goes on one liter of gas at velocity $v$. What is the relationship between $f(v)$ and $g(v)$? Find $g(80)$ and $g'(80)$.

b. Let $h(v)$ be the gas consumption in liters per hour. In other words, $h(v)$ tell you how many liters of gas the car uses in one hour if it is going at velocity $v$. What is the relationship between $h(v)$ and $f(v)$? Find $h(80)$ and $h'(80)$.

c. How would you explain the practical meaning of the values of these functions and their derivatives to a driver who knows no calculus?