# Calculus Homework #8 Due 4-20

1. A spherical snowball is melting. Its radius is decreasing at 0.2 cm per hour when the radius is 15 cm. How fast is its volume decreasing at that time?

2. A  potter forms a piece of clay into a cylinder. As he rolls it, the length, $L$, of the cylinder increases and the radius, $r$, decreases. If the length of the cylinder is increasing at 0.1 cm per second, find the rate at which the radius is changing when the radius is 1 cm and the length is 5 cm.

3. A certain quantity of gas occupies a volume of 20cc at a pressure of 1 atmosphere. The gas expands without the addition of heat, so, for some constant $k$, its pressure, $P$, and volume, $V$, satisfy the equation $PV^{1.4}=k$. (a) Find the rate of change of pressure with respect to volume. (b) The volume is increasing at 2cc/min when the volume is 30cc. At that moment, is the pressure increasing or decreasing? How fast?

4. (a) A hemispherical bowl of radius 10cm contains water to a depth of $h$cm. Find the radius of the surface of the water as a function of $h$. (b) The water level drops at a rate of 0.1 cm per hour. At the what rate is the radius of the water decreasing when the depth is 5 cm?

# Calculus Homework #6 due 3-30

Find the derivative of each of the following functions.

1. $f(x)=\sin{x}\cdot{x^3}+e^x$

2. $f(x)=4e^{x}\tan{x}$

3. $f(x)=\sec{x}-\frac{1}{x^2}$

4. $f(x)=\sqrt{x^2-2x+1}+x^2$

# Calculus homework #5 due 3-23

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 changes 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?

# Calculus Homework Due 5-14-15

1. A rectangle has a vertex on the $x$-axis between 0 and 9, a vertex at the point (9,0), a vertex on the line $x=9$, and a vertex on the curve $y=\sqrt{x}$. Find the dimensions of the rectangle having the maximum possible area.

2. To get the best view of the Statue of Liberty you should be at the position where $\theta$ is a maximum. If the statue stands 92 meters high, including the pedestal, which is 46 meters high, how far from the base should you be?

# Calculus Homework Due 5-7-15

Evaluate each limit.

1. $\lim_{x \to 1}\frac{\ln{x}}{x^2-1}$

2. $\lim_{t \to\pi}\frac{\sin^2{t}}{t-\pi}$

3. $\lim_{x \to 0}\left ( \frac{1}{x}-\frac{1}{\sin{x}} \right )$

4. $\lim_{x \to 0^+}x^a\ln{x},\;a>0$

# Calculus homework due 4-23-15

Find the derivative of each of the following functions.

1. $f(x)=\sin\left(\frac{x^3+1}{e^x}\right)$

2. $f(x)=e^{\tan(4x)}$

3. $f(x)=\sec\left(\frac{x^3}{3}-x^2\right)$

4. $f(x)=\sqrt{x^4+x^2}+x^2$