# Calculus Homework #12 Due 5-25

Calculate the following definite integrals

1. $\int_0^3 (x^2+4x+3)dx$

2. $\int_0^{\frac{\pi}{4}}\sin xdx$

3. $\int_2^5 (x^3-\pi x^2)dx$

4. $\int_1^2\frac{1+y^2}{y}dy$

# Calculus Homework #11 due 5-18

Given the graph of $f(x)$ above, calculate the following definite integrals.

1. $\int_{-3}^{-5}f(x)dx$
2. $\int_2^4f(x)dx$
3. $\int_{-3}^3f(x)dx$
4. $\int_3^2f(x)dx$

# Calculus Homework #10 Due 5-11

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?

3. You run a small furniture business. You sign a deal with a customer to deliver some number of chairs, the exact number to be determined by the customer later. The price will be $90 per chair up to 300 chairs, and above 300, the price will be reduced by$0.25 per chair (on the whole order) for every additional chair over 300 ordered. What is the maximum amount of money your company can make on this order?

4. For which positive number $x$ is $x^{\frac{1}{x}}$ largest?

# Calculus Homework #9 Due 5-4

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 #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 #7 due 4-6

Calculate $\frac{dy}{dx}$ for each of the following equations.

1. $yx=x^{\sin{x}}$
2. $\cos\ln{y}=e^x$
3. $\frac{x}{y}=\log_2{x}$
4. $y = \frac{\tan{x}\sqrt{x^{-5x+3}}}{(\log_3{x}+2)^2}$