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<t<20 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?

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Classwork 9-23

The speed that a sailboat is capable of sailing is determined by three factors: its total length L, the surface area A of its sails, and its displacement V. In general, a sailboat is capable of greater speed if it is longer, has a larger sail area, or displaces less water. To make sailing races fair, only boats in the same “class” can qualify to race together. For a certain race a boat is considered to qualify if 0.30L+0.38A^\frac{1}{2}-3V^\frac{1}{3}\leq 16 where L is measured in feet, A in square feet, and V in cubic feet. Use this inequality to answer the following questions:
1. A sailboat has length 60 feet, sail area 3400 square feet, and displacement 650 cubic feet. Does this boat qualify for the race?
2. A sailboat has length 65 feet and displaces 600 cubic feet. What is the largest possible sail area that could be used and still allow the boat to qualify for this race?

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<t<20 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?