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Find the derivative of each of the following functions.

1.

2.

3.

4.

There are 14 questions, broken down as follows.

3 graphing rational functions

2 simplifying logs

3 log equations

2 exponential equations

1 inverse function

1 graphing exponential function

1 graphing log function

1 write the equation of an exponential function given certain conditions

Here are some good review questions.

- Solve for :
- Solve for and express your solution as a ratio of natural logs:
- Sketch the graph of
- Sketch the graph of
- Sketch the graph of
- has a horizontal asymptote at . Where is the asymptote for ?
- . Write the equation for .
- Solve for and express your solution as a ratio of natural logs: .
- Re-write as an exponential equation: .
- Sketch the graph of

Also study quizzes 2, 3, and 4.

For each equation below, determine whether it is true for all possible values of the variables or not and explain your answer. Ignore values of the variables for which any term is undefined.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

1. When an electric current passes through two resistors with resistance and , connected in parallel, the combined resistance, , can be calculated from the equation . Find the rate at which the combined resistance changes with respect to changes in . Assume that 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 , the balance in the year 2020, depends on the year, , in which the painting is sold and the sale price . If is measured from the year 2000 so that then .

a. Explain why is given by this formula.

b. Show that the formula for is equivalent to .

c. Find , given that and .

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

a. Let be the distance the same car goes on one liter of gas at velocity . What is the relationship between and ? Find and .

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

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

One member of your group is a zombie. How long will it take everyone in New York City to become a zombie? Will it take 28 days? How long before everyone in the world is a zombie? Find a function that will model this and express it graphically and algebraically.

While in chem lab, you find a sandwich behind some beakers. You’re very hungry, having missed lunch, and are thinking about eating this sandwich that you found, but you don’t know how long it’s been there. Using a handy Geiger counter, you find that the sandwich has 98% of its carbon-14 remaining. Would you eat the sandwich? Explain your answer algebraically.