I. Do the following graphs without the aid of a calculator or computer.

1. y = 3e^-2x

2. y = e^-xcos (x)

3. y = 1/(1+e^x)

4. y = (1/2) sin (3x + Pi)

5. y = e^{2 ln|x|}

6. y = x sin (x)

II. Do the following integrals. Take the derivative to check your answer.

1. Int 1/(x+1) dx

2. Int 1/sqrt(x+1) dx

3. Int(from e to e²) 1/(x ln x) dx

4. Int (e^x+e^-x)/(e^x-e^-x) dx

5. Int x² e^x dx

6. Int(from 0 to 4) 5/(3x+1) dx

III. Find all functions y(x) such that ...

1. y' = 5x/y

2. y' = sqrt(x)/2y

3. y' = 3y

4. y' = x(1+y)

5. y' = 3/4 sqrt(x) and y(0) = 10

6. y' = 3y and y(0) = 10

7. y'' = g and y(0) = 4, y(10) = 12 (g is a constant.)

8. y'' = sin x and y(0) = 1, y'(0) = 7

9. If radium has a half-life of 1620 years, what percentage of a sample will be left after 500 years?

10. If a sample of radioactive material decays at a rate of 10% per day, what is its half-life?

IV. Read about Taylor series if you need to.

1. Find the Taylor series of y=e^x2, centered about x=0. What is the radius of convergence?

2. Find the Taylor series of y=1/x centered about x=1. What is the radius of convergence?