Solve the initial value problems in Exercises 55–58.


57. d²y/dx² = 2e^(−x),y(0) = 1,y′(0) = 0

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.

16. y = (ln x)³

Evaluate the integrals in Exercises 31–78.

71. ∫(from √2/3 to 2/3)dy/(|y|√(9y²-1))

7. Order the following functions from slowest growing to fastest growing as x→∞.

a. e^x

b. x^x

c. (ln x)^x

d. e^(x/2)

In Exercises 7–10, determine from its graph if the function is one-to-one.

f(x) = 1 - x/2, x ≤ 0

x/(x + 2), x > 0

Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.

2. sinh x = 4/3

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = (x^2 - 2x + 2)e^(x)