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)

Solve the initial value problems in Exercises 55–58.

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

Evaluate the integrals in Exercises 53–76.

61. ∫(from 0 to 2)dt/√(8+2t²)

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

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

69. lim (x → ∞) (√(9x + 1)) / (√(x + 1))

Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.

f(x)=x³-1

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)