In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
a. Plot the function y=f(x) together with its derivative over the given interval. Explain why you know that f is one-to-one over the interval.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

a. x² + √x

In Exercises 41–44:

a. Find f⁻¹(x).

41. f(x) = 2x + 3, a = −1

In Exercises 41–44:

a. Find f⁻¹(x).

42. f(x) = (x + 2) / (1 − x), a = 1/2

In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.

1. 2y' + 3y = e^(-x)

a. y = e^(-x)

155. Which is bigger, πᵉ or e^π?

Calculators have taken some of the mystery out of this once-challenging question.

(Go ahead and check; you will see that it is a very close call.)

You can answer the question without a calculator, though.

a. Find an equation for the line through the origin tangent to the graph of

y = ln(x).

Evaluate the integrals in Exercises 67–74 in terms of

a. inverse hyperbolic functions.

69. ∫(from 5/4 to 2)dx/(1-x²)