110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.

a. f(x) = 3^(-x), g(x) = 2^(-x)

Evaluate the integrals in Exercises 31–78.

55. ∫(from -2 to -1)e^(-(x+1)) dx

Evaluate the integrals in Exercises 31–78.

35. ∫sec²x e^(tan x)dx

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

11. y = 5x^(3.6)

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

3. y = (1/4)xe^(4x) - (1/16)e^(4x)

23. What roles do the functions e^x and ln(x) play in growth comparisons?

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).

82. Use the definitions of the hyperbolic functions to find each of the following limits.

a. lim(x→∞) tanh x

In Exercises 41–44:

a. Find f⁻¹(x).

41. f(x) = 2x + 3, a = −1

a. Show that h(x) = x³ / 4 and k(x) = (4x)^(1/3) are inverses of one another.

In Exercises 41–44:

a. Find f⁻¹(x).

43. f(x) = 5 − 4x, a = 1/2

5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

a. log_3(x)

88. Given that x>0, find the maximum value, if any, of

a. x^(1/x)

89. Use limits to find horizontal asymptotes for each function.

a. y = x tan(1/x)

[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.

141. a. y=arccos(cos x)

Verify the integration formulas in Exercises 37–40.

37. a. ∫sech(x)dx = tan⁻¹(sinh x) + C

78. Which one is correct, and which one is wrong? Give reasons for your answers.

a. lim (x → 0) (x² - 2x) / (x² - sin x) = lim (x → 0) (2x - 2) / (2x - cos x) = lim (x → 0) 2 / (2 + sin x) = 2 / (2 + 0) = 1

Find the volumes of the solids in Exercises 135 and 136.

135. The solid lies between planes perpendicular to the x-axis at x=-1 and x=1. The cross-sections perpendicular to the x-axis are

a. circles whose diameters stretch from the curve y=-1/√(1+x²) to the curve y=1/√(1+x²).

[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.

139. a. y=arctan(tan x)

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

1. a. arctan 1

6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

a. log_2(x²)

153. The linearization of 2ˣ

a. Find the linearization of f(x) = 2ˣ at x = 0. Then round its coefficients to two decimal places.

In Exercises 1–4, solve for t.

1. a. e^(-0.3t) = 27

Suppose that the function g and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.

Assuming the inverse function g^(-1) is differentiable, find the slope of g^(-1)(x) at

a. x=1

75. a. Find the open intervals on which the function is increasing and decreasing.

g(x) = x(ln x)²

21. a. Show that ln(x) grows slower as x→∞ than x^(1/n) for any positive integer n, even x^(1/1,000,000).

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

7. a. sec^(-1)(-√2)

77. Which one is correct, and which one is wrong? Give reasons for your answers.

a. lim (x → 3) (x - 3) / (x² - 3) = lim (x → 3) 1 / (2x) = 1/6

In Exercises 41–44:

a. Find f⁻¹(x).

42. f(x) = (x + 2) / (1 − x), a = 1/2

154. The linearization of log₃x

a. Find the linearization of

f(x) = log₃x at x = 3.

Then round its coefficients to two decimal places.