Back
Problem 7.2.56
Evaluate the integrals in Exercises 39–56.
56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))
Problem 7.3.83
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
83. y = 3^(log₂ t)
Problem 7.7.4
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.
4. cosh x = 13/5, x>0
Problem 7.2.87
Solve the initial value problems in Exercises 87 and 88.
87. dy/dx = 1 + 1/x, y(1) = 3
Problem 7.3.109
Evaluate the integrals in Exercises 97–110.
109. ∫ (dx / (x log₁₀x))
Problem 7.2.52
Evaluate the integrals in Exercises 39–56.
52. ∫(from π/4 to π/2)cot(t)dt
Problem 7.6.11
Find the values in Exercises 9–12.
11. tan(arcsin(-1/2))
Problem 7.5.30
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
30. lim (θ → 0) ((1/2)^θ - 1) / θ
Problem 7.8.17
17. Show that √(10x+1) and √(x+1) grow at the same rate as x→∞ by showing that they both grow at the same rate as √x as x→∞.
Problem 7.3.61
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
61. y = 5√s
Problem 7.7.27
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
27. y = (1 - θ)tanh⁻¹(θ)
Problem 7.5.66
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
66. lim (x → 0⁺) x (ln x)²
Problem 7.1.29
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = 1/x², x > 0
Problem 7.6.63
Evaluate the integrals in Exercises 53–76.
63. ∫(from -1 to -√2/2)dy/(y√(4y²-1))
Problem 7.3.37
Evaluate the integrals in Exercises 33–54.
∫8e^(x+1) dx
Problem 7.6.37
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
37. y=s√(1-s²) + arccos(s)
Problem 7.8.7
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)
Problem 7.3.43
Evaluate the integrals in Exercises 33–54.
∫ 2t e^(-t²) dt
Problem 7.5.41
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
41. lim (x → 0⁺) (ln x)² / ln(sin x)
Problem 7.6.71
Evaluate the integrals in Exercises 53–76.
71. ∫(from -π/2 to π/2) 2cosθ dθ/(1+(sinθ)²)
Problem 7.3.130
130. Where does the periodic function f(x) = 2e^(sin(x/2)) take on its extreme values, and what are these values?
Problem 7.6.61
Evaluate the integrals in Exercises 53–76.
61. ∫(from 0 to 2)dt/√(8+2t²)
Problem 7.1.27
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x³ + 1
Problem 7.3.103
Evaluate the integrals in Exercises 97–110.
103. ∫₁⁴ (ln 2 · log₂x / x) dx
Problem 7.3.128
For Exercises 127 and 128 find a function f satisfying each equation.
128. f(x) = e² + ∫₁ˣ f(t) dt
Problem 7.6.84
Evaluate the integrals in Exercises 77–90.
84. ∫(from 2 to 4)2dx/(x²-6x+10)
Problem 7.3.17
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = cos(e^(-θ^2))
Problem 7.7.31
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
31. y = cos⁻¹(x) - x sech⁻¹(x)
Problem 7.6.53
Evaluate the integrals in Exercises 53–76.
53. ∫dx/√(9-x²)
Problem 7.4.12
Solve the differential equation in Exercises 9–22.
12. (dy/dx) = 3x²e^(-y)
