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Problem 7.7.41
Evaluate the integrals in Exercises 41–60.
41. ∫sinh(2x)dx
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 7.3.107
Evaluate the integrals in Exercises 97–110.
107. ∫₀⁹ (2 log₁₀(x + 1) / (x + 1)) dx
Problem 7.1.31
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) = (x + 3) / (x − 2)
Problem 7.2.66
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
66. y = θsin(θ)/√(sec(θ))
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 7.4.12
Solve the differential equation in Exercises 9–22.
12. (dy/dx) = 3x²e^(-y)
Problem 7.2.35
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
35. y = ln((x²+1)^5/√(1-x))
Problem 7.5.37
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
37. lim (y → 0) (√(5y + 25) - 5) / y
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 7.7.4
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 7.3.103
Evaluate the integrals in Exercises 97–110.
103. ∫₁⁴ (ln 2 · log₂x / x) dx
Problem 7.3.122
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
122. y = (ln x)^(ln x)
Problem 7.3.49
Evaluate the integrals in Exercises 33–54.
49. ∫ e^(sec πt) sec πt tan πt dt
Problem 7.3.109
Evaluate the integrals in Exercises 97–110.
109. ∫ (dx / (x log₁₀x))
Problem 7.6.84
Evaluate the integrals in Exercises 77–90.
84. ∫(from 2 to 4)2dx/(x²-6x+10)
Problem 7.1.35
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) = (x + b) / (x − 2), b > −2 and constant
Problem 7.6.130
130. Use the identity arccot(u)=π/2 - arctan(u) to derive the formula for the derivative of arccot(u) in Table 7.4 from the formula for the derivative of arctan(u).
Problem 7.3.85
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
85. y = log₂(8t^(ln 2))
Problem 7.5.90
90. Find f'(0) for
f(x) = e^(-1/x²), x≠0
= 0, x = 0.
Problem 7.1.51
Let f(x) = x³ − 3x² − 1, x ≥ 2. Find the value of df⁻¹/dx at the point x = −1 = f(3).
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 7.3.95
Evaluate the integrals in Exercises 87–96.
95. ∫₂⁴ x^(2x) (1 + ln x) dx
Problem 7.3.73
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
73. y = log₄ x + log₄ x²
Problem 7.7.25
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
25. y = sinh⁻¹(√x)
Problem 7.6.47
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
47. y=(arccot(x³))³
Problem 7.6.77
Evaluate the integrals in Exercises 77–90.
77. ∫dx/√(-x²+4x-3)
Problem 7.2.37
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
37. ∫(from x²/2 to x²)ln(√t)dt
Problem 7.3.143
143.
b. Find the average value of ln(x) over [1, e].
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)