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Problem 7.3.57
Solve the initial value problems in Exercises 55–58.
57. d²y/dx² = 2e^(−x), y(0) = 1, y′(0) = 0
Problem 7.6.49
For problems 49–52 use implicit differentiation to find dy/dx at the given point P.
49. 3arctan(x) + arcsin(y) = π/4; P(1, -1)
Problem 7.7.29
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
29. y = (1 - t)coth⁻¹(√t)
Problem 7.8.13
13. When is a polynomial f(x) of at most the order of a polynomial g(x) as x→∞? Give reasons for your answer.
Problem 7.3.41
Evaluate the integrals in Exercises 33–54.
∫ (e^(√r) / √r) dr
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 7.2.45
Evaluate the integrals in Exercises 39–56.
45. ∫(from 1 to 2)(2ln x)/x dx
Problem 7.6.51
For problems 49–52 use implicit differentiation to find dy/dx at the given point P.
51. y arccos(xy) = -3√2/4 π; P(1/2, -√2)
Problem 7.2.81
81. Find the lengths of the following curves.
a. y = (x²/8) - ln(x), 4≤x≤8
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 7.5.27
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
27. lim (x → (π/2)^-) (x - π/2) sec x
Problem 7.3.105
Evaluate the integrals in Exercises 97–110.
105. ∫₀² (log₂(x + 2) / (x + 2)) dx
Problem 7.6.63
Evaluate the integrals in Exercises 53–76.
63. ∫(from -1 to -√2/2)dy/(y√(4y²-1))
Problem 7.5.1
In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.
1. lim (x → -2) (x + 2) / (x² - 4)
Problem 7.7.45
Evaluate the integrals in Exercises 41–60.
45. ∫tanh(x/7)dx
Problem 7.6.84
Evaluate the integrals in Exercises 77–90.
84. ∫(from 2 to 4)2dx/(x²-6x+10)
Problem 7.3.47
Evaluate the integrals in Exercises 33–54.
∫₀^(π/4) (1 + e^(tan θ)) sec²θ dθ
Problem 7.3.143
143.
b. Find the average value of ln(x) over [1, e].
Problem 7.2.8
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
8. y = ln kx, k constant
Problem 7.7.33
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
33. y = csch⁻¹(1/2)^θ
Problem 7.3.139
In Exercises 139–142, find the length of each curve.
139. y = (1/2)(e^x + e^(−x)) from x = 0 to x = 1.
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 7.7.65
Since the hyperbolic functions can be expressed in terms of exponential functions, it is possible to express the inverse hyperbolic functions in terms of logarithms, as shown in the following table.
sinh⁻¹x = ln(x + √(x² + 1)), -∞ < x < ∞
cosh⁻¹x = ln(x + √(x² - 1)), x ≥ 1
tanh⁻¹x = (1/2)ln((1+x)/(1-x)), |x| < 1
sech⁻¹x = ln((1+√(1-x²))/x), 0 < x ≤ 1
csch⁻¹x = ln(1/x + √(1+x²)/|x|), x ≠ 1
coth⁻¹x = (1/2)ln((x+1)/(x-1)), |x| > 1
Use these formulas to express the numbers in Exercises 61–66 in terms of natural logarithms.
65. sech⁻¹(3/5)
Problem 7.3.141
In Exercises 139–142, find the length of each curve.
141. y = ln(cos(x)) from x = 0 to x = π/4.
Problem 7.3.51
Evaluate the integrals in Exercises 33–54.
51. ∫ from ln(π/6) to ln(π/2) 2e^v cos(e^v) dv
Problem 7.6.59
Evaluate the integrals in Exercises 53–76.
59. ∫(from 0 to 1)4ds/√(4-s²)
Problem 7.2.49
Evaluate the integrals in Exercises 39–56.
49. ∫3sec²t/(6 + 3tan(t)) dt
Problem 7.6.9
Find the values in Exercises 9–12.
9. sin(arccos((√2)/2))
Problem 7.5.39
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
39. lim (x → ∞) (ln 2x - ln(x + 1))