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Problem 7.2.41
Evaluate the integrals in Exercises 39–56.
41. ∫2y dy/(y²-25)
Problem 7.3.35
Evaluate the integrals in Exercises 33–54.
∫(from ln3 to ln2) (e^x) dx
Problem 7.3.21
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = ln(e^(θ)/(1+e^θ))
Problem 7.6.45
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
45. y=cos(x-arccos(x))
Problem 7.3.31
In Exercises 27–32, find dy/dx.
3+siny = y-x^3
Problem 7.6.69
Evaluate the integrals in Exercises 53–76.
69. ∫dx/((2x-1)√((2x-1)²-4))
Problem 7.3.15
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(θ)(sinθ + cosθ)
Problem 7.7.73
Evaluate the integrals in Exercises 31–78.
73. ∫dx/√(-2x-x²)
Problem 7.3.107
Evaluate the integrals in Exercises 97–110.
107. ∫₀⁹ (2 log₁₀(x + 1) / (x + 1)) dx
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 7.5.37
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
37. lim (y → 0) (√(5y + 25) - 5) / y
Problem 7.3.63
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
63. y = x^π"
Problem 7.6.77
Evaluate the integrals in Exercises 77–90.
77. ∫dx/√(-x²+4x-3)
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 7.5.71
Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
Try it—you just keep on cycling. Find the limits some other way.
71. lim (x → (π/2)⁻) sec x / tan x
Problem 7.3.29
In Exercises 27–32, find dy/dx.
e^(2x)=sin(x+3y)
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 7.2.56
Evaluate the integrals in Exercises 39–56.
56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))
Problem 7.5.22
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
22. lim (x → 1) (x - 1) / (ln x - sin πx)
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 7.5.57
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
57. lim (x → 0⁺) x^(-1/ln x)
Problem 7.6.126
126. Show that the sum arctan(x)+arctan(1/x) is constant.
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 7.1.55
Show that increasing functions and decreasing functions are one-to-one. That is, show that for any x₁ and x₂ in I, x₂ ≠ x₁ implies f(x₂) ≠ f(x₁).
Problem 7.3.113
Evaluate the integrals in Exercises 111–114.
113. ∫₁^(1/x) (1 / t) dt, x > 0
Problem 7.3.19
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = ln(3te^(-t))
Problem 7.5.44
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
44. lim (x → 0⁺) (csc x - cot x + cos x)
Problem 7.5.3
In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.
3. lim (x → ∞) (5x² - 3x) / (7x² + 1)
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 7.5.39
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
39. lim (x → ∞) (ln 2x - ln(x + 1))