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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 7.7.31
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 7.3.122
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 7.1.9
In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 1 - x/2, x ≤ 0
x/(x + 2), x > 0
Problem 7.6.63
Evaluate the integrals in Exercises 53–76.
63. ∫(from -1 to -√2/2)dy/(y√(4y²-1))
Problem 7.6.84
Evaluate the integrals in Exercises 77–90.
84. ∫(from 2 to 4)2dx/(x²-6x+10)
Problem 7.5.73
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.
73. lim (x → ∞) (2^x - 3^x) / (3^x + 4^x)
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 7.3.43
Evaluate the integrals in Exercises 33–54.
∫ 2t e^(-t²) dt
Problem 7.6.13
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
13. lim(x → 1⁻)arcsin(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 7.3.95
Evaluate the integrals in Exercises 87–96.
95. ∫₂⁴ x^(2x) (1 + ln x) dx
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 7.7.9
Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
9. (sinh(x)+cosh(x))⁴
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 7.3.130
130. Where does the periodic function f(x) = 2e^(sin(x/2)) take on its extreme values, and what are these values?
Problem 7.3.39
Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)dx
Problem 7.2.12
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
12. y = ln(10/x)
Problem 7.5.24
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
24. lim (x → π/2) (ln(csc x)) / (x - (π/2))²
Problem 7.3.87
Evaluate the integrals in Exercises 87–96.
87. ∫ 5ˣ dx
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 7.2.37
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 7.3.81
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
81. y = log₁₀(e^x)
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 7.6.9
Find the values in Exercises 9–12.
9. sin(arccos((√2)/2))
Problem 7.3.99
Evaluate the integrals in Exercises 97–110.
99. ∫₀³ (√2 + 1)x^(√2) dx
Problem 7.7.43
Evaluate the integrals in Exercises 41–60.
43. ∫6cosh(x/2 - ln3)dx
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 7.3.45
Evaluate the integrals in Exercises 33–54.
∫ (e^(1/x) / x²) dx
Problem 7.6.79
Evaluate the integrals in Exercises 77–90.
79. ∫(from -1 to 0)6dt/√(3-2t-t²)
