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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 7.2.18
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
18. y = t√(ln t)
Problem 7.3.112
Evaluate the integrals in Exercises 111–114.
112. ∫₁^(eˣ) (1 / t) dt
Problem 7.3.109
Evaluate the integrals in Exercises 97–110.
109. ∫ (dx / (x log₁₀x))
Problem 7.6.25
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
25. y=arcsec(2s+1)
Problem 7.3.71
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
71. y = log₂(5θ)
Problem 7.1.1
Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?
Problem 7.1.53
Suppose that the differentiable function y = f(x) has an inverse and that the graph of f passes through the point (2, 4) and has a slope of 1/3 there. Find the value of df⁻¹/dx at x = 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 7.6.21
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
21. y=arccos(x²)
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 7.6.49
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 7.3.15
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(θ)(sinθ + cosθ)
Problem 7.5.19
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
19. lim (θ → π/6) (sin θ - 1/2) / (θ - π/6)
Problem 7.2.10
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
10. y = ln(t^(3/2))
Problem 7.3.89
Evaluate the integrals in Exercises 87–96.
89. ∫₀¹ 2^(−θ) dθ
Problem 7.6.17
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
17. lim(x→∞)arcsec(x)
Problem 7.4.47
47. Carbon-14 The oldest known frozen human mummy, discovered in the Schnalstal glacier of the Italian Alps in 1991 and called Otzi, was found wearing straw shoes and a leather coat with goat fur, and holding a copper ax and stone dagger. It was estimated that Otzi died 5000 years before he was discovered in the melting glacier. How much of the original carbon-14 remained in Otzi at the time of his discovery?
Problem 7.5.17
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
17. lim (θ → π/2) (2θ - π) / cos(2π - θ)
Problem 7.8.19
19. Show that e^x grows faster as x→∞ than x^n for any positive integer n, even x^1,000,000. (Hint: What is the nth derivative of x^n?)
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 7.3.113
Evaluate the integrals in Exercises 111–114.
113. ∫₁^(1/x) (1 / t) dt, x > 0
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 7.2.88
Solve the initial value problems in Exercises 87 and 88.
88. d²y/dx² = sec²x, y(0)=0 and y'(0)=1
Problem 7.3.127
For Exercises 127 and 128 find a function f satisfying each equation.
127. ∫₂ˣ √(f(t)) dt = x ln x
Problem 7.6.19
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
19. lim(x→∞)arccsc(x)
Problem 7.2.86
86. Use a derivative to show that g(x)=√(x² + ln x) is one-to-one.
Problem 7.5.53
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
53. lim (x → 1⁺) x^(1/(1 - x))
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 7.5.39
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
39. lim (x → ∞) (ln 2x - ln(x + 1))
