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Problem 7.6.84
Evaluate the integrals in Exercises 77–90.
84. ∫(from 2 to 4)2dx/(x²-6x+10)
Problem 7.3.3
In Exercises 1–4, solve for t.
e^(sqrt(t)) = x^2
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 7.3.143
143.
b. Find the average value of ln(x) over [1, e].
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 7.1.23
Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.
f(x)=(x+1)², x≥-1
Problem 7.7.43
Evaluate the integrals in Exercises 41–60.
43. ∫6cosh(x/2 - ln3)dx
Problem 7.4.10
Solve the differential equation in Exercises 9–22.
10. (dy/dx) = x²√y, y > 0
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 7.6.75
Evaluate the integrals in Exercises 53–76.
75. ∫y dy/√(1-y^4)
Problem 7.3.77
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
77. y = log₃(((x + 1)/(x − 1))^(ln 3))
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 7.3.95
Evaluate the integrals in Exercises 87–96.
95. ∫₂⁴ x^(2x) (1 + ln x) dx
Problem 7.4.37
37. Plutonium-239 The half-life of the plutonium isotope is 24,360 years. If 10 g of plutonium is released into the atmosphere by a nuclear accident, how many years will it take for 80% of the isotope to decay?
Problem 7.6.9
Find the values in Exercises 9–12.
9. sin(arccos((√2)/2))
Problem 7.6.102
Evaluate the integrals in Exercises 91–102.
102. ∫(from -1/3 to 1/√3)(cos(arctan 3x))/(1+9x²) 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 7.3.67
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
67. y = 7^(sec θ) ln 7"
Problem 7.7.59
Evaluate the integrals in Exercises 41–60.
59. ∫(from -ln2 to 0)cosh²(x/2) dx
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 7.7.9
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 7.3.135
135. Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y = e^(2x), below by the curve y = e^x, and on the right by the line x = ln(3).
Problem 7.5.51
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
51. lim (θ → 0) (θ - sin θ cos θ) / (tan θ - θ)
Problem 7.2.64
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
64. y = 1/(t(t+1)(t+2))
Problem 7.2.61
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
61. y = √(θ + 3) sin θ
Problem 7.1.63
Suppose that the range of g lies in the domain of f so that the composition fog is defined. If f and g are one-to-one, can anything be said about fog? Give reasons for your answer.
Problem 7.5.63
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
63. lim (x → ∞) ((x + 2)/(x - 1))^x
Problem 7.6.93
Evaluate the integrals in Exercises 91–102.
93. ∫(arcsin x)²dx/√(1-x²)
Problem 7.2.29
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
29. y = ln(1/(x√(x+1)))
Problem 7.3.5
5. e^(2t)-3e^t = 0
