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Problem 7.6.55
Evaluate the integrals in Exercises 53–76.
55. ∫dx/(17+x²)
Problem 7.5.53
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
53. lim (x → 1⁺) x^(1/(1 - x))
Problem 7.3.99
Evaluate the integrals in Exercises 97–110.
99. ∫₀³ (√2 + 1)x^(√2) dx
Problem 7.5.32
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
32. lim (x → 0) (3^x - 1) / (2^x - 1)
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 7.6.27
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
27. y=arccsc(x²+1)
Problem 7.7.59
Evaluate the integrals in Exercises 41–60.
59. ∫(from -ln2 to 0)cosh²(x/2) dx
Problem 7.4.43
43. Surrounding medium of unknown temperature A pan of warm water (46°C) was put in a refrigerator. Ten minutes later, the water’s temperature was 39°C; 10 min after that, it was 33°C. Use Newton’s Law of Cooling to estimate how cold the refrigerator was.
Problem 7.7.51
Evaluate the integrals in Exercises 41–60.
51. ∫(from ln2 to ln4)coth(x)dx
Problem 7.6.93
Evaluate the integrals in Exercises 91–102.
93. ∫(arcsin x)²dx/√(1-x²)
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 7.3.63
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
63. y = x^π"
Problem 7.2.45
Evaluate the integrals in Exercises 39–56.
45. ∫(from 1 to 2)(2ln x)/x dx
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 7.2.37
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 7.4.3
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
3. y = 1/x ∫(from 1 to x) e^t/t dt, x²y' + xy = e^x
Problem 7.3.89
Evaluate the integrals in Exercises 87–96.
89. ∫₀¹ 2^(−θ) dθ
Problem 7.6.108
L’Hôpital’s Rule
Find the limits in Exercises 103–110.
108. lim(x→∞)(e^x arctan(e^x))/(e^(2x)+x)
Problem 7.1.33
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) = x² − 2x, x ≤ 1
Problem 7.2.86
86. Use a derivative to show that g(x)=√(x² + ln x) is one-to-one.
Problem 7.1.19
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)=x²+1, x≥0
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 7.5.47
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
47. lim (t → ∞) (e^t + t²) / (e^t - t)
Problem 7.7.1
Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.
1. sinh x = -3/4
Problem 7.7.23
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
23. y = (x²+1)sech(ln x)
(Hint: Before differentiating, express in terms of exponentials and simplify.)
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 7.7.80
80. Volume The region enclosed by the curve y=sech(x), the x-axis, and the lines x=±ln√3 is revolved about the x-axis to generate a solid. Find the volume of the solid.
Problem 7.6.15
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
15. lim(x→∞)arctan(x)
Problem 7.3.83
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
83. y = 3^(log₂ 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))
