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Problem 7.PE.61
Evaluate the integrals in Exercises 31–78.
61. ∫(from 1 to 3)(ln(v+1))²/(v+1) dv
Problem 7.PE.99
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
99. lim(x→0) (2^sin(x) - 1)/(e^x - 1)
Problem 7.PE.52
Evaluate the integrals in Exercises 31–78.
52. ∫(from 1 to 32)(1/5x) dx
Problem 7.PE.116
In Exercises 115 and 116, find the absolute maximum and minimum values of each function on the given interval.
116. y = 10x (2 - ln(x)), (0, e²]"133. Find the absolute maximum value of
f(x) = x^2 * ln(1/x)
and say where it is assumed
Problem 7.PE.37
Evaluate the integrals in Exercises 31–78.
37. ∫(from -1 to 1)dx/(3x-4)
Problem 7.PE.17
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
17. y = ln(arccos(x))
Problem 7.PE.83
In Exercises 79–84, solve for y.
83. ln(y-1) = x + ln(y)
Problem 7.PE.45
Evaluate the integrals in Exercises 31–78.
45. ∫(ln x)^(-3)/x dx
Problem 7.PE.43
Evaluate the integrals in Exercises 31–78.
43. ∫tan(ln v)/v dv
Problem 7.PE.109b
109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
b. f(x)=x, g(x)=x + 1/x
Problem 7.PE.102
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
102. lim(x→0) (x sin(x²))/(tan³x)
Problem 7.PE.127
In Exercises 125–128 solve the differential equation.
127. yy' = sec(y²)sec²(x)
Problem 7.PE.104
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
104. lim(x→4) (sin²(πx))/(e^(x-4) + 3 - x)
Problem 7.PE.39
Evaluate the integrals in Exercises 31–78.
39. ∫(from 0 to π)tan(x/3)dx
Problem 7.PE.23
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
23. y = arccsc(secθ), 0<θ<π/2
Problem 7.PE.31
Evaluate the integrals in Exercises 31–78.
31. ∫e^x sin(e^x)dx
Problem 7.PE.109e
109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
e. f(x) = arccsc(x), g(x) = 1/x
Problem 7.PE.15
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
15. y = sin⁻¹√(1-u²), 0<u<1
Problem 7.PE.88
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
88. lim(x→0) (tan x)/(x + sin(x))
Problem 7.PE.7
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
7. y = log₂(x²/2)
Problem 7.PE.49
Evaluate the integrals in Exercises 31–78.
49. ∫x3^(x²)dx
Problem 7.PE.97
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
97. lim(x→0) (10^x - 1)/x
Problem 7.PE.1
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
1. y = 10e^(-x/5)
Problem 7.PE.93
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
93. lim(x→0) (csc(x) - cot(x))
Problem 7.PE.133
133. What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed?
Problem 7.PE.64
Evaluate the integrals in Exercises 31–78.
64. ∫(from 1 to e)(8ln3 log_3(θ))/θ dθ
Problem 7.PE.81
In Exercises 79–84, solve for y.
81. 9e^(2y) = = x^2
Problem 7.PE.129
In Exercises 129–132 solve the initial value problem.
129. dy/dx = e^(-x-y-2), y(0) = -2
Problem 7.PE.110c
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
c. f(x) = 10x^3 + 2x^2, g(x) = e^x
Problem 7.PE.47
Evaluate the integrals in Exercises 31–78.
47. ∫(1/r)csc²(1+ln(r))dr