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Problem 7.PE.110a
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
a. f(x) = 3^(-x), g(x) = 2^(-x)
Problem 7.PE.11
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
11. y = 5x^(3.6)
Problem 7.PE.35
Evaluate the integrals in Exercises 31–78.
35. ∫sec²x e^(tan x)dx
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.3
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
3. y = (1/4)xe^(4x) - (1/16)e^(4x)
Problem 7.PE.67
Evaluate the integrals in Exercises 31–78.
67. ∫(from -2 to 2)3dt/(4+3t²)
Problem 7.PE.41
Evaluate the integrals in Exercises 31–78.
41. ∫(from 0 to 4)2t/(t² - 25)dt
Problem 7.PE.9
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
9. y = 8^(-t)
Problem 7.PE.91
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
91. lim(x→π/2⁻) (sec(7x))(cos(3x))
Problem 7.PE.110f
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
f. f(x) = sech(x), g(x) = e^(-x)
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.55
Evaluate the integrals in Exercises 31–78.
55. ∫(from -2 to -1)e^(-(x+1)) dx
Problem 7.PE.33
Evaluate the integrals in Exercises 31–78.
33. ∫e^x sec²(e^x - 7)dx
Problem 7.PE.131
In Exercises 129–132 solve the initial value problem.
131. x dy - (y + √y)dx = 0, y(1) = 1
Problem 7.PE.58
Evaluate the integrals in Exercises 31–78.
58. ∫(from 0 to ln9)e^θ(e^θ-1)^(1/2) dθ
Problem 7.PE.95
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
95. lim(x→∞) (√(x² + x + 1) - √(x² - x))
Problem 7.PE.79
In Exercises 79–84, solve for y.
79. 3^y = 2^(y+1)
Problem 7.PE.85
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
85. lim(x→1) (x² + 3x - 4)/(x - 1)
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.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.52
Evaluate the integrals in Exercises 31–78.
52. ∫(from 1 to 32)(1/5x) dx
Problem 7.PE.77
Evaluate the integrals in Exercises 31–78.
77. ∫dt/((t+1)√(t²+2t-8))
Problem 7.PE.19
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
19. y = t arctan(t) - 1/2 ln(t)
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.43
Evaluate the integrals in Exercises 31–78.
43. ∫tan(ln v)/v dv
Problem 7.PE.45
Evaluate the integrals in Exercises 31–78.
45. ∫(ln x)^(-3)/x dx
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.127
In Exercises 125–128 solve the differential equation.
127. yy' = sec(y²)sec²(x)
Problem 7.PE.125
In Exercises 125–128 solve the differential equation.
125. dy/dx = √y cos(√y)