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Problem 7.AAE.11
Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?
Problem 7.AAE.5
Find the limits in Exercises 1–6.
5. lim(n→∞) (1/(n+1) + 1/(n+2) + ... + 1/(2n))
Problem 7.AAE.19
19. Center of mass Find the center of mass of a thin plate of constant density covering the region in the first and fourth quadrants enclosed by the curves y=1/(1+x²) and y=-1/(1+x²) and by the lines x=0 and x=1.
Problem 7.AAE.9
In Exercises 9 and 10, use implicit differentiation to find dy/dx.
9. y^e^x = x^y + 1
Problem 7.AAE.3
Find the limits in Exercises 1–6.
3. lim(x→0⁺) (cox(√x))^(1/x)
Problem 7.AAE.17
17. Even-odd decompositions
b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that
f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2
Problem 7.GYR.7
7. What integrals lead to logarithms? Give examples. What are the integrals of tan x, cot x, sec x, and csc x?
Problem 7.P.25
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
25. y = 2(x² + 1)/√(cos 2x)
Problem 7.P.5
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
5. y = ln(sin²θ)
Problem 7.P.29
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
29. y = (sin θ)^√θ
Problem 7.P.111c
111. True, or false? Give reasons for your answers.
c. x = o(x + ln(x))
Problem 7.P.27
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
27. y = (((t+1)(t-1))/((t-2)(t+3)))^5, t>2
Problem 7.P.118
118. A particle is traveling upward and to the right along the curve y=ln(x). Its x-coordinate is increasing at the rate (dx/dt)=√x m/sec. At what rate is the y-coordinate changing at the point (e², 2)?
Problem 7.P.13
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
13. y = (x+2)^(x+2)
Problem 7.P.112a
112. True, or false? Give reasons for your answers.
a. 1/x⁴ = O(1/x² + 1/x⁴)
Problem 7.P.111e
111. True, or false? Give reasons for your answers.
e. arctan x = O(1)
Problem 7.P.112c
112. True, or false? Give reasons for your answers.
c. ln x = o(x+1)
Problem 7.PE.41
Evaluate the integrals in Exercises 31–78.
41. ∫(from 0 to 4)2t/(t² - 25)dt
Problem 7.PE.33
Evaluate the integrals in Exercises 31–78.
33. ∫e^x sec²(e^x - 7)dx
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.75
Evaluate the integrals in Exercises 31–78.
75. ∫(from -2 to -1)2dv/(v²+4v+5)
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.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.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.7
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
7. y = log₂(x²/2)
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.52
Evaluate the integrals in Exercises 31–78.
52. ∫(from 1 to 32)(1/5x) dx