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문제 7.2.88
Solve the initial value problems in Exercises 87 and 88.
88. d²y/dx² = sec²x, y(0)=0 and y'(0)=1
문제 7.7.41
Evaluate the integrals in Exercises 41–60.
41. ∫sinh(2x)dx
문제 7.6.21
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
21. y=arccos(x²)
문제 7.6.57
Evaluate the integrals in Exercises 53–76.
57. ∫dx/(x√(25x²-2))
문제 7.4.15
Solve the differential equation in Exercises 9–22.
15. √x (dy/dx) = e^(y+√x), x > 0
문제 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
문제 7.3.51
Evaluate the integrals in Exercises 33–54.
51. ∫ from ln(π/6) to ln(π/2) 2e^v cos(e^v) dv
문제 7.3.5
5. e^(2t)-3e^t = 0
문제 7.A.15
15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.
문제 7.AAE.1
Find the limits in Exercises 1–6.
1. lim(b→1⁻) ∫(from 0 to b) dx/√(1-x²)
문제 7.AAE.20b
20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.
b. Find the centroid of the region.
문제 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.
문제 7.AAE.13
13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.
문제 7.AAE.7a
7. Let A(t) be the area of the region in the first quadrant enclosed by the coordinate axes, the curve y=e^(-x), and the vertical line x=t, t>0. Let V(t) be the volume of the solid generated by revolving the region about the x-axis. Find the following limits.
a. lim(x→∞)A(t)
문제 7.AAE.20a
20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.
a. Find the volume of the solid.
문제 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?
문제 7.AAE.5
Find the limits in Exercises 1–6.
5. lim(n→∞) (1/(n+1) + 1/(n+2) + ... + 1/(2n))
문제 7.AAE.9
In Exercises 9 and 10, use implicit differentiation to find dy/dx.
9. y^e^x = x^y + 1
문제 7.AAE.3
Find the limits in Exercises 1–6.
3. lim(x→0⁺) (cox(√x))^(1/x)
문제 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
문제 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?
문제 7.P.5
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
5. y = ln(sin²θ)
문제 7.P.111c
111. True, or false? Give reasons for your answers.
c. x = o(x + ln(x))
문제 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
문제 7.P.13
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
13. y = (x+2)^(x+2)
문제 7.P.111e
111. True, or false? Give reasons for your answers.
e. arctan x = O(1)
문제 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)?
문제 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)
문제 7.P.112c
112. True, or false? Give reasons for your answers.
c. ln x = o(x+1)
문제 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