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Problem 7.2.14
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
14. y = ln(2θ+2)
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 7.1.57
Use the results of Exercise 55 to show that the functions in Exercises 56–60 have inverses over their domains. Find a formula for df⁻¹/dx using Theorem 1.
f(x) = 27x³
Problem 7.2.39
Evaluate the integrals in Exercises 39–56.
39. ∫(from -3 to -2)dx/x
Problem 7.1.51
Let f(x) = x³ − 3x² − 1, x ≥ 2. Find the value of df⁻¹/dx at the point x = −1 = f(3).
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))
Problem 7.3.11
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = xe^x-e^x
Problem 7.5.44
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
44. lim (x → 0⁺) (csc x - cot x + cos x)
Problem 7.4.10
Solve the differential equation in Exercises 9–22.
10. (dy/dx) = x²√y, y > 0
Problem 7.3.55
Solve the initial value problems in Exercises 55–58.
55. dy/dt = e^t sin(e^t − 2), y(ln 2) = 0
Problem 7.7.39
Verify the integration formulas in Exercises 37–40.
39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C
Problem 7.5.14
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
14. lim (t → 0) sin 5t / 2t
Problem 7.3.53
Evaluate the integrals in Exercises 33–54.
53. ∫ (e^r / (1 + e^r)) dr
Problem 7.6.51
For problems 49–52 use implicit differentiation to find dy/dx at the given point P.
51. y arccos(xy) = -3√2/4 π; P(1/2, -√2)
Problem 7.7.21
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
21. y = ln(cosh v) - 1/2 tanh²v
Problem 7.3.29
In Exercises 27–32, find dy/dx.
e^(2x)=sin(x+3y)
Problem 7.7.41
Evaluate the integrals in Exercises 41–60.
41. ∫sinh(2x)dx
Problem 7.5.17
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
17. lim (θ → π/2) (2θ - π) / cos(2π - θ)
Problem 7.8.13
13. When is a polynomial f(x) of at most the order of a polynomial g(x) as x→∞? Give reasons for your answer.
Problem 7.3.126
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
126. eʸ = y^(ln x)
Problem 7.3.65
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
65. y = (cos θ)^(√2)
Problem 7.2.70
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
70. y = ∛(x(x+1)(x-2)/(x²+1)(2x+3))
Problem 7.5.69
Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
Try it—you just keep on cycling. Find the limits some other way.
69. lim (x → ∞) (√(9x + 1)) / (√(x + 1))
Problem 7.1.35
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) = (x + b) / (x − 2), b > −2 and constant
Problem 7.7.19
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
19. y = (sech θ)(1-ln(sech θ))
Problem 7.A.15
15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.
Problem 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.
Problem 7.AAE.13
13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.
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))