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Problem 7.5.24
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
24. lim (x → π/2) (ln(csc x)) / (x - (π/2))²
Problem 7.6.105
L’Hôpital’s Rule
Find the limits in Exercises 103–110.
105. lim(x→∞) x arctan(2/x)
Problem 7.6.117
Solve the initial value problems in Exercises 115–120.
117. dy/dx = 1/(x√(x² - 1)), x > 1; y(2) = π
Problem 7.2.59
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
59. y = √(t/(t+1))
Problem 7.1.84
84. Find lim(x→∞) (√(x² + 1) - √x).
Problem 7.2.73
73. Find the area between the curves y=ln(x) and y=ln(2x) from x=1 to x=5.
Problem 7.4.12
Solve the differential equation in Exercises 9–22.
12. (dy/dx) = 3x²e^(-y)
Problem 7.6.128
128. Derive the formula dy/dx = 1/(1+x²) for the derivative of y = arctan(x) by differentiating both sides of the equivalent equation tan(y)=x.
Problem 7.6.71
Evaluate the integrals in Exercises 53–76.
71. ∫(from -π/2 to π/2) 2cosθ dθ/(1+(sinθ)²)
Problem 7.2.45
Evaluate the integrals in Exercises 39–56.
45. ∫(from 1 to 2)(2ln x)/x dx
Problem 7.7.63
Since the hyperbolic functions can be expressed in terms of exponential functions, it is possible to express the inverse hyperbolic functions in terms of logarithms, as shown in the following table.
sinh⁻¹x = ln(x + √(x² + 1)), -∞ < x < ∞
cosh⁻¹x = ln(x + √(x² - 1)), x ≥ 1
tanh⁻¹x = (1/2)ln((1+x)/(1-x)), |x| < 1
sech⁻¹x = ln((1+√(1-x²))/x), 0 < x ≤ 1
csch⁻¹x = ln(1/x + √(1+x²)/|x|), x ≠ 1
coth⁻¹x = (1/2)ln((x+1)/(x-1)), |x| > 1
Use these formulas to express the numbers in Exercises 61–66 in terms of natural logarithms.
63. tanh⁻¹(-1/2)
Problem 7.2.49
Evaluate the integrals in Exercises 39–56.
49. ∫3sec²t/(6 + 3tan(t)) dt
Problem 7.3.9
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(5-7x)
Problem 7.3.67
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
67. y = 7^(sec θ) ln 7"
Problem 7.7.27
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
27. y = (1 - θ)tanh⁻¹(θ)
Problem 7.1.73
In Exercises 73 and 74, repeat the steps above to solve for the functions y=f(x) and x=f^(-1)(y) defined implicitly by the given equations over the interval.
73. y^(1/3) - 1 = (x+2)³, -5 ≤ x ≤ 5, x_0 = -3/2
Problem 7.6.126
126. Show that the sum arctan(x)+arctan(1/x) is constant.
Problem 7.3.77
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
77. y = log₃(((x + 1)/(x − 1))^(ln 3))
Problem 7.5.66
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
66. lim (x → 0⁺) x (ln x)²
Problem 7.5.27
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
27. lim (x → (π/2)^-) (x - π/2) sec x
Problem 7.6.45
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
45. y=cos(x-arccos(x))
Problem 7.4.19
Solve the differential equation in Exercises 9–22.
19. y²(dy/dx) = 3x²y³ - 6x²
Problem 7.4.21
Solve the differential equation in Exercises 9–22.
21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)
Problem 7.6.90
Evaluate the integrals in Exercises 77–90.
90. ∫dx/((x-2)√(x²-4x+3))
Problem 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
Problem 7.6.35
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
35. y=arccsc(e^t)
Problem 7.4.3
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
3. y = 1/x ∫(from 1 to x) e^t/t dt, x²y' + xy = e^x
Problem 7.4.9
Solve the differential equation in Exercises 9–22.
9. 2√(xy) (dy/dx) = 1, x, y > 0
Problem 7.6.99
Evaluate the integrals in Exercises 91–102.
99. ∫1/(√x (x+1)((arctan√x)²+9)) dx
Problem 7.3.33
Evaluate the integrals in Exercises 33–54.
∫(e^(3x) + 5e^(-x)) dx