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Problem 7.6.61
Evaluate the integrals in Exercises 53–76.
61. ∫(from 0 to 2)dt/√(8+2t²)
Problem 7.7.29
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
29. y = (1 - t)coth⁻¹(√t)
Problem 7.6.31
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
31. y=arccot(√t)
Problem 7.1.33
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) = x² − 2x, x ≤ 1
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 7.6.25
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
25. y=arcsec(2s+1)
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 7.3.63
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
63. y = x^π"
Problem 7.3.53
Evaluate the integrals in Exercises 33–54.
53. ∫ (e^r / (1 + e^r)) dr
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 7.3.21
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = ln(e^(θ)/(1+e^θ))
Problem 7.3.112
Evaluate the integrals in Exercises 111–114.
112. ∫₁^(eˣ) (1 / t) dt
Problem 7.6.69
Evaluate the integrals in Exercises 53–76.
69. ∫dx/((2x-1)√((2x-1)²-4))
Problem 7.3.73
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
73. y = log₄ x + log₄ x²
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 7.1.1
Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?
Problem 7.5.37
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
37. lim (y → 0) (√(5y + 25) - 5) / y
Problem 7.6.81
Evaluate the integrals in Exercises 77–90.
81. ∫dy/(y²-2y+5)
Problem 7.7.55
Evaluate the integrals in Exercises 41–60.
55. ∫(from -π/4 to π/4)cosh(tanθ)sec²θ dθ
Problem 7.6.99
Evaluate the integrals in Exercises 91–102.
99. ∫1/(√x (x+1)((arctan√x)²+9)) dx
Problem 7.2.47
Evaluate the integrals in Exercises 39–56.
47. ∫(from 2 to 4)dx/(x(ln x)²)
Problem 7.1.31
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 + 3) / (x − 2)
Problem 7.1.9
In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 1 - x/2, x ≤ 0
x/(x + 2), x > 0
Problem 7.3.29
In Exercises 27–32, find dy/dx.
e^(2x)=sin(x+3y)
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 7.5.14
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
14. lim (t → 0) sin 5t / 2t
Problem 7.5.53
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
53. lim (x → 1⁺) x^(1/(1 - x))
Problem 7.3.45
Evaluate the integrals in Exercises 33–54.
∫ (e^(1/x) / x²) dx
Problem 7.5.22
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
22. lim (x → 1) (x - 1) / (ln x - sin πx)
Problem 7.4.21
Solve the differential equation in Exercises 9–22.
21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)
