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Problem 7.6.63
Evaluate the integrals in Exercises 53–76.
63. ∫(from -1 to -√2/2)dy/(y√(4y²-1))
Problem 7.7.31
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
31. y = cos⁻¹(x) - x sech⁻¹(x)
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 7.3.139
In Exercises 139–142, find the length of each curve.
139. y = (1/2)(e^x + e^(−x)) from x = 0 to x = 1.
Problem 7.3.5
5. e^(2t)-3e^t = 0
Problem 7.6.13
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
13. lim(x → 1⁻)arcsin(x)
Problem 7.5.3
In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.
3. lim (x → ∞) (5x² - 3x) / (7x² + 1)
Problem 7.5.32
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
32. lim (x → 0) (3^x - 1) / (2^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 7.3.143
143.
b. Find the average value of ln(x) over [1, e].
Problem 7.1.38
Show that the graph of the inverse of f(x)=mx+b, where m and b are constants and m≠0, is a line with slope 1/m and y-intercept -b/m.
Problem 7.3.95
Evaluate the integrals in Exercises 87–96.
95. ∫₂⁴ x^(2x) (1 + ln x) dx
Problem 7.6.113
Verify the integration formulas in Exercises 111–114.
113. ∫ (arcsin x)² dx = x(arcsin x)² - 2x + 2 √(1 - x²) arcsin x + C
Problem 7.7.9
Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
9. (sinh(x)+cosh(x))⁴
Problem 7.7.43
Evaluate the integrals in Exercises 41–60.
43. ∫6cosh(x/2 - ln3)dx
Problem 7.3.130
130. Where does the periodic function f(x) = 2e^(sin(x/2)) take on its extreme values, and what are these values?
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 7.2.12
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
12. y = ln(10/x)
Problem 7.4.5
In Exercises 5–8, show that each function is a solution of the given initial value problem.
5. Differential Equation: 2y + y' = 4x + 2
Initial condition: y(-1) = e² - 2
Solution candidate: y = e^(-2x) + 2x
Problem 7.2.37
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
37. ∫(from x²/2 to x²)ln(√t)dt
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 7.3.39
Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)dx
Problem 7.5.73
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.
73. lim (x → ∞) (2^x - 3^x) / (3^x + 4^x)
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 7.3.99
Evaluate the integrals in Exercises 97–110.
99. ∫₀³ (√2 + 1)x^(√2) dx
Problem 7.7.17
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
17. y = ln(sinh z)
Problem 7.3.7
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(-5x)
Problem 7.3.81
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
81. y = log₁₀(e^x)
Problem 7.3.45
Evaluate the integrals in Exercises 33–54.
∫ (e^(1/x) / x²) dx
Problem 7.3.41
Evaluate the integrals in Exercises 33–54.
∫ (e^(√r) / √r) dr
