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Problem 7.1.70
Behavior at the origin Using calculus and accurate sketches, explain how the graphs of f(x) = xᵖ ln x differ as x → 0⁺ for p = 1/2, 1, and 2.
Problem 7.3.45
37–56. Integrals Evaluate each integral.
∫₀ ˡⁿ ² tanh x dx
Problem 7.3.42
37–56. Integrals Evaluate each integral.
∫ sinh²z dz (Hint: Use an identity.)
Problem 7.1.30
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ (x²) / (4x³ + 7) dx
Problem 7.1.29
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₀³ (2x - 1) / (x + 1) dx
Problem 7.3.50
37–56. Integrals Evaluate each integral.
∫ dx/x√(16 + x²)
Problem 7.1.12
7–28. Derivatives Evaluate the following derivatives.
d/dx (ln³(3x² + 2))
Problem 7.1.35
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ₑᵉ^³ dx / (x ln x ln²(ln x))
Problem 7.3.13
11–15. Identities Prove each identity using the definitions of the hyperbolic functions.
cosh 2x = cosh²x + sinh²x (Hint: Begin with the right side of the equation.)
Problem 7.3.103
101–104. Proving identities Prove the following identities.
cosh (x + y) = cosh x cosh y + sinh x sinh y
Problem 7.3.65
63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.
∫₋₂² dt/(t² – 9)
Problem 7.3.100
Surface area of a catenoid When the catenary y = a cosh x/a is revolved about the x-axis, it sweeps out a surface of revolution called a catenoid. Find the area of the surface generated when y = cosh x on [–ln 2, ln 2] is rotated about the x-axis.
Problem 7.1.40
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₋₂² (e^{z/2}) / (e^{z/2} + 1) dz
Problem 7.1.33
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ e^{2x} / (4 + e^{2x}) dx
Problem 7.2.7
Suppose a quantity described by the function y(t) = y₀eᵏᵗ, where t is measured in years, has a doubling time of 20 years. Find the rate constant k.
Problem 7.1.67
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.
d. 2ˣ = 2² ˡⁿ ˣ
Problem 7.3.17
16–18. Identities Use the given identity to prove the related identity.
Use the identity cosh 2x = cosh²x + sinh²x to prove the identities cosh²x = (cosh 2x + 1)/2 and sinh²x = (cosh 2x − 1)/2.
Problem 7.2.29
27–30. Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time.
Valium metabolism The drug Valium is eliminated from the bloodstream with a half-life of 36 hr. Suppose a patient receives an initial dose of 20 mg of Valium at midnight. How much Valium is in the patient’s blood at noon the next day? When will the Valium concentration reach 10% of its initial level?
Problem 7.2.52
"General relative growth rates Define the relative growth rate of the function f over the time interval T to be the relative change in f over an interval of length T:
R_T = [f(t + T) − f(t)] / f(t)
Show that for the exponential function y(t) = y₀ e^{kt}, the relative growth rate R_T, for fixed T, is constant for all t."
Problem 7.3.5
Express sinh⁻¹ x in terms of logarithms.
Problem 7.1.13
7–28. Derivatives Evaluate the following derivatives.
d/dx ((2x)⁴ˣ)
Problem 7.3.25
22–36. Derivatives Find the derivatives of the following functions.
f(x) = tanh²x
Problem 7.1.25
Evaluate the following derivatives.
d/dx ((1/x)ˣ)
Problem 7.2.19
15–20. Designing exponential growth functions Complete the following steps for the given situation.
a. Find the rate constant k and use it to devise an exponential growth function that fits the given data.
b. Answer the accompanying question.
Rising costs Between 2010 and 2016, the average rate of inflation was about 1.6%/yr. If a cart of groceries cost $100 in 2010, what will it cost in 2025, assuming the rate of inflation remains constant at 1.6%?
Problem 7.2.12
After the introduction of foxes on an island, the number of rabbits on the island decreases by 4.5% per month. If y(t) equals the number of rabbits on the island t months after foxes were introduced, find the rate constant k for the exponential decay function y(t) = y₀eᵏᵗ.
Problem 7.1.69
Average value What is the average value of f(x) = 1/x on the interval [1, p] for p > 1? What is the average value of f as p → ∞?
Problem 7.1.63
63–66. Calculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l’Hôpital’s Rule.
limₕ→₀ (1 + 2h)^{1/h}
Problem 7.3.8
On what interval is the formula d/dx (tanh⁻¹ x) = 1/(1 - x²) valid?
Problem 7.1.45
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₁² (1 + ln x) x^x dx
Problem 7.3.34
22–36. Derivatives Find the derivatives of the following functions.
f(x) = csch⁻¹(2/x)