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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 7.3.50
37–56. Integrals Evaluate each integral.
∫ dx/x√(16 + x²)
Problem 7.1.29
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₀³ (2x - 1) / (x + 1) dx
Problem 7.1.30
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ (x²) / (4x³ + 7) dx
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 7.1.77
Harmonic sum In Chapter 10, we will encounter the harmonic sum 1 + 1/2 + 1/3 + ⋯ + 1/n. Use a left Riemann sum to approximate ∫[1 to n+1] (dx/x) (with unit spacing between the grid points) to show that 1 + 1/2 + 1/3 + ⋯ + 1/n > ln(n + 1). Use this fact to conclude that lim (n → ∞) (1 + 1/2 + 1/3 + ⋯ + 1/n) does not exist.
Problem 7.3.103
101–104. Proving identities Prove the following identities.
cosh (x + y) = cosh x cosh y + sinh x sinh y
Problem 7.1.72
Derivative of ln|x| Differentiate ln x, for x > 0, and differentiate ln(−x), for x < 0, to conclude that d/dx (ln|x|) = 1/x
Problem 7.3.88
88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.
lim x → ∞ (1 − coth x) / (1 − tanh x)
Problem 7.3.86
Solid of revolution Compute the volume of the solid of revolution that results when the region in Exercise 85 is revolved about the x-axis.
Problem 7.1.33
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ e^{2x} / (4 + e^{2x}) dx
Problem 7.2.52
"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 7.1.67
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 7.3.56
37–56. Integrals Evaluate each integral.
∫₂₅²²⁵ dx / (x² + 25x) (Hint: √(x² + 25x) = √x √(x + 25).)
Problem 7.1.25
Evaluate the following derivatives.
d/dx ((1/x)ˣ)
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 7.3.25
22–36. Derivatives Find the derivatives of the following functions.
f(x) = tanh²x
Problem 7.1.75
Bounds on e Use a left Riemann sum with at least n = 2 subintervals of equal length to approximate ln 2 = ∫[1 to 2] (dt/t) and show that ln 2 < 1. Use a right Riemann sum with n = 7 subintervals of equal length to approximate ln 3 = ∫[1 to 3] (dt/t) and show that ln 3 > 1.
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 7.1.13
7–28. Derivatives Evaluate the following derivatives.
d/dx ((2x)⁴ˣ)
Problem 7.1.26
Evaluate the following derivatives.
d/dx (x^{x¹⁰})
Problem 7.1.3
Evaluate ∫ 4ˣ dx.
Problem 7.2.25
Population of Texas Texas was the third fastest growing state in the United States in 2016. Texas grew from 25.1 million in 2010 to 26.47 million in 2016. Use an exponential growth model to predict the population of Texas in 2025.
Problem 7.3.91
88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.
lim x → 0⁺ (tanh x)ˣ