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Problem 7.3.32
22–36. Derivatives Find the derivatives of the following functions.
f(t) = 2 tanh⁻¹ √t
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 7.53
37–56. Integrals Evaluate each integral.
∫ (cosh z) / (sinh² z) dz
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 7.2.19
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 7.1.45
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₁² (1 + ln x) x^x dx
Problem 7.3.23
22–36. Derivatives Find the derivatives of the following functions.
f(x) = cosh²x
Problem 7.1.14
7–28. Derivatives Evaluate the following derivatives.
d/dx (x^{π})
Problem 7.3.8
On what interval is the formula d/dx (tanh⁻¹ x) = 1/(1 - x²) valid?
Problem 7.3.105
Inverse identity Show that cosh⁻¹(cosh x) = |x| by using the formula cosh⁻¹ t = ln (t + √(t² – 1)) and considering the cases x ≥ 0 and x < 0.
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 7.1.51
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ 3^{-2x} 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.R.18
10–19. Derivatives Find the derivatives of the following functions.
g(t) = sinh⁻¹(√t)
Problem 7.R.17
10–19. Derivatives Find the derivatives of the following functions.
f(x) = tanh⁻¹(cos x)
Problem 7.R.31
Linear approximation Find the linear approximation to ƒ(x) = cosh x at a = ln 3 and then use it to approximate the value of cosh 1.
Problem 7.R.29e
Log-normal probability distribution A commonly used distribution in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is
f(x) = 1/xσ√(2π) e⁻ˡⁿ^² ˣ / ²σ^², for x ≥ 0
where ln x has zero mean and standard deviation σ > 0.
e. For what value of σ > 0 in part (d) does ƒ(x*) have a minimum?
Problem 7.R.13
10–19. Derivatives Find the derivatives of the following functions.
f(t) = cosh t sinh t
Problem 7.R.10
10–19. Derivatives Find the derivatives of the following functions.
f(x) = ln(3 sin² 4x)
Problem 7.R.33b
Derivatives of hyperbolic functions Compute the following derivatives.
b. d/dx (x sech x)
Problem 7.R.29b
Log-normal probability distribution A commonly used distribution in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is
f(x) = 1/xσ√(2π) e⁻ˡⁿ^² ˣ / ²σ^², for x ≥ 0
where ln x has zero mean and standard deviation σ > 0.
b. Evaluate lim x → 0 ƒ(x). (Hint: Let x = eʸ.)
Problem 7.R.32
Limit Evaluate lim x → ∞ (tanh x)ˣ.
Problem 7.R.12
10–19. Derivatives Find the derivatives of the following functions.
f(x) = (sinh x) / (1 + sinh x)
Problem 7.R.1c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. ln xy = (ln x)(ln y)
Problem 7.R.28
27–28. Curve sketching Use the graphing techniques of Section 4.4 to graph the following functions on their domains. Identify local extreme points, inflection points, concavity, and end behavior. Use a graphing utility only to check your work.
f(x) = ln x – ln² x
Problem 7.RE.4
2–9. Integrals Evaluate the following integrals.
∫₁⁴ (10^{√x} / √x) dx
Problem 7.RE.20a
Population growth The population of a large city grows exponentially with a current population of 1.3 million and a predicted population of 1.45 million 10 years from now.
a. Use an exponential model to estimate the population in 20 years. Assume the annual growth rate is constant.
Problem 7.RE.7
2–9. Integrals Evaluate the following integrals.
∫ dx / √(x² − 9), x > 3
Problem 7.RE.21
Caffeine An adult consumes an espresso containing 75 mg of caffeine. If the caffeine has a half-life of 5.5 hours, when will the amount of caffeine in her bloodstream equal 30 mg?
Problem 7.RE.1a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The variable y = t + 1 doubles in value whenever t increases by 1 unit.