Back
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 7.1.45
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₁² (1 + ln x) x^x dx
Problem 7.1.5
Express 3ˣ, x^{π}, and x^{sin x} using the base e.
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 7.2.39
39–40. LED lighting LED (light-emitting diode) bulbs are rapidly decreasing in cost, and they are more energy-efficient than standard incandescent light bulbs and CFL (compact fluorescent light) bulbs. By some estimates, LED bulbs last more than 40 times longer than incandescent bulbs and more than 8 times longer than CFL bulbs. Haitz’s law, which is explored in the following two exercises, predicts that over time, LED bulbs will exponentially increase in efficiency and exponentially decrease in cost.
Haitz’s law predicts that the cost per lumen of an LED bulb decreases by a factor of 10 every 10 years. This means that 10 years from now, the cost of an LED bulb will be 1/10 of its current cost. Predict the cost of a particular LED bulb in 2021 if it costs 4 dollars in 2018.
Problem 7.2.33
Atmospheric pressure The pressure of Earth’s atmosphere at sea level is approximately 1000 millibars and decreases exponentially with elevation. At an elevation of 30,000 ft (approximately the altitude of Mt. Everest), the pressure is one-third the sea-level pressure. At what elevation is the pressure half the sea-level pressure? At what elevation is it 1% of the sea-level pressure?
Problem 7.3.8
On what interval is the formula d/dx (tanh⁻¹ x) = 1/(1 - x²) valid?
Problem 7.1.26
Evaluate the following derivatives.
d/dx (x^{x¹⁰})
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 7.2.27
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.
Crime rate The homicide rate decreases at a rate of 3%/yr in a city that had 800 homicides/yr in 2018. At this rate, when will the homicide rate reach 600 homicides/yr?
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.R.18
10–19. Derivatives Find the derivatives of the following functions.
g(t) = sinh⁻¹(√t)
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.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.33b
Derivatives of hyperbolic functions Compute the following derivatives.
b. d/dx (x sech x)
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.17
10–19. Derivatives Find the derivatives of the following functions.
f(x) = tanh⁻¹(cos x)
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.