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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.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.10
10–19. Derivatives Find the derivatives of the following functions.
f(x) = ln(3 sin² 4x)
Problem 7.R.12
10–19. Derivatives Find the derivatives of the following functions.
f(x) = (sinh x) / (1 + sinh x)
Problem 7.R.13
10–19. Derivatives Find the derivatives of the following functions.
f(t) = cosh t sinh t
Problem 7.R.17
10–19. Derivatives Find the derivatives of the following functions.
f(x) = tanh⁻¹(cos x)
Problem 7.R.18
10–19. Derivatives Find the derivatives of the following functions.
g(t) = sinh⁻¹(√t)
Problem 7.R.33b
Derivatives of hyperbolic functions Compute the following derivatives.
b. d/dx (x sech 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.32
Limit Evaluate lim x → ∞ (tanh x)ˣ.
Problem 7.RE.26b
Savings account A savings account advertises an annual percentage yield (APY) of 5.4%, which means that the balance in the account increases at an annual growth rate of 5.4%/yr.
b. What is the doubling time of the balance?
Problem 7.RE.24
Radioactive decay The mass of radioactive material in a sample has decreased by 30% since the decay began. Assuming a half-life of 1500 years, how long ago did the decay begin?
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.
Problem 7.RE.2
2–9. Integrals Evaluate the following integrals.
∫ (eˣ / (4eˣ + 6)) dx
Problem 7.RE.4
2–9. Integrals Evaluate the following integrals.
∫₁⁴ (10^{√x} / √x) dx
Problem 7.RE.5
2–9. Integrals Evaluate the following integrals.
∫ (x + 4) / (x² + 8x + 25) dx
Problem 7.RE.7
2–9. Integrals Evaluate the following integrals.
∫ dx / √(x² − 9), x > 3
Problem 7.RE.9
2–9. Integrals Evaluate the following integrals.
∫₀¹ (x² / (9 − x⁶)) dx
Problem 7.RE.20b
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.
b. Find the doubling time of the population.
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.23a
Moore’s Law In 1965, Gordon Moore observed that the number of transistors that could be placed on an integrated circuit was approximately doubling each year, and he predicted that this trend would continue for another decade. In 1975, Moore revised the doubling time to every two years, and this prediction became known as Moore’s Law.
a. In 1979, Intel introduced the Intel 8088 processor; each of its integrated circuits contained 29,000 transistors. Use Moore’s revised doubling time to find a function y(t) that approximates the number of transistors on an integrated circuit t years after 1979.
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.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.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 7.1.33
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ e^{2x} / (4 + e^{2x}) dx
Problem 7.1.10
7–28. Derivatives Evaluate the following derivatives.
d/dx (ln (cos² x))
Problem 7.1.1
What are the domain and range of ln x?
Problem 7.1.4
What is the inverse function of ln x, and what are its domain and range?
Problem 7.1.9
7–28. Derivatives Evaluate the following derivatives.
d/dx (sin (ln x))
Problem 7.1.25
Evaluate the following derivatives.
d/dx ((1/x)ˣ)