Textbook Question
2–9. Integrals Evaluate the following integrals.
∫₁⁴ (10^{√x} / √x) dx
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.RE.20b
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.RE.20bChapter 7, 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.
Verified step by step guidance1
Recognize that the population growth follows an exponential model of the form \(P(t) = P_0 \cdot e^{kt}\), where \(P_0\) is the initial population, \(k\) is the growth rate, and \(t\) is time in years.
Use the given information to find the growth rate \(k\). You know \(P_0 = 1.3\) million and \(P(10) = 1.45\) million, so set up the equation \(1.45 = 1.3 \cdot e^{10k}\).
Solve for \(k\) by dividing both sides by 1.3 and then taking the natural logarithm: \(\ln\left(\frac{1.45}{1.3}\right) = 10k\), which gives \(k = \frac{1}{10} \ln\left(\frac{1.45}{1.3}\right)\).
Recall that the doubling time \(T\) is the time it takes for the population to double, so \(P(T) = 2P_0\). Substitute into the model: \(2P_0 = P_0 \cdot e^{kT}\), which simplifies to \(2 = e^{kT}\).
Take the natural logarithm of both sides to solve for \(T\): \(\ln(2) = kT\), so \(T = \frac{\ln(2)}{k}\). Use the value of \(k\) found in step 3 to express the doubling time.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Growth Model
Exponential growth describes a process where the quantity increases at a rate proportional to its current value, often modeled by P(t) = P_0 * e^(kt). Here, P_0 is the initial population, k is the growth rate, and t is time. Understanding this model is essential to relate population changes over time.
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Exponential Growth & Decay
Doubling Time
Doubling time is the period required for a quantity growing exponentially to double in size. It can be found using the formula T = ln(2)/k, where k is the growth rate. This concept helps determine how quickly the population will reach twice its current size.
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Natural Logarithm and Growth Rate Calculation
The natural logarithm (ln) is used to solve for the growth rate k in the exponential model by rearranging P(t) = P_0 * e^(kt). Calculating k involves taking ln of the ratio of populations and dividing by time, which is crucial for finding doubling time and making predictions.
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Related Practice
Textbook Question
10–19. Derivatives Find the derivatives of the following functions.
g(t) = sinh⁻¹(√t)
Textbook Question
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.
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Textbook Question
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?
Textbook Question
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?
Textbook Question
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?
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