Textbook Question
Zero net area Consider the function f(x) = (1 − x)/x
a. Are there numbers 0 < a < 1 such that ∫₁₋ₐ¹⁺ᵃ f(x) dx = 0?
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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 7.3.59a
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.59aChapter 7, Problem 7.3.59a
Visual approximation
a. Use a graphing utility to sketch the graph of y = coth x and then explain why ∫₅¹⁰ coth x dx ≈ 5.
Verified step by step guidance1
Recall the definition of the hyperbolic cotangent function: \(y = \coth x = \frac{\cosh x}{\sinh x}\), where \(\sinh x\) and \(\cosh x\) are the hyperbolic sine and cosine functions respectively.
Use a graphing utility to plot the function \(y = \coth x\) over the interval \([5, 10]\). Observe the behavior of the graph in this range, noting that \(\coth x\) approaches 1 as \(x\) becomes large.
Understand that the definite integral \(\int_5^{10} \coth x \, dx\) represents the area under the curve of \(y = \coth x\) from \(x=5\) to \(x=10\).
Since \(\coth x\) is close to 1 for large \(x\), the graph between 5 and 10 is near the horizontal line \(y=1\). Therefore, the area under the curve is approximately the area of a rectangle with height 1 and width \(10 - 5 = 5\).
Conclude that this approximation explains why \(\int_5^{10} \coth x \, dx \approx 5\), because the integral sums values close to 1 over an interval of length 5.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbolic Cotangent Function (coth x)
The hyperbolic cotangent function, coth x, is defined as cosh x divided by sinh x. It behaves similarly to the reciprocal of the tangent function but for hyperbolic angles. Understanding its shape and asymptotic behavior helps in visualizing the graph and estimating integrals involving coth x.
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Integrals of Natural Exponential Functions (e^x)
Definite Integral as Area Under the Curve
A definite integral ∫_a^b f(x) dx represents the net area between the graph of f(x) and the x-axis from x = a to x = b. Visualizing this area on the graph of coth x allows approximation of the integral's value by estimating the region's size.
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Using Graphing Utilities for Approximation
Graphing utilities plot functions accurately, revealing key features like asymptotes and behavior over intervals. By sketching y = coth x from 5 to 10, one can visually assess the area under the curve, supporting an approximate value for the integral without exact calculation.
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Related Practice
Textbook Question
Velocity of falling body Refer to Exercise 95, which gives the position function for a falling body. Use m = 75 kg and k = 0.2.
a. Confirm that the BASE jumper’s velocity t seconds after jumping is v(t) = d'(t) = √(mg/k) tanh (√(kg/m) t).
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Textbook Question
ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.
a. Make a sketch of the function f(x) = 1/x on the interval [1, 2]. Explain why the area of the region bounded by y = f(x) and the x-axis on [1, 2] is ln 2.
Textbook Question
Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.
a. coth 4
Textbook Question
Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.
a. Based on these figures, find an exponential growth function for the power (rate of electricity use) for the city.
Textbook Question
Wave velocity Use Exercise 73 to do the following calculations.
a. Find the velocity of a wave where λ = 50 m and d = 20 m.