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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions문제 7.3.59a
7장, 문제 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.

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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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
03:39
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.
추천 영상:
05:43
Definition of the Definite Integral

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.
추천 영상:
5:37
Introduction to Cotangent Graph
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교과서 질문

Chemotherapy In an experimental study at Dartmouth College, mice with tumors were treated with the chemotherapeutic drug Cisplatin. Before treatment, the tumors consisted entirely of clonogenic cells that divide rapidly, causing the tumors to double in size every 2.9 days. Immediately after treatment, 99% of the cells in the tumor became quiescent cells which do not divide and lose 50% of their volume every 5.7 days. For a particular mouse, assume the tumor size is 0.5 cm³ at the time of treatment.

a. Find an exponential decay function V₁(t) that equals the total volume of the quiescent cells in the tumor t days after treatment.

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교과서 질문

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.

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교과서 질문

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

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교과서 질문

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.

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교과서 질문

Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.


a. cosh 0

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교과서 질문

Falling body When an object falling from rest encounters air resistance proportional to the square of its velocity, the distance it falls (in meters) after t seconds is given by d(t) = (m/k) ln (cosh (√(kg/m) t)), where m is the mass of the object in kilograms, g = 9.8 m/s² is the acceleration due to gravity, and k is a physical constant.


a. A BASE jumper (m = 75 kg) leaps from a tall cliff and performs a ten-second delay (she free-falls for 10 s and then opens her chute). How far does she fall in 10 s? Assume k = 0.2.

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