Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problema 7.3.45

Capitolo 7, Problema 7.3.45

37–56. Integrals Evaluate each integral.

∫₀ ˡⁿ ² tanh x dx

Guida verificata passo dopo passo

Recognize that the integral to evaluate is \(\int_0^{\ln 2} \tanh x \, dx\), where \(\tanh x = \frac{\sinh x}{\cosh x}\).

Recall the definition of hyperbolic tangent: \(\tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}\), but it is often easier to work with the derivative of \(\ln(\cosh x)\) since \(\frac{d}{dx} \ln(\cosh x) = \tanh x\).

Use the fact that \(\frac{d}{dx} \ln(\cosh x) = \tanh x\) to rewrite the integral as \(\int_0^{\ln 2} \tanh x \, dx = \left[ \ln(\cosh x) \right]_0^{\ln 2}\).

Evaluate the expression \(\ln(\cosh x)\) at the upper limit \(x = \ln 2\) and the lower limit \(x = 0\) separately.

Subtract the value at the lower limit from the value at the upper limit to find the value of the definite integral: \(\ln(\cosh(\ln 2)) - \ln(\cosh(0))\).

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Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.

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05:43

Definition of the Definite Integral

Hyperbolic Functions and Their Properties

Hyperbolic functions like tanh(x) are analogs of trigonometric functions but based on exponential functions. The function tanh(x) = (e^x - e^{-x}) / (e^x + e^{-x}) is continuous and differentiable, with known derivatives and integrals that simplify integration tasks.

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06:21

Properties of Functions

Integration Techniques for Hyperbolic Functions

Integrating hyperbolic functions often involves recognizing standard integral forms or using substitution. For tanh(x), the integral can be expressed in terms of logarithmic functions, since d/dx [ln(cosh x)] = tanh x, which helps in evaluating definite integrals efficiently.

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5:50

Asymptotes of Hyperbolas

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Compounded inflation The U.S. government reports the rate of inflation (as measured by the consumer index) both monthly and annually. Suppose for a particular month, the monthly rate of inflation is reported as 0.8%. Assuming this rate remains constant, what is the corresponding annual rate of inflation? Is the annual rate 12 times the monthly rate? Explain.

100

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29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫₀^{π/2} 4^{sin x} cos x dx

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22–36. Derivatives Find the derivatives of the following functions.

f(x) = √coth 3x

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Points of inflection Find the x-coordinate of the point(s) of inflection of f(x) = tanh² x.