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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.108b
Chapter 7, Problem 7.3.108b

"Integral formula Carry out the following steps to derive the formula ∫ csch x dx = ln |tanh(x / 2)| + C (Theorem 7.6).


b. Use the identity for sinh(2u) to show that 2 / sinh(2u) = sech² u / tanh u."

Verified step by step guidance
1
Recall the double-angle identity for hyperbolic sine: \(\sinh(2u) = 2 \sinh u \cosh u\).
Start with the expression \(\frac{2}{\sinh(2u)}\) and substitute the identity: \(\frac{2}{2 \sinh u \cosh u} = \frac{1}{\sinh u \cosh u}\).
Express \(\frac{1}{\sinh u \cosh u}\) in terms of \(\tanh u\) and \(\operatorname{sech} u\) by rewriting the denominator: \(\sinh u = \frac{\tanh u}{\operatorname{sech} u}\) and \(\cosh u = \frac{1}{\operatorname{sech} u}\).
Use the definitions \(\tanh u = \frac{\sinh u}{\cosh u}\) and \(\operatorname{sech} u = \frac{1}{\cosh u}\) to rewrite the expression \(\frac{1}{\sinh u \cosh u}\) as \(\frac{\operatorname{sech}^2 u}{\tanh u}\).
Conclude that \(\frac{2}{\sinh(2u)} = \frac{\operatorname{sech}^2 u}{\tanh u}\), as required.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hyperbolic Functions and Their Identities

Hyperbolic functions like sinh, cosh, and tanh are analogs of trigonometric functions but for a hyperbola. Key identities, such as sinh(2u) = 2 sinh u cosh u, help simplify expressions and are essential for manipulating integrals involving hyperbolic functions.
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Verifying Trig Equations as Identities

Integration of Hyperbolic Functions

Integrating hyperbolic functions often involves substitution and using their identities to rewrite the integrand. For example, integrating csch x requires expressing it in terms of sinh x and applying logarithmic integration techniques to arrive at the formula ∫ csch x dx = ln |tanh(x/2)| + C.
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Algebraic Manipulation of Hyperbolic Expressions

Proving identities like 2 / sinh(2u) = sech² u / tanh u requires careful algebraic manipulation using definitions: sech u = 1/cosh u and tanh u = sinh u / cosh u. Rearranging and substituting these expressions is crucial to verify the given identity.
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Related Practice
Textbook Question

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.

Textbook Question

Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:


b. exp(x − y) = exp(x) / exp(y)

Textbook Question

37–38. Caffeine After an individual drinks a beverage containing caffeine, the amount of caffeine in the bloodstream can be modeled by an exponential decay function, with a half-life that depends on several factors, including age and body weight. For the sake of simplicity, assume the caffeine in the following drinks immediately enters the bloodstream upon consumption.


An individual consumes two cups of coffee, each containing 90 mg of caffeine, two hours apart. Assume the half-life of caffeine for this individual is 5.7 hours.


b. Determine the amount of caffeine in the bloodstream 1 hour after drinking the second cup of coffee.

Textbook Question

Depreciation of equipment A large die-casting machine used to make automobile engine blocks is purchased for \$2.5 million. For tax purposes, the value of the machine can be depreciated by 6.8% of its current value each year.


a. What is the value of the machine after 10 years?

Textbook Question

Terminal velocity Refer to Exercises 95 and 96.


a. Compute a jumper’s terminal velocity, which is defined as lim t → ∞ v(t) = lim t → ∞ √(mg/k) tanh (√(kg/m) t).

Textbook Question

A slowing race Starting at the same time and place, Abe and Bob race, running at velocities u(t) = 4 / (t + 1) mi/hr and v(t) = 4e^(−t/2) mi/hr, respectively, for t ≥ 0.

b. Find and graph the position functions of both runners. Which runner can run only a finite distance in an unlimited amount of time?