Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The variable y = t + 1 doubles in value whenever t increases by 1 unit.
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 7.3.79a
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.79aChapter 7, Problem 7.3.79a
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
Verified step by step guidance1
Recall the definition of the hyperbolic cosine function: \(\cosh x = \frac{e^{x} + e^{-x}}{2}\).
Substitute \(x = 0\) into the definition: \(\cosh 0 = \frac{e^{0} + e^{-0}}{2}\).
Evaluate the exponentials: \(e^{0} = 1\) and \(e^{-0} = 1\).
Add the values in the numerator: \(1 + 1 = 2\).
Divide by 2 to simplify: \(\cosh 0 = \frac{2}{2}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of Hyperbolic Cosine (cosh)
The hyperbolic cosine function, cosh(x), is defined as (e^x + e^(-x)) / 2. It is an even function and describes the shape of a hanging cable or chain (catenary). Understanding this definition allows direct evaluation of cosh at any point, including zero.
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Definition of the Definite Integral
Properties of Exponential Functions
Exponential functions e^x and e^(-x) are fundamental in defining hyperbolic functions. Knowing that e^0 = 1 simplifies calculations, especially when evaluating cosh(0), since it involves e^0 and e^(-0). This property helps in simplifying expressions without a calculator.
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Evaluating Functions at Specific Points
Evaluating a function at a specific point means substituting the value into the function's formula and simplifying. For cosh(0), substituting x=0 into the definition and simplifying using known values of exponentials yields the exact result without approximation.
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A power line is attached at the same height to two utility poles that are separated by a distance of 100 ft; the power line follows the curve ƒ(x) = a cosh x/a. Use the following steps to find the value of a that produces a sag of 10 ft midway between the poles. Use a coordinate system that places the poles at x = ±50.
a. Show that a satisfies the equation cosh 50/a − 1 = 10/a.
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Textbook Question
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.