Textbook Question
Linear approximation Find the linear approximation to ƒ(x) = cosh x at a = ln 3 and then use it to approximate the value of cosh 1.
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.R.13
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.R.13Chapter 7, Problem 7.R.13
10–19. Derivatives Find the derivatives of the following functions.
f(t) = cosh t sinh t
Verified step by step guidance1
Recall the product rule for derivatives: if you have a function \(f(t) = u(t) v(t)\), then its derivative is \(f'(t) = u'(t) v(t) + u(t) v'(t)\).
Identify the two functions in the product: \(u(t) = \cosh t\) and \(v(t) = \sinh t\).
Find the derivatives of each function separately: \(\frac{d}{dt} \cosh t = \sinh t\) and \(\frac{d}{dt} \sinh t = \cosh t\).
Apply the product rule: \(f'(t) = (\sinh t)(\sinh t) + (\cosh t)(\cosh t)\).
Simplify the expression if possible, using hyperbolic identities such as \(\cosh^2 t - \sinh^2 t = 1\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbolic Functions
Hyperbolic functions, such as sinh(t) and cosh(t), are analogs of trigonometric functions but based on hyperbolas. They have unique properties and identities, like cosh²(t) - sinh²(t) = 1, which are useful in differentiation and integration.
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Asymptotes of Hyperbolas
Product Rule for Differentiation
The product rule states that the derivative of a product of two functions u(t) and v(t) is u'(t)v(t) + u(t)v'(t). This rule is essential when differentiating functions like f(t) = cosh(t) * sinh(t), where both factors depend on t.
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05:18The Product Rule
Derivatives of Hyperbolic Functions
The derivatives of basic hyperbolic functions are: d/dt[sinh(t)] = cosh(t) and d/dt[cosh(t)] = sinh(t). Knowing these derivatives allows you to apply the product rule correctly to find the derivative of functions involving hyperbolic terms.
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5:50Asymptotes of Hyperbolas
Related Practice
Textbook Question
Limit Evaluate lim x → ∞ (tanh x)ˣ.
Textbook Question
Log-normal probability distribution A commonly used distribution in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is
f(x) = 1/xσ√(2π) e⁻ˡⁿ^² ˣ / ²σ^², for x ≥ 0
where ln x has zero mean and standard deviation σ > 0.
b. Evaluate lim x → 0 ƒ(x). (Hint: Let x = eʸ.)
Textbook Question
27–28. Curve sketching Use the graphing techniques of Section 4.4 to graph the following functions on their domains. Identify local extreme points, inflection points, concavity, and end behavior. Use a graphing utility only to check your work.
f(x) = ln x – ln² x
Textbook Question
10–19. Derivatives Find the derivatives of the following functions.
f(x) = tanh⁻¹(cos x)
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. ln xy = (ln x)(ln y)