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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.
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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.R.10
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.R.10Capitolo 7, Problema 7.R.10
10–19. Derivatives Find the derivatives of the following functions.
f(x) = ln(3 sin² 4x)
Guida verificata passo dopo passo1
Recognize that the function is a composition of functions: \(f(x) = \ln(3 \sin^{2}(4x))\). We will need to use the chain rule and properties of logarithms to differentiate it.
Use the logarithm property to simplify the function before differentiating: \(\ln(3 \sin^{2}(4x)) = \ln(3) + \ln(\sin^{2}(4x)) = \ln(3) + 2 \ln(\sin(4x))\). Since \(\ln(3)\) is a constant, its derivative is zero.
Focus on differentiating \(2 \ln(\sin(4x))\). Apply the constant multiple rule: the derivative is \(2\) times the derivative of \(\ln(\sin(4x))\).
Differentiate \(\ln(\sin(4x))\) using the chain rule: the derivative of \(\ln(u)\) is \(\frac{1}{u} \cdot \frac{du}{dx}\). Here, \(u = \sin(4x)\), so \(\frac{du}{dx} = \cos(4x) \cdot 4\) by the chain rule.
Combine all parts to write the derivative: \(f'(x) = 2 \cdot \frac{1}{\sin(4x)} \cdot \cos(4x) \cdot 4\). Simplify this expression as much as possible to get the final derivative.
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Chain Rule
The chain rule is used to differentiate composite functions by taking the derivative of the outer function evaluated at the inner function and multiplying it by the derivative of the inner function. For example, if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). This rule is essential when differentiating functions like ln(3 sin² 4x), which involve nested functions.
Video consigliato:
05:02
Intro to the Chain Rule
Derivative of the Natural Logarithm Function
The derivative of ln(u), where u is a differentiable function of x, is (1/u) * du/dx. This means you first find the derivative of the inside function u and then divide by u itself. This property simplifies differentiating logarithmic functions such as ln(3 sin² 4x).
Video consigliato:
05:18Derivative of the Natural Logarithmic Function
Derivative of Trigonometric Functions and Power Rule
To differentiate expressions like sin²(4x), you apply the power rule and the chain rule together. The power rule states that d/dx [u^n] = n u^(n-1) du/dx, and the derivative of sin(4x) involves the chain rule: cos(4x) * 4. Combining these helps find the derivative of sin²(4x).
Video consigliato:
06:35Derivatives of Other Inverse Trigonometric Functions
Pratica correlata
Domanda del libro di testo
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.
e. For what value of σ > 0 in part (d) does ƒ(x*) have a minimum?
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Limit Evaluate lim x → ∞ (tanh x)ˣ.
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Domanda del libro di testo
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.)
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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
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Domanda del libro di testo
10–19. Derivatives Find the derivatives of the following functions.
f(x) = (sinh x) / (1 + sinh x)
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