Domanda del libro di testo
Tangent lines Determine an equation of the line tangent to the graph of y=(x²−1)² / x³−6x−1 at the point (0,−1).
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Ch. 3 - Derivatives
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
Capitolo 3, Problema 91a
If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>
a. (f^-1)'(7)
Guida verificata passo dopo passo1
To find the derivative of the inverse function at a point, we use the formula: \((f^{-1})'(b) = \frac{1}{f'(a)}\), where \(f(a) = b\).
Identify the point \(b = 7\) on the graph of \(f\). Find the corresponding \(a\) such that \(f(a) = 7\).
Once you have found \(a\), locate \(f'(a)\) on the graph of \(f'\). This is the slope of the tangent to \(f\) at \(a\).
Substitute \(f'(a)\) into the formula \((f^{-1})'(7) = \frac{1}{f'(a)}\) to find the derivative of the inverse function at 7.
Ensure that \(f'(a) \neq 0\) to avoid division by zero, which would indicate that the inverse function is not differentiable at that point.
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Inverse Function Theorem
The Inverse Function Theorem states that if a function f is continuous and differentiable, and its derivative f' is non-zero at a point, then the inverse function f^-1 exists locally around that point. The derivative of the inverse function can be calculated using the formula (f^-1)'(y) = 1 / f'(f^-1(y)), which relates the derivatives of the function and its inverse.
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4:49
Inverse Cosine
Derivative Interpretation
The derivative of a function at a point represents the slope of the tangent line to the graph of the function at that point. In the context of the question, understanding how to interpret the derivative graphically is crucial for evaluating (f^-1)'(7), as it involves analyzing the behavior of f and its inverse at specific values.
Video consigliato:
Percorso guidato
05:44Derivatives
Graphical Analysis of Functions
Graphical analysis involves examining the graphs of functions and their derivatives to understand their behavior. For the given question, one must analyze the graph of f to find the corresponding x-value for f(x) = 7, and then use the graph of f' to determine the slope at that point, which is essential for calculating the derivative of the inverse function.
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Percorso guidato
5:20Relations and Functions
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