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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 3 - Derivatives문제 91a
3장, 문제 91a

If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>
a. (f^-1)'(7)

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1
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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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
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.
추천 영상:
가이드 코스
05:44
Derivatives

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.
추천 영상:
가이드 코스
5:20
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