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Math
Calculus
Ch 2
Definition (9)

Definition (9)

Expert-verified

Calculus

Found in: Page 236

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.
- what it means to say that y is an implicit function of x, and the meaning of implicit differentiation

It means that the relation between $x, y$ is not described by expressing $y$ in terms of $x$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Information

It is mentioned that $y$ is an implicit function of $x$ .

Step 2: Simplification

It means that relation between $x$ and $y$ is not described by expressing $y$ explicitly in terms of $x$ .

In Implicit differentiation, we differentiate each side of the equation with two variables by treating one of the variables as a function of the other.

Most popular questions for Math Textbooks

Suppose $u (x) = \sqrt{3 x^{2} + 1}$ and $f (u) = \frac{u^{2} + 3 u^{5}}{1 - u}$ . Use the chain rule to find role="math" localid="1648356625815" $\frac{d}{d x} (f (u (x)))$ without first finding the formula for $f (u (x))$ .

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$f (x) = x^{3} + 1, [a, b] = [- 2, 1]$

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(a) In real-world terms, what does h(12) represent and what are its units? What does h' (12) represent, and what are its units?

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For each function $f (x)$ and interval localid="1648297458718" $[a, b]$ in Exercises localid="1648297462718" $81 - 86$ , use the Intermediate Value Theorem to argue that the function must have at least one real root on localid="1648297466951" $[a, b]$ . Then apply Newton’s method to approximate that root.

localid="1648297471865" $f (x) = x^{3} - 3 x + 1, [a, b] = [1, 2]$

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