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Calculus

Found in: Page 384

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Read the section and make your own summary of the material.

1. Mean Value Theorem

2. Signed and Absolute Area

3. Average value of a function

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information is:

We need to prepare a summary

Step 2. Summary

$1 . The mean value theorem is defined to be for c \in [a, b] . 2 . For any integral function f on the interval [a, b], a) The signed area between the graph of f and x - axis is given by the definite integral \int_{a}^{b} f (x) dx b) The absolute area between the graph of f and x - axis is given by the definite integral \int_{a}^{b} |f (x)| dx 3 . The average value of a function on the interval [a, b] is defined to be \frac{1}{b - a} \int_{a}^{b} f (x) dx 4 . The mean value theorem is defined to be for c \in [a, b] such that f' (c) = \frac{f (b) - f (a)}{b - a}$

Most popular questions for Math Textbooks

If $\int_{- 2}^{3} f (x) d x = 4, \int_{- 2}^{6} f (x) d x = 9, \int_{- 2}^{3} g (x) d x = 2$ and $\int_{3}^{6} g (x) d x = 3$ ,then find the values of each definite integral in Exercises $29 - 40$ . If there is not enough information, explain why.

$\int_{3}^{6} f (x) d x$ .

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating .

$\int (x^{2} - 1) (3 x + 5) d x$

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A function f for which the signed area between f and the x-axis on [0, 4] is zero, and a different function g for which the absolute area between g and the x-axis on [0, 4] is zero.

(b) A function f whose signed area on [0, 5] is less than its signed area on [0, 3].

(c) A function f whose average value on [−1, 6] is negative while its average rate of change on the same interval is positive.

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.

$\int (x^{2} + 2^{x} + 2^{2}) d x$

Calculate the exact value of each definite integral in Exercises 47–52 by using properties of definite integrals and the formulas in Theorem 4.13.

$\int_{0}^{6} (3 x + 2) d x$

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