Login Registrieren

Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 4: Q. 0 (page 384)

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

Short Answer

Expert verified

1. Mean Value Theorem

2. Signed and Absolute Area

3. Average value of a function

Step by step solution

01

Step 1. Given information is:

We need to prepare a summary

02

Step 2. Summary

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Explain why we call the collection of antiderivatives of a function f a family. How are the antiderivatives of a function related?

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The absolute area between the graph of f and the x-axis on [a, b] is equal to $| \int_{a}^{b} f (x) d x |$ .

(b) True or False: The area of the region between f(x) = x − 4 and g(x) = $- x^{2}$ on the interval [−3, 3] is negative.

(c) True or False: The signed area between the graph of f on [a, b] is always less than or equal to the absolute area on the same interval.

(d) True or False: The area between any two graphs f and g on an interval [a, b] is given by $\int_{a}^{b} (f (x) - g (x)) d x$ .

(e) True or False: The average value of the function f(x) = $x^{2} - 3$ on [2, 6] is

\frac{f (6) + f (2)}{2}

\frac{33 + 1}{2}

(f) True or False: The average value of the function f(x) = $x^{2} - 3$ on [2, 6] is $\frac{f (6) - f (2)}{4}$ = $\frac{33 - 1}{4}$ = 8.

(g) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 2] and the average value of f on [2, 5].

(h) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 3] and the average value of f on [3, 5].

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.

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.

$\int_{6}^{1} (3 (1 - 2 x)^{2} + 4 x) d x$

For each function f and interval [a, b] in Exercises 27–33, use the given approximation method to approximate the signed area between the graph of f and the x-axis on [a, b]. Determine whether each of your approximations is likely to be an over-approximation or an under-approximation of the actual area.

$f (x) = \sin (x), [a, b] = [0, π]$ , n = 3 with

a) Trapezoid sim b) Upper sum

See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free