# 1.7 Intermediate Value Theoremap Calculus

The Intermediate Value Theorem. Definition of a Critical Number. A critical number of a function f is a number c in the domain of f such that either f ‘(c) = 0 or f ‘(c) does not exist. Rolle’s Theorem. Let f be a function that satisfies the following three hypotheses. Use the Intermediate Value Theorem again. Look for a sign change. Looking down the table, there is a sign change between -1.8 and -1.7. With this information we now know the zero is between these two values. Repeat this process again with two decimal places between -1.8 and -1.7.

## The Intermediate Value Theorem

1.7 Intermediate Value Theorem. Powered by Create your own unique website with customizable templates. In fact, the intermediate value theorem is equivalent to the least upper bound property. Suppose the intermediate value theorem holds, and for a nonempty set S S S with an upper bound, consider the function f f f that takes the value 1 1 1 on all upper bounds of S S S and − 1-1 − 1 on the rest of R. CALCULUS AB WORKSHEET ON CONTINUITY AND INTERMEDIATE VALUE THEOREM Work the following on notebook paper. On problems 1 – 4, sketch the graph of a function f that satisfies the stated conditions. F has a limit at x = 3, but it is not continuous at x = 3. F is not continuous at x = 3, but if its value at x = 3 is changed from f 31 to f 30.

Your teacher probably told you that you can draw the graph of acontinuous function without lifting your pencil off the paper.This is made precise by the following result:

Intermediate Value Theorem. Let f (x) be a continuousfunction on the interval [a, b]. If d [f (a), f (b)], thenthere is a c [a, b] such that f (c) = d.

In the case where f (a) > f (b), [f (a), f (b)] is meant to be thesame as [f (b), f (a)]. Another way to state the IntermediateValue Theorem is to say that the image of a closed interval undera continuous function is a closed interval. We will present anoutline of the proof of the Intermediate Value Theorem on thenext page.

Here is a classical consequence of the Intermediate Value Theorem:

Example. Every polynomial of odd degree has at least onereal root.

We want to show that if P(x) = anxn + an - 1xn - 1 + ... + a1x + a0 is a polynomial with n odd and an 0,then there is a real number c, such that P(c) = 0.

First let me remind you that it follows from the results inprevious pages that every polynomial is continuous on the realline. There you also learned that

Consequently for x large enough, P(x) and anxn havethe same sign. But anxn has opposite signs for positive xand negative x. Thus it follows that if an > 0, there are realnumbers x0 < x1 such that P(x0) < 0 and P(x1) > 0. Similarlyif an < 0, we can find x0 < x1 such that P(x0) > 0 andP(x1) < 0. In either case, it now follows directly from theIntermediate Value Theorem that (for d = 0) there is a realnumber c [x0, x1] with P(c) = 0.

The natural question arises whether every function whichsatisfies the conclusion of the Intermediate Value Theorem mustbe continuous. Unfortunately, the answer is no andcounterexamples are quite messy. The easiest counterexample isthe function

f (x) =
As you found out on an earlier page, this function fails to becontinuous at x = 0. On the other hand, it is not too hard to seethat f (

## Intermediate Value Theorem Formula

x) has the 'Intermediate Value Property' even on closedintervals containing x = 0.

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## 1.7 Intermediate Value Theorem Ap Calculus Algebra

Apostol, Tom M. Calculus, Volume 1: One-Variable Calculus, with An Introduction to Linear Algebra. Waltham, Mass: Blaisdell, 1967. ISBN: 9780471000051.

Additional course notes by James Raymond Munkres, Professor of Mathematics, Emeritus, are also provided.

Real numbers
0Proof writing and set theoryI 2.1-2.4
1Axioms for the real numbersI 3.1-3.7
2Integers, induction, sigma notationI 4.1-4.6Course Notes A
3Least upper bound, triangle inequalityI 3.8-3.10, I 4.8Course Notes B
4Functions, area axioms1.2-1.10
The integral
5Definition of the integral1.12-1.17
6Properties of the integral, Riemann conditionCourse Notes C
7Proofs of integral properties88-90, 113-114Course Notes D
8Piecewise, monotonic functions1.20-1.21Course Notes E
Limits and continuity
9Limits and continuity defined3.1-3.4Course Notes F
10Proofs of limit theorems, continuity3.5-3.7
11Hour exam I
12Intermediate value theorem3.9-3.11
13Inverse functions3.12-3.14Course Notes G
14Extreme value theorem and uniform continuity3.16-3.18Course Notes H
Derivatives
15Definition of the derivative4.3-4.4, 4.7-4.8
16Composite and inverse functions4.10, 6.20Course Notes I
17Mean value theorem, curve sketching4.13-4.18
18Fundamental theorem of calculus5.1-5.3Course Notes K
19Trigonometric functionsCourse Notes L
Elementary functions; integration techniques
20Logs and exponentials6.3-6.7, 6.12-6.16Course Notes M
21IBP and substitution5.7, 5.9Course Notes N
22Inverse trig; trig substitution6.21
23Hour exam II
24Partial fractions6.23Course Notes N
Taylor's formula and limits
25Taylor's formula7.1-7.2
26Proof of Taylor's formulaCourse Notes O
27L'Hopital's rule and infinite limits7.12-7.16Course Notes P
Infinite series
28Sequences and series; geometric series10.1-10.6, 10.8 (first page only)
29Absolute convergence, integral test10.11, 10.13, 10.18
30Tests: comparison, root, ratio10.12, 10.15Course Notes Q
31Hour exam III
32Alternating series; improper integrals10.17, 10.23
Series of functions
33Sequences of functions, convergence11.1-11.2
34Power series11.3-11.4Course Notes R
35Properties of power seriesCourse Notes R
36Taylor series11.9Course Notes S
37Fourier seriesCourse Notes T

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