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
Intermediate Value Theorem Formulax) has the 'Intermediate Value Property' even on closedintervals containing x = 0.
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All textbook readings are from:
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.
|SES #||TOPICS||TEXTBOOK READINGS||COURSE NOTES READINGS|
|0||Proof writing and set theory||I 2.1-2.4|
|1||Axioms for the real numbers||I 3.1-3.7|
|2||Integers, induction, sigma notation||I 4.1-4.6||Course Notes A|
|3||Least upper bound, triangle inequality||I 3.8-3.10, I 4.8||Course Notes B|
|4||Functions, area axioms||1.2-1.10|
|5||Definition of the integral||1.12-1.17|
|6||Properties of the integral, Riemann condition||Course Notes C|
|7||Proofs of integral properties||88-90, 113-114||Course Notes D|
|8||Piecewise, monotonic functions||1.20-1.21||Course Notes E|
|Limits and continuity|
|9||Limits and continuity defined||3.1-3.4||Course Notes F|
|10||Proofs of limit theorems, continuity||3.5-3.7|
|11||Hour exam I|
|12||Intermediate value theorem||3.9-3.11|
|13||Inverse functions||3.12-3.14||Course Notes G|
|14||Extreme value theorem and uniform continuity||3.16-3.18||Course Notes H|
|15||Definition of the derivative||4.3-4.4, 4.7-4.8|
|16||Composite and inverse functions||4.10, 6.20||Course Notes I|
|17||Mean value theorem, curve sketching||4.13-4.18|
|18||Fundamental theorem of calculus||5.1-5.3||Course Notes K|
|19||Trigonometric functions||Course Notes L|
|Elementary functions; integration techniques|
|20||Logs and exponentials||6.3-6.7, 6.12-6.16||Course Notes M|
|21||IBP and substitution||5.7, 5.9||Course Notes N|
|22||Inverse trig; trig substitution||6.21|
|23||Hour exam II|
|24||Partial fractions||6.23||Course Notes N|
|Taylor's formula and limits|
|26||Proof of Taylor's formula||Course Notes O|
|27||L'Hopital's rule and infinite limits||7.12-7.16||Course Notes P|
|28||Sequences and series; geometric series||10.1-10.6, 10.8 (first page only)|
|29||Absolute convergence, integral test||10.11, 10.13, 10.18|
|30||Tests: comparison, root, ratio||10.12, 10.15||Course Notes Q|
|31||Hour exam III|
|32||Alternating series; improper integrals||10.17, 10.23|
|Series of functions|
|33||Sequences of functions, convergence||11.1-11.2|
|34||Power series||11.3-11.4||Course Notes R|
|35||Properties of power series||Course Notes R|
|36||Taylor series||11.9||Course Notes S|
|37||Fourier series||Course Notes T|
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