AP®MathsCollege BoardCalculus ABRevision NotesUnit 1: Limits & ContinuityContinuityIntermediate Value Theorem

Intermediate Value Theorem (College Board AP® Calculus AB)

Revision Note

Roger B

Author

Roger B

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Maths

Intermediate value theorem

What is the intermediate value theorem?

  • The intermediate value theorem says that:

    • If f is a continuous function on the closed interval open square brackets a comma space b close square brackets

    • and if d is a number between f open parentheses a close parentheses and f open parentheses b close parentheses

    • then there is at least one number c between a and b such that f open parentheses c close parentheses equals d

  • In practical terms this means that

    • If a function continuous on an interval open square brackets a comma space b close square brackets starts with value f open parentheses a close parentheses and ends with value f open parentheses b close parentheses

    • Then somewhere between a and b the function takes on every value between f open parentheses a close parentheses and f open parentheses b close parentheses

  • This seems really obvious if you think about the 'a function whose graph I can sketch without taking my pencil off the paper' way of describing continuity

    • But it is an incredibly important result in mathematics

Exam Tip

On the exam it's important to justify any use of the intermediate value theorem.

  • Often this means explaining how you know the function in question is continuous

  • Remember that if a function is differentiable, then it is continuous

    • And if a function is twice-differentiable then both the function and its derivative are continuous

Worked Example

A social sciences researcher is using a function m to model the total mass of all the garden gnomes appearing on lawns in a particular neighborhood at time t. The function m is twice-differentiable, with m open parentheses t close parentheses measured in kilograms and t measured in days.

The table below gives selected values of m to the power of apostrophe open parentheses t close parentheses, the rate of change of the mass, over the time interval 0 less or equal than t less or equal than 12.

t

(days)

0

3

7

10

12

m to the power of apostrophe open parentheses t close parentheses

(kilograms per day)

2.6

4.8

12.2

0.7

-1.3

Is there a time t, 0 less or equal than t less or equal than 3, for which m to the power of apostrophe open parentheses t close parentheses equals 4? Justify your answer.

Answer:

This is a job for the intermediate value theorem, but first you have to justify why m to the power of apostrophe open parentheses t close parentheses is continuous

m is twice-differentiable, which means m and m to the power of apostrophe are both continuous

m open parentheses t close parentheses is twice-differentiable, which means m to the power of apostrophe open parentheses t close parentheses is differentiable, which means m to the power of apostrophe open parentheses t close parentheses is continuous

m to the power of apostrophe open parentheses 0 close parentheses equals 2.6 less than 4 less than 4.8 equals m to the power of apostrophe open parentheses 3 close parentheses

Therefore by the intermediate value theorem there is a time t, 0 less or equal than t less or equal than 3, for which m to the power of apostrophe open parentheses t close parentheses equals 4

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    Roger B

    Author: Roger B

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.