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Table of Contents 8. Section 2.7 Intermediate Value Theorem.

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Presentation on theme: "Table of Contents 8. Section 2.7 Intermediate Value Theorem."— Presentation transcript:

1 Table of Contents 8. Section 2.7 Intermediate Value Theorem

2 Essential Question What is the IVT used for?

3 Intermediate value theorem
A function that is continuous on a closed interval takes on every value between f(a) and f(b).

4 In other words….. A functions that is continuous can’t skip values

5 Example Prove that sin x = 0.3 has at least one solution.
We can apply the IVT because sin x is continuous Choose an interval (0, 𝜋 2 ) Sin(0) = 0, sin( 𝜋 2 ) = 1 0.3 is between these 2 values Therefore, the IVT tells us that sin x = 0.3 has a solution in (0, 𝜋 2 ) Could choose other intervals as well

6 Existence of zeros The IVT can be used to show the existence of zeros
If one value of the function is negative and another is positive, there must be a zero somewhere in between

7 Bisection Method Cut intervals in half successively to narrow down where zero is

8 Example Show that 𝑐𝑜𝑠 2 𝑥 −2𝑠𝑖𝑛 𝑥 4 has a zero in (0,2)
f(0) = 1 f(2) = Opposite signs, therefore a zero

9 Example cont. Now use bisection method to find zero more accurately
Find value halfway in between f(1) = Use interval that changes signs (0,1) f(1/2) = (1/2,1) f(3/4) = (3/4,1) f(7/8) = (3/4, 7/8) Function has zero in this interval

10 Assignment Pg 106: #1-7 odd, odd


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