Intermediate Value Theorem By Daria Formal Definition Suppose

Intermediate Value Theorem By Daria

Formal Definition ● Suppose that f is a function that is continuous at every point on the interval [a, b] ○ f will take on every value between f(a) and f(b) over the interval ○ For any w between the values of f(a) and f(b) there exists at least one number c in the closed interval [a, b] for which f(c) = w

Informal Definition

How is this Applicable? The IVT can concretely prove that ● there is a point above a line ● and a point below that line ● that the function given is continuous

Try It Try to draw on a scrap piece of paper a graph that is continuous on the closed interval that does not follow the rules of the intermediate value theorem.

Example Question 1 Let f be a continuous function on the closed interval [− 1, 5], where f(-1)= -1 and h(5)=2 Which of the following is guaranteed by the Intermediate Value Theorem? A. B. C. D. f(c)=1 for at least one c between -1 and 5 f(c)=-2 for at least one c between -1 and 2 f(c)=-2 for at least one c between -1 and 5 f(c) = 1 for at least one c between -1 and 2

Plug it in ● Suppose that f is a function that is continuous at every point on the interval [-1, 5] ○ f will take on every value between -1 and 2 over the interval ○ For any w between the values of -1 and 2 there exists at least one number c in the closed interval [-1, 5] for which f(c) = w

Correct Answer Is. . . A. f(c)=1 for at least one c between -1 and 5

Application Problem Two cars are having a race. Five minutes after the race begins the red car is traveling 80 mph and a green car is traveling 75 mph. Fifteen minutes into the race the red car has slowed down to 70 mph and the green car has sped up to 79 mph. Can you prove that there is a moment in which the cars are going the same speed?

Answer Calculate the difference between the cars’ times f(5) = 80 - 75 = 5 f(15) = 70 - 79 = -9 Because one is positive and one is negative, there must be a moment in which f(t) = 0

Works Cited “Intermediate Value Theorem Review. ” Khan Academy, www. khanacademy. org/math/ap-calculus-ab/abexistence-theorems/ab-ivt-evt/a/intermediate-value-theorem-review “Intermediate Value Theorem. ” Khan Academy, www. khanacademy. org/math/ap-calculus-ab/abexistence-theorems/ab-ivt-evt/v/intermediate-value-theorem. “Intermediate Value Theorem. ” Definition Of, www. mathsisfun. com/algebra/intermediate-value-theorem. html. Foerster, Paul. “Calculus Concepts and Applications”

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