Intermediate Value Theorem

Austin and Max

Formal Definition

If the function f is continuous for all x in the closed interval [a,b], and y is a number between f(a) and f(b), then there is a number x = c in (a,b) for which f(c) = y.

Informal Definition

 Solution to certain functions is known  Continuous  Two values in-between solution

Proof

 Completeness axiom  Real number corresponding to every point on a number line  No “holes”

Visual

Example

F(x)= x 3 – 4x 2 + 2x + 7 Conclude that x=c is between 1 and 3 F(c) is exactly equal to 5 Approximate value of c

Example

F is continuous between [1,3] Therefore x=c occurs within the parameters Plot y = 5 The intersection point between [1,3] is 1.3111

Real World

 Max and Austin run the 1600m race. After one minute in the race, Max is running at 20 km/h while Austin is running a 15 km/h. After 3 minutes, Max is at 17 km/h while Austin is at 19 km/h.

 If the runners’ speed is a continuous function of time, during the race between 1 and 3 minutes are the runners running at exactly the same speed?

Solution

 F(t) = Max’s speed – Austin’s speed   F(1) = 20 – 15 = 5(+) F(3) = 17 - 19 = -2(-)  Speeds are continuous so the function is also  IVT applies  Solution: There is a value between 1 and 3 minutes that both runners are going at the same speed

Resource

Foerster, Paul A. "2-6 The Intermediate Value Theorem and Its Consquences." Calculus Concepts and Applications. Emeryville: Key Curriculum, 2005. 60-63. Print.