3.4: Intermediate Value Theorem

3.4: Intermediate Value Theorem Pre Calculus II (CP) November 29, 2011

First, lets define... • continuous function: • unbroken curve • can be drawn without taking your pencil off the paper. • closed interval: • includes the endpoints [a, b] • root • a solution to an equation – the place where the graph crosses the x-axis.

EXAMPLES 1,2 • EX 1: Consider (a) Is it continuous on [0, 5]? (b) Is it continuous on [-3. 5] (c) Is it continuous on the set of all reals? Is this a closed interval? • EX 2: Consider (a) Is it continuous on [-1, 1]? (b) Is it continuous on [1, 5]?

Intermediate Value Theorem • ...says a function will never take on 2 values without taking on all the values in between • Thm: If a function f(x) is a continuous function on a closed interval [a,b], it takes on every (y) value between f(a) and f(b).

Illustration - IVT

EXAMPLE 3 • Use the IVT to explain whether has a solution between x =2 and x = 3. f(2) = f(3) = Conclusion:

EXAMPLE 4 Suppose . A table of values is given below. How many roots are there on the interval [-4, 5]? Justify your answer with a theorem.

EXAMPLE 5 Is any real number exactly 1 less than its cube? Let x = the number. Equation: f(0) f(1) f(2) Answer?

EXAMPLE 6 • EX: How many solutions does have on [0, 10]? Justify your answer.

Homework • Section 3.4 (p. 172-174) # 4, 5, 7-13, 15, 16.

3.4: Intermediate Value Theorem