Discovering the Intermediate Value Theorem

Calculus AB

Discovery Worksheet 1.4b

Discovering the Intermediate Value Theorem

Name ______

I. Intermediate Value Theorem – Specific case of finding zeros of polynomials

1. Consider the function described by on the interval [1, 2].

a.  Find f(1) ______

b.  Find f(2) ______

c.  Is there a place between x = 1 and x = 2 where f(x) = 0? Explain. If you are having problems answering, you may wish to skip this question for now and answer part d. Return to this question when you finish part d.

d.  Using your graphing calculator, sketch the graph of the above function. On this sketch, label the values of f(1), f(2), and where f(x)=0. Round approximations to three decimal places.

2. Consider the function described by on the interval [-3, 2].

a.  Find g(-3) ______

b.  Find g(1) ______

c.  Find g(2) ______

d.  Is there a place between x = -3 and x = 2 where g(x) = 0? Explain. If you are having problems answering, you may wish to skip this question for now and answer part e. Return to this question when you finish part e.

e.  Using your graphing calculator, sketch the graph of the above function. On this sketch, label the values of g(-3), g(1), g(2) and where g(x)=0. Round approximations to three decimal places.

3. Consider the function described by on the interval [-3, 2].

a. Find h(-3) ______

b. Find h(1) ______

c. Find h(2) ______

d. Is there a place between x = -3 and x = 2 where h(x) = 0? Explain. If you are having problems answering, you may wish to skip this question for now and answer part e. Return to this question when you finish part e.

e. Using your graphing calculator, sketch the graph of the above function. On this sketch, label the values of h(-3), h(1), h(2) and where h(x)=0. Round approximations to three decimal places.

4. At this point, we have enough information to arrive at the Intermediate Value Theorem and how it may be used to find zeros of functions. Please complete the following:

5. Without using a calculator, use the Intermediate Value Theorem to determine whether the function will have a zero on the interval [0, ].

II. Intermediate Value Theorem – General case

6. Let’s take a look at a more general version of the Intermediate Value Theorem. Using your results from question #1 above, answer each of the following:

a.  Does an x value exist on the interval [1, 2] so that f(x) must equal 7? Explain.

b.  What other values must f(x) equal on the interval [1, 2]?

7. At this point, we have enough information to arrive at the general version of the Intermediate Value Theorem. Please complete the following:

8. Use the Intermediate Value Theorem to determine if your oven is at , as it cools down after turning it off, at some instant must its temperature be exactly ?