The Intermediate Value Theorem

A continuous graph cannot skip a value.

12 problems · about 21 minutes · FUN-1.A

What this lesson teaches

The student applies the Intermediate Value Theorem to guarantee the existence of a root.

A continuous function on a closed interval attains every value between its endpoint values.
A sign change guarantees a root.
The theorem promises existence only, never location or count.

Straightforward practice. Get the method working first.

5 problems

f(1) = −3 and f(4) = 6. Does the theorem guarantee a root between them? 1 or 0.
Answer 1
Why Yes, assuming continuity.
Why does the theorem require continuity?
Answer A jump lets the function step over the value entirely.
Why A discontinuous function can skip values.
f(x) = x² − 2. What is f(1)?
Answer -1
Why 1 − 2.
Same function. What is f(2)?
Answer 2
Why 4 − 2.
Does the theorem guarantee a root between 1 and 2 there? 1 for yes, 0 for no.
Answer 1
Why The values change sign.

The same ideas with more to keep track of.

3 problems

f(a) = 3 and f(b) = 9, both positive. Does the theorem guarantee a root? 1 or 0.
Answer 0
Why Zero is not between 3 and 9.
f(a) = −4 and f(b) = 4. Does it guarantee a root? 1 or 0.
Answer 1
Why Zero lies between them.
Does the theorem tell you how many roots there are? 1 for yes, 0 for no.
Answer 0
Why It only promises at least one.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Put the argument in order.
Answer 1. Confirm the function is continuous on the closed interval. 2. Evaluate the function at both endpoints. 3. Check that the target value lies between them. 4. Conclude the value is attained somewhere inside.
Why Continuity has to be established before anything else.
f(x) = x³. What is f(−2)?
Answer -8
Why (−2)³.
The Endpoints: f(x) = x³ − x − 1. What is f(1)?
Answer -1
Why −1.
The Other End: Same function. What is f(2)?
Answer 5
Why 5.