Skip to lesson

Choose “Save as PDF” in the print dialog to keep a copy.

Cogito

Calculus · Chapter 2 · Lesson 3

Practice Worksheet

The Intermediate Value Theorem

A continuous graph cannot skip a value.

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

NameDate

Figure — use these to answer the problems

Move the coefficients so the curve starts below the axis and ends above it. It must cross somewhere.
-8-8-6-6-4-4-2-222446688
y = 1x³ − 2x − 1
a1
b-2
c-1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. f(1) = −3 and f(4) = 6. Does the theorem guarantee a root between them? 1 or 0.

    Answer

  2. Why does the theorem require continuity?

    • A jump lets the function step over the value entirely.
    • Because the graph must be smooth.

  3. f(x) = x² − 2. What is f(1)?

    Answer

  4. Same function. What is f(2)?

    Answer

  5. Does the theorem guarantee a root between 1 and 2 there? 1 for yes, 0 for no.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. f(a) = 3 and f(b) = 9, both positive. Does the theorem guarantee a root? 1 or 0.

    Answer

  2. f(a) = −4 and f(b) = 4. Does it guarantee a root? 1 or 0.

    Answer

  3. Does the theorem tell you how many roots there are? 1 for yes, 0 for no.

    Answer

Stretch Yourself

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

4 problems

  1. Put the argument in order.

    Write 1 to 4 in the boxes to put these in order.

    • Confirm the function is continuous on the closed interval.
    • Evaluate the function at both endpoints.
    • Check that the target value lies between them.
    • Conclude the value is attained somewhere inside.

  2. f(x) = x³. What is f(−2)?

    Answer

  3. The Endpoints

    f(x) = x³ − x − 1. What is f(1)?

    Answer

  4. The Other End

    Same function. What is f(2)?

    Answer