Mean value theorem for f(x) = x^2 - 2x on [0, 3] Step by Step

Step by step

The question

Question: find the number c from the mean value theorem for f(x) = x² − 2x on [0, 3].

f(x) = x² − 2x

The theorem: for some c between a and b, the slope at c equals the average slope from a to b.

f′(c) = f(b) − f(a)b − a

for some c between a and b

1 Check the conditions

f is a polynomial.

Its graph has no breaks and no sharp corners.

So it is continuous on [0, 3], and smooth between.

The theorem can be used.

a polynomial: no breaks, no corners ✓

The ends: f of 0 is 0.

f(0) = 0² − 2(0) = 0 + 0 = 0

And f of 3 is 3.

f(3) = 3² − 2(3) = 9 − 6 = 3

2 The average slope

The average slope from 0 to 3: the change in f over the change in x.

It is 1.

f(3) − f(0)3 − 0 = 3 − 03 − 0 = 33 = 1

Careful!

Divide by the width, 3 minus 0.

The average slope is 1, not 3.

Careful: not f(3) − f(0) = 3 ✗

3 Find c

The slope: use the power rule.

f′(x) = 2x^(2 − 1) − 2 × 1

Now we have f′(x) = 2x − 2.

= 2x − 2

At c the slope is the average slope: f′(c) = 1.

2c − 2 = 1

Add 2 on both sides, so they stay equal.

2c = 3

Divide by 2: c = 3/2.

c = 32 = 32

4 Keep the ones inside

c = 3/2 is between 0 and 3.

Keep it.

c = 32 ≈ 1.5 between 0 and 3 ✓

So c = 3/2.

= c = 32 ≈ 1.5

5 Check

Let's check: put c into f′.

We get 1, the average slope.

Right!

f′(32) = 1

And in a picture.

The line through the two ends has the average slope.

At c, the tangent line is parallel to it: the same slope.

Right!

Let's look back.

First: check the conditions.

Then: the average slope.

Then: find c.

Then: keep the ones inside.

Last: check.

Answer