Mean value theorem for f(x) = x^2 on [1, 3] Step by Step

Step by step

The question

Question: find the number c from the mean value theorem for f(x) = x² on [1, 3].

f(x) = x²

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 [1, 3], and smooth between.

The theorem can be used.

a polynomial: no breaks, no corners ✓

The ends: f of 1 is 1.

f(1) = 1² = 1

And f of 3 is 9.

f(3) = 3² = 9

2 The average slope

The average slope from 1 to 3: the change in f over the change in x.

It is 4.

f(3) − f(1)3 − 1 = 9 − 13 − 1 = 82 = 4

Careful!

Divide by the width, 3 minus 1.

The average slope is 4, not 8.

Careful: not f(3) − f(1) = 8 ✗

3 Find c

The slope: use the power rule.

f′(x) = 2x^(2 − 1)

Now we have f′(x) = 2x.

= 2x

At c the slope is the average slope: f′(c) = 4.

2c = 4

Divide by 2: c = 2.

c = 42 = 2

4 Keep the ones inside

c = 2 is between 1 and 3.

Keep it.

c = 2 between 1 and 3 ✓

So c = 2.

= c = 2

5 Check

Let's check: put c into f′.

We get 4, the average slope.

Right!

f′(2) = 4

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