Average value of f(x) = 4x - 2 on [0, 3], and find c Step by Step

Step by step

The question

Question: find the average value of f(x) = 4x − 2 on [0, 3], and the number c where f(c) is the average.

f(x) = 4x − 2

1 The formula

The average value: the area under the curve, divided by the width of the interval.

average = 1b − a∫ from a to b of f(x) dx

Here the width is 3 minus 0: 3.

= 13 − 0∫ from 0 to 3 of (4x − 2) dx

Think of it as the height of a rectangle with the same area as under the curve.

average height × width = area

2 The integral

The antiderivative: add 1 to each power, divide by the new power.

Now we have 2 x² − 2x.

∫ from 0 to 3 of (4x − 2) dx = (2x² − 2x) from 0 to 3

Put in 3, then take away the same with 0.

= (2(3)² − 2(3)) − (2(0)² − 2(0))

That is 12 minus 0: 12.

= 12 − 0 = 12

3 Divide by the width

Divide by the width, 3.

The average value is 4.

average = 123 = 4

Careful!

12 is the area.

To get the average height, divide it by the width, 3.

Careful: not average = 12 ✗

4 Where f(c) is the average

Now find c where f is exactly the average.

f(c) = 4

4c − 2 = 4

Add 2 on both sides.

4c = 6

Divide by 4: c = 3/2.

c = 32

5 Check

Let's check: a rectangle 4 high and 3 wide has area 12: the same as the integral.

Right!

4 × 3 = 12 = area

Let's look back.

First: the formula.

Then: the integral.

Then: divide by the width.

Then: where f(c) is the average.

Last: check.

Answer