f′(x) = 6x^2 - 4x, f(1) = 3, find f(x)

Step by step, with the reason for every step · Initial value problems · Calculus 1

Step by step

The question

Question: find f(x), if f′(x) = 6x² − 4x and f(1) = 3.

f′(x) = 6x² − 4x

1 Find the antiderivative

Go backwards: the antiderivative.

Add 1 to each power, divide by the new power.

And + C.

f(x) = 6 × x³3 − 4 × x²2 + C

Now we have f(x) = 2x³ − 2x² + C.

= 2x³ − 2x² + C

Every antiderivative has + C.

The point will tell us what C is.

+ C: we do not know it yet

2 Find C

Put in x = 1.

f(1) = 2(1)³ − 2(1)² + C

The other terms are 0 at x = 1.

So C = 3.

C = 3

3 Write the function

So f(x) = 2x³ − 2x² + 3.

f(x) = 2x³ − 2x² + 3

4 Check

Let's check: its derivative is 6x² − 4x, the slope we were given.

f′(x) = 6x² − 4x

And at x = 1 it gives 3.

Right!

x = 1: 0 + 3 = 3

Let's look back.

First: find the antiderivative.

Then: find C.

Then: write the function.

Last: check.

Answer

f(x) = 2x³ − 2x² + 3

Your turn

Try this one yourself, then check:

Find f(x), if f′(x) = 6x² − 4x

Show the answer

f(x) = 2x³ − 2x² − 1