f′(x) = 2x, f(0) = 4, 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) = 2x and f(0) = 4.

f′(x) = 2x

1 Find the antiderivative

Go backwards: the antiderivative.

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

And + C.

f(x) = 2 × x²2 + C

Now we have f(x) = x² + C.

= x² + 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 = 0.

f(0) = 0² + C

The other terms are 0 at x = 0.

So C = 4.

C = 4

3 Write the function

So f(x) = x² + 4.

f(x) = x² + 4

4 Check

Let's check: its derivative is 2x, the slope we were given.

f′(x) = 2x

And at x = 0 it gives 4.

Right!

x = 0: 0 + 4 = 4

Let's look back.

First: find the antiderivative.

Then: find C.

Then: write the function.

Last: check.

Answer

f(x) = x² + 4

Your turn

Try this one yourself, then check:

Find f(x), if f′(x) = 6x

Show the answer

f(x) = 3x² + 1