Step by step
The question
Question: find f(x), if f′(x) = 12x² − 2x and f(1) = 1.
f′(x) = 12x2 − 2x
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = 12 × x33 − 2 × x22 + C
Now we have f(x) = 4x³ − x² + C.
= 4x3 − 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 = 1.
f(1) = 4(1)3 − 12 + C
Work it out: 3 + C.
= 3 + C
It must equal 1.
3 + C = 1
Take away 3 on both sides: C = −2.
C = 1 − 3 = −2
Careful!
C is not just the f value.
The other terms at x = 1 are not 0 here.
Careful: not C = 1 ✗
3 Write the function
So f(x) = 4x³ − x² − 2.
f(x) = 4x3 − x2 − 2
4 Check
Let's check: its derivative is 12x² − 2x, the slope we were given.
f′(x) = 12x2 − 2x
And at x = 1 it gives 1.
Right!
x = 1: 3 − 2 = 1
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
f(x) = 4x3 − x2 − 2