Step by step
The question
Question: find f(x), if f′(x) = 3x² − 4x + 1 and f(2) = 5.
f′(x) = 3x2 − 4x + 1
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = 3 × x33 − 4 × x22 + x + C
Now we have f(x) = x³ − 2x² + x + C.
= x3 − 2x2 + x + 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 = 2.
f(2) = 23 − 2(2)2 + 2 + C
Work it out: 2 + C.
= 2 + C
It must equal 5.
2 + C = 5
Take away 2 on both sides: C = 3.
C = 5 − 2 = 3
Careful!
C is not just the f value.
The other terms at x = 2 are not 0 here.
Careful: not C = 5 ✗
3 Write the function
So f(x) = x³ − 2x² + x + 3.
f(x) = x3 − 2x2 + x + 3
4 Check
Let's check: its derivative is 3x² − 4x + 1, the slope we were given.
f′(x) = 3x2 − 4x + 1
And at x = 2 it gives 5.
Right!
x = 2: 2 + 3 = 5
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
f(x) = x3 − 2x2 + x + 3