Step by step
The question
Question: find f(x), if f′(x) = 1 − 3x² and f(2) = 0.
f′(x) = 1 − 3x2
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = x − 3 × x33 + C
Now we have f(x) = x − x³ + C.
= x − x3 + 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) = 2 − 23 + C
Work it out: −6 + C.
= −6 + C
It must equal 0.
−6 + C = 0
Add 6 on both sides: C = 6.
C = 0 + 6 = 6
Careful!
C is not just the f value.
The other terms at x = 2 are not 0 here.
Careful: not C = 0 ✗
3 Write the function
So f(x) = x − x³ + 6.
f(x) = x − x3 + 6
4 Check
Let's check: its derivative is 1 − 3x², the slope we were given.
f′(x) = 1 − 3x2
And at x = 2 it gives 0.
Right!
x = 2: −6 + 6 = 0
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
f(x) = x − x3 + 6