Step by step
The question
Question: find f(x), if f′(x) = x² and f(3) = 10.
f′(x) = x2
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = x33 + C
Now we have f(x) = x³ over 3 + C.
= x33 + 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 = 3.
f(3) = 333 + C
Work it out: 9 + C.
= 9 + C
It must equal 10.
9 + C = 10
Take away 9 on both sides: C = 1.
C = 10 − 9 = 1
Careful!
C is not just the f value.
The other terms at x = 3 are not 0 here.
Careful: not C = 10 ✗
3 Write the function
So f(x) = x³ over 3 + 1.
f(x) = x33 + 1
4 Check
Let's check: its derivative is x², the slope we were given.
f′(x) = x2
And at x = 3 it gives 10.
Right!
x = 3: 9 + 1 = 10
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
f(x) = x33 + 1