Step by step
The question
Question: find f(x), if f′(x) = 2x + 3 and f(1) = 6.
f′(x) = 2x + 3
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = 2 × x22 + 3x + C
Now we have f(x) = x² + 3x + C.
= x2 + 3x + 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) = 12 + 3(1) + C
Work it out: 4 + C.
= 4 + C
It must equal 6.
4 + C = 6
Take away 4 on both sides: C = 2.
C = 6 − 4 = 2
Careful!
C is not just the f value.
The other terms at x = 1 are not 0 here.
Careful: not C = 6 ✗
3 Write the function
So f(x) = x² + 3x + 2.
f(x) = x2 + 3x + 2
4 Check
Let's check: its derivative is 2x + 3, the slope we were given.
f′(x) = 2x + 3
And at x = 1 it gives 6.
Right!
x = 1: 4 + 2 = 6
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
f(x) = x2 + 3x + 2