Worked examples
The question
Question: find f(x), if f′(x) = 2x and f(0) = 4.
f′(x) = 2x
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = 2 × x22 + C
Now we have f(x) = x² + C.
= 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 = 0.
f(0) = 02 + C
The other terms are 0 at x = 0.
So C = 4.
C = 4
3 Write the function
So f(x) = x² + 4.
f(x) = x2 + 4
4 Check
Let's check: its derivative is 2x, the slope we were given.
f′(x) = 2x
And at x = 0 it gives 4.
Right!
x = 0: 0 + 4 = 4
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
▶ Watch it solved
The question
Question: find f(x), if f′(x) = 3x² + 2 and f(1) = 4.
f′(x) = 3x2 + 2
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = 3 × x33 + 2x + C
Now we have f(x) = x³ + 2x + C.
= x3 + 2x + 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) = 13 + 2(1) + C
Work it out: 3 + C.
= 3 + C
It must equal 4.
3 + C = 4
Take away 3 on both sides: C = 1.
C = 4 − 3 = 1
Careful!
C is not just the f value.
The other terms at x = 1 are not 0 here.
Careful: not C = 4 ✗
3 Write the function
So f(x) = x³ + 2x + 1.
f(x) = x3 + 2x + 1
4 Check
Let's check: its derivative is 3x² + 2, the slope we were given.
f′(x) = 3x2 + 2
And at x = 1 it gives 4.
Right!
x = 1: 3 + 1 = 4
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer f(x) = x3 + 2x + 1
▶ Watch it solved
The question
Question: find f(x), if f′(x) = 6x² − 4x and f(1) = 3.
f′(x) = 6x2 − 4x
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = 6 × x33 − 4 × x22 + C
Now we have f(x) = 2x³ − 2x² + C.
= 2x3 − 2x2 + 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) = 2(1)3 − 2(1)2 + C
The other terms are 0 at x = 1.
So C = 3.
C = 3
3 Write the function
So f(x) = 2x³ − 2x² + 3.
f(x) = 2x3 − 2x2 + 3
4 Check
Let's check: its derivative is 6x² − 4x, the slope we were given.
f′(x) = 6x2 − 4x
And at x = 1 it gives 3.
Right!
x = 1: 0 + 3 = 3
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer f(x) = 2x3 − 2x2 + 3
▶ Watch it solved
