Step by step
The question
Question: find y, if dy/dx = 3x² − 2x and y = 4 when x = 2.
dydx = 3x2 − 2x
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
y = 3 × x33 − 2 × x22 + C
Now we have y = x³ − x² + C.
= x3 − 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 = 2 and y = 4.
4 = 23 − 22 + C
Work it out: 4 + C.
= 4 + C
It must equal 4.
4 + C = 4
Take away 4 on both sides: C = 0.
C = 4 − 4 = 0
Careful!
C is not just the y value.
The other terms at x = 2 are not 0 here.
Careful: not C = 4 ✗
3 Write the function
So y = x³ − x².
y = x3 − x2
4 Check
Let's check: its derivative is 3x² − 2x, the slope we were given.
dy/dx = 3x2 − 2x
And at x = 2 it gives 4.
Right!
x = 2: 4 + 0 = 4
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
y = x3 − x2