Step by step
The question
Question: find y, if dy/dx = 6x² − 6x + 5 and y = 1 when x = 0.
dydx = 6x2 − 6x + 5
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
y = 6 × x33 − 6 × x22 + 5x + C
Now we have y = 2x³ − 3x² + 5x + C.
= 2x3 − 3x2 + 5x + 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 and y = 1.
1 = 2(0)3 − 3(0)2 + 5(0) + C
The other terms are 0 at x = 0.
So C = 1.
C = 1
3 Write the function
So y = 2x³ − 3x² + 5x + 1.
y = 2x3 − 3x2 + 5x + 1
4 Check
Let's check: its derivative is 6x² − 6x + 5, the slope we were given.
dy/dx = 6x2 − 6x + 5
And at x = 0 it gives 1.
Right!
x = 0: 0 + 1 = 1
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
y = 2x3 − 3x2 + 5x + 1