Question

Write the function that has derivative y prime = `x^2` and passes through (1,1) ?

Answer 1

`implies y = 1/3 (x^3 + 2)`

Explanation:

I'd follow the instructions on the packet.

`y' = x^2`

`implies y = x^3/3 + C`

Applying the IV's:

`y(1) = 1 implies 1 = 1/3 + C implies C = 2/3`

So:

`implies y = x^3/3 + 2/3 = 1/3 (x^3 + 2)`

Answer 2

`y=(x^3)/3+2/3`
`color(white)("XXXXXXXXX")`although, maybe, that's only my opinion.

Explanation:

If `y' = x^2`
then
`color(white)("XXX")y=(x^3)/3+c` for some constant `c`

If `(color(red)x,color(blue)y)=(color(red)1,color(blue)1)` is a solution to this equation
`color(white)("XXX")color(blue)1=color(red)1^3/3+c=1/3+c`

`color(white)("XXXXXX")rarr c=2/3`

and the function is
`color(white)("XXX")y=x^3/3+2/3`