Hello!

Denote this given but unknown `f ( 0 ) ` as `y_0 .`

If a derivative of a function `f ( x ) ` is given, the function itself can be found by integration as `f ( x ) = int_( x_0 )^x f ( t ) dt ,...

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Hello!

Denote this given but unknown `f ( 0 ) ` as `y_0 .`

If a derivative of a function `f ( x ) ` is given, the function itself can be found by integration as `f ( x ) = int_( x_0 )^x f ( t ) dt , ` where `x_0 ` is an arbitrary point in the function's domain.

Note that for different `x_0 ` the results will differ by some constant. Indeed, it is impossible to determine `f ( x ) ` uniquely knowing only its derivative, the result is always up to some constant. For example, `f ( x ) = 0 ` and `f ( x ) = 1 ` both have the same derivative `0 .`

But if we are given some function value at a specific point, we can determine the constant, too.

Now integrate our function starting, say, at the given point `x_0 = 0 :`

`int_0^x f ( t ) dt = int_0^x ( 3 t - 1 ) dt = ( 3 / 2 t^2 - t )_( t = 0 )^x = 3 / 2 x^2 - x .`

What is the constant? The function `g ( x ) = 3 / 2 x^2 - x ` that we found has the value `0 ` at `x_0 = 0 , ` so it is not exact what we seek for. Add `y_0 ` to it and

`f ( x ) = 3 / 2 x^2 - x + y_0` will be the needed function.