For solving initial value problems is there any way to do it without a calculator?
For example: f"(x)=12x^2-6x+8, f(-1)=5 f'(-1)=-21
Yes, just integrate the f''(x) twice, and use the values of f' and f at a particular x value (x=-1 in this case) for the constants.
I will do the first step for you.
f'(x) = 4x^3/3 -3x^2 +8x + C
f'(-1) = -4/3 -3 -8 + C = 5
C = 17 1/3
f(x) = x^4/3 -x^3 + 4x^2 + 17 1/3 x = C'
Use f'(-1) = -21 to solve for C'