Antiderivatives
If two polynomial functions \(f(x)\) and \(g(x)\) satisfy \[\frac{d}{dx}\{f(x)+g(x)\}=7, \frac{d}{dx}\{f(x)g(x)\}=12x-1, f(0)=0, g(0)=-1,\] what is \(f(1)-g(1)?\)
If \[ f(x)=\int \left\{\frac{d}{dx} \left(x^3-5x^2+18x\right)\right\} dx \text{ and } f(1)=2,\] what is \(f(3)?\)
If \(a ,b,\) and \(c\) are constants satisfying \[\int \left( 12x^3 +ax-9\right)dx=bx^4+3x^2+cx+C,\] what is the value of \(a+b+c?\)
Details and assumptions
\(C\) is the constant of integration.
If \(a, b\) and \(c\) are constants and \[\int (18x^2+ax+1)dx=bx^3+8x^2+cx+C,\] where \(C\) is the constant of integration, what is \(a+b+c?\)
If \(f(x)\) satisfies \[\int (2x+1)f(x)dx = x^4-2x^3-2x^2+C,\] what is the value of \(f(5)?\)
Details and assumptions
\(C\) is the constant of integration.