The Algebra of Functions

Question

For f(x)=4x^3, construct and simplify the difference quotient f(x+h)-f(x)/h

 

 

 

 

Just so there is no confusion

 

f(x+h)-f(x)/h means f(x+h)-f(x) is all over h

Answer 1

f(x) = 4x³ + 3

 

You want to find the "difference quotient":

 

(f(x+h) - f(x))/h

 

     f(x+h) = 4(x+h)³ + 3 = 4(x³ + 3x²h +  3xh² + h³) + 3 = 4x³ + 12x²h + 12xh² + h³ + 3

 

f(x+h)-f(x)/h = (4x³ + 12x²h + 12xh² + h³ + 3 - 4x³ - 3)/h

 

    = (12x²h + 12xh² + h³)/h

 

    = 12x² + 12xh + h²