Question

find the difference quotient of f;that is, find f(x+h)-f(x)/h, h=0, for the following function f(x)=2×^2+×+3f(×+h)-f(×)/h

Answer 1

Let's begin by listing out the given information:

`\begin{gathered} f\mleft(x\mright)=2x^2+x+3 \\ (f\mleft(x+h\mright)-f\mleft(x\mright))/(h) \\ f(x+h)=2(x+h)^2+(x+h)+3 \\ f(x+h)=2(x^2+2xh+h^2)+(x+h)+3 \\ f(x+h)=2x^2+4xh+2h^2+x+h+3 \\ We\text{ find the difference, we have:} \\ (f\mleft(x+h\mright)-f\mleft(x\mright))/(h)\Rightarrow(2x^2+4xh+2h^2+x+h+3-\lbrack2x^2+x+3\rbrack)/(h) \\ \Rightarrow(4xh+2h^2+h)/(h) \\ We\text{ factorise, we have:} \\ (h(4x+2h+1))/(h) \\ \Rightarrow4x+2h+1 \\ \\ \therefore(f(x+h)-f(x))/(h)=4x+2h+1 \end{gathered}`