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:

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