Question

(c) Compute the difference quotienttor is canceled.f(x + h) - f(x)hSimplify until the h in the denominator is canceled

Answer 1

Given:

The difference quotient;

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

where;

`f(x)=x^2-2x+3`
`\begin{gathered} \text{If} \\ f(x)=x^2-2x+3,\text{ then} \\ f(x+h)=(x+h)^2-2(x+h)+3 \end{gathered}`

Substituting the two functions above in the difference quotient formula;

`\begin{gathered} (f(x+h)-f(x))/(h)=((x+h)^2-2(x+h)+3-(x^2-2x+3))/(h) \\ =\frac{(x+h)(x+h)-2x-2h+3-x^2+2x-3^{}}{h} \\ =\frac{(x^2+2xh+h^2)-2x-2h+3-x^2+2x-3^{}}{h} \\ \text{Collecting the like terms,} \\ =\frac{x^2-x^2-2x+2x+3-3+2xh+h^2-2h^{}}{h} \\ =(2xh+h^2-2h)/(h) \\ \text{Factorizing h from the numerator;} \\ =(h(2x+h-2))/(h) \\ \text{Cancelling out the h in the numerator and denominator, we have;} \\ 2x+h-2 \\ =2x-2+h \end{gathered}`

Therefore, the difference quotient is;

`2x-2+h`