Question

Find the rate of change of f on [x,x+h] if the function is f(x)=3x^2+4x-6

Answer 1

$\bf slope = m = \cfrac{rise}{run} \implies \cfrac{ f(x_2) - f(x_1)}{ x_2 - x_1}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array}\\\\[-0.35em] \rule{34em}{0.25pt}\\\\ f(x)= 3x^2+4x-6\qquad \begin{cases} x_1=x\\ x_2=x+h \end{cases}\implies \cfrac{f(x+h)-f(x)}{(x+h)-x} \\\\\\ \cfrac{[3(x+h)^2+4(x+h)-6]~~-~~[3x^2+4x-6]}{h} \\\\\\ \cfrac{[3(x^2+2hx+h^2)+4x+4h-6]~~~~-3x^2-4x+6}{h}$

$\bf \cfrac{[\underline{3x^2}+6hx+3h^2\underline{+4x}+4h\underline{-6}]~~~~\underline{-3x^2-4x+6}}{h}\implies \cfrac{6hx+3h^2+4h}{h} \\\\\\ \cfrac{\underline{h}(6x+3h+4)}{\underline{h}}\implies 6x+3h+4$