Find the indicated difference quotient and simplify your answer f ( x ) = x 3 + 3 x , f ( x + h ) f ( x h , h 0 f ( x ) = 1 / x 2 , g ( x ) g ( 3 ) / ( x 3 ) , x 3 f ( t ) = 1 / ( t 2 ) , f ( t ) f ( 1 ) / ( t 1 ) , t 1

Difference quotient is defined as the formula used to find the average rate of change of a function over a given interval. For the first function, $$ begin{align} \frac{f(x+h)-f(x)}{h} &= \frac{(x+h)^3+3(x+h) - (x^3+3x)}{h} \ &= \frac{(x+h)^3-x^3+3(x+h)-3x}{h} \ &= \frac{3h^2x+3h^2+3h}{h} \ &= 3x+3h+3 end{align} $$ For the second function, $$ begin{align} \frac{g(x)-g(3)}{x-3} &= \frac{1/x^2 - 1/9}{x-3} \ &= \frac{9-x^2}{9x^2(x-3)} end{align} $$ For the third function, $$ begin{align} \frac{f(t)-f(1)}{(t-1)} &= \frac{1/(t-2) - 1}{t-1} \ &= \frac{-1}{(t-2)(t-1)} end{align} $$ Therefore, the indicated difference quotients have been found and simplified.