Question

Suppose that an object is moving along the ground and that the distance (in centimeters) it has travelled after t seconds is given by the formula f(t)=t2 cm. Calculate the average speed during the 1 millionth of a second after t=3.

Answer 1

$\bf slope = m = \cfrac{rise}{run} \implies \cfrac{ f(x_2) - f(x_1)}{ x_2 - x_1}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array}\\\\ -------------------------------\\\\ f(t)= t^2 \qquad \begin{cases} t_1=3\\ t_2=3+\frac{1}{1,000,000}\\ \qquad 3+0.000001\\ \qquad 3.000001 \end{cases}\implies \cfrac{f(3.000001)-f(3)}{3.000001-3} \\\\\\ \cfrac{9.000006000001~~-~~9}{0.000001}\implies \cfrac{0.000006000001}{0.000001}\implies 6.000001$