Question

The velocity graph of a braking car is shown. Use it to estimate the distance, d traveled by the car while the brakes are applied. (Use M₆ to get the most precise estimate.)
d = _____ ft

a = 30

Answer 1

(Midpoint rule)

From the graph,

$$ \text { width }=\Delta x=\frac{6-0}{6}=1 $$

the interval [0,6] can be divided into 6 equal parts of width 1

$$ [0,1],[1,2],[2,3],[3,4],[4,5], \text { and }[5,6] $$

\(x_{1}^{*}=0.5, x_{2}^{*}=1.5, x_{3}^{*}=2.5, x_{4}^{*}=3.5, x_{5}^{*}=4.5, x_{6}^{*}=5.5\)

By using midpoint rule, the area under the curve is,

\(\begin{aligned} M_{6} & \approx \Delta x\left[f\left(x_{1}^{*}\right)+f\left(x_{2}^{+}\right)+f\left(x_{3}^{*}\right)+f\left(x_{4}^{*}\right)+f\left(x_{5}^{+}\right)+f\left(x_{6}^{*}\right)\right] \\ & \approx 1[f(0.5)+f(1.5)+f(2.5)+f(3.5)+f(4.5)+f(5.5)] \\ & \approx 1[80+60+45+25+15+5] \quad\left(\begin{array}{l}\text { From the graph, we can write } \\ \text { approximate value of } f \text { at particular } x\end{array}\right) \\ &=230 \mathrm{ft} . \end{aligned}\)