# 009B Sample Final 1, Problem 1

Suppose the speed of a bee is given in the table.

 Time (s) Speed (cm/s) $0.0$ $125.0$ $2.0$ $118.0$ $4.0$ $116.0$ $6.0$ $112.0$ $8.0$ $120.0$ $10.0$ $113.0$ (a) Using the given measurements, find the left-hand estimate for the distance the bee moved during this experiment.

(b) Using the given measurements, find the midpoint estimate for the distance the bee moved during this experiment.

Foundations:
1. The height of each rectangle in the left-hand Riemann sum is given by choosing
the left endpoints of each interval.
3. The height of each rectangle in the midpoint Riemann sum is given by
${\frac {f(a)+f(b)}{2}}$ where  $a$ is the left endpoint of the interval and  $b$ is the right endpoint of the interval.

Solution:

(a)

Step 1:
To estimate the distance the bee moved during this experiment,
we need to calculate the left-hand Riemann sum over the interval  $[0,10].$ Based on the information given in the table, we will have  $5$ rectangles and
each rectangle will have width  $2.$ Step 2:
Let  $s(t)$ be the speed of the bee during the experiment.
Then, the left-hand Riemann sum is

${\begin{array}{rcl}\displaystyle {2(s(0)+s(2)+s(4)+s(6)+s(8))}&=&\displaystyle {2(125+118+116+112+120)}\\&&\\&=&\displaystyle {1182{\text{ cm}}.}\end{array}}$ (b)

Step 1:
To estimate the distance the bee moved during this experiment,
we need to calculate the Riemann sum using the midpoint rule over the interval  $[0,10].$ Based on the information given in the table, we will have  $5$ rectangles and
each rectangle will have width  $2.$ Step 2:
Let  $s(t)$ be the speed of the bee during the experiment.
Then, the Riemann sum using the midpoint rule is

${\begin{array}{rcl}\displaystyle {2{\bigg (}{\frac {s(0)+s(2)}{2}}+{\frac {s(2)+s(4)}{2}}+{\frac {s(4)+s(6)}{2}}+{\frac {s(6)+s(8)}{2}}+{\frac {s(8)+s(10)}{2}}{\bigg )}}&=&\displaystyle {1170{\text{ cm}}.}\end{array}}$ (a)    $1182{\text{ cm}}$ (b)    $1170{\text{ cm}}$ 