Answer
a.
(i) $\approx 86.8$
(ii) $\approx 70.6$
(iii) $\approx 81.4$
b. overestimate
c. underestimate
d. M gives the best estimate because it balances the errors introduced by the L and R sums, as the rectangles takes the in between values compared to the values on the very left or very right
Work Step by Step
a.
(i) The length of each rectangle is 2 because 12/6 is 2. Then, look at the graph and add the corresponding estimated function values at each point, starting from when x =0. Skip f(12) because it's asking for the sample points from the left. $2* (9+ 8.8 + 8.2 + 7.4 + 6 + 4) \approx 86.8$
(ii) Look at the graph and add the corresponding estimated function values at each point, starting from when x =2 until x = 12. Skip f(0) because it's asking for the sample points from the right. $2* (8.8 + 8.2 + 7.4 + 6 + 4 + 1)\approx 70.6$
(iii) Take the midpoints of the 6 rectangles, and add the corresponding function points, then times everything by 2 because of the length of each rectangle remains 2.$ 2* (9+ 8.6+ 7.8+ 6.6+5+3.7)\approx 81.4$
b. Overestimate because the function is concave downwards and decreasing
c. Underestimate because the function is concave downwards and decreasing
d. M gives the best estimate because it balances the errors introduced by the L and R sums, as the rectangles takes the in between values compared to the values on the very left or very right
