Question

(i) How many subintervals are created when [-1,3] is subdivided into four equal parts?

a) 2
b) 3
c) 4
d) 5

(ii) What is the process for constructing upper and lower rectangles for each subinterval?
a) Use triangles
b) Divide by 2
c) Choose random heights
d) Use the midpoint of each subinterval
iii) What does summing the areas of upper and lower rectangles represent?
a) Total area
b) Average area
c) Overestimation
d) Underestimation

Answer 1

(i) c) 4

(ii) d) Use the midpoint of each subinterval

(iii) c) Overestimation

Step-by-step explanation:

In calculus, when the interval [-1, 3] is subdivided into four equal parts, it creates four subintervals. Each subinterval is determined by the endpoints and can be expressed as [-1, -0.5], [-0.5, 0], [0, 0.5], and [0.5, 3]. Therefore, the correct answer to (i) is option c) 4.

For constructing upper and lower rectangles in the context of Riemann sums, the process involves using the midpoint of each subinterval to determine the height of the rectangles. This is because using the midpoint provides a more accurate representation of the function over the given interval. The correct answer to (ii) is option d) Use the midpoint of each subinterval.

Summing the areas of upper and lower rectangles represents an estimation of the total area under the curve. In particular, when the upper rectangles are used, it results in an overestimation of the actual area, making option (iii) c) Overestimation the correct choice. This discrepancy arises because the upper rectangles reach higher points on the curve, capturing more area than the actual function.

In conclusion, subdividing the interval into four equal parts leads to four subintervals, and using the midpoint of each subinterval is the appropriate method for constructing upper and lower rectangles. Summing the areas of these rectangles provides an estimate of the total area, with the use of upper rectangles leading to an overestimation of the actual area under the curve.