Let f (x) = x2. Calculate the following quantities.
Let f (x) = x2 on the interval [0,12]. How many rectangles must we use to guarantee that the right- and left-hand sums are within 10
of each other?
Since a, b, f (a), and f (b) are the same as in the example, we get the inequality
We solve this inequality for n:
We conclude that we need to use at least 173 rectangles.
Remember that if LHS(n) and RHS(n) are close to each other, they must also be close to the exact area between f and the x-axis on [a,b].
Let f (x) = 4x + 6 on the interval [2,8]. How many rectangles must we use to ensure that the left-hand sum is within 0.25 of the exact area between f and the x-axis on this interval?
Plugging this into the inequality, we get
We solve the inequality and get:
We must use at least 37 rectangles.
Let f (x) = x4 on the interval [-2,-1]. For what values of n are LHS(n) and RHS(n) within 0.5 of each other on this interval?
We must have n = 31 or greater.
Let f (x) = cos(x) on the interval How large must n be to guarantee that LHS(n) is within 0.3 of the exact area between f and the x-axis on this interval?
We solve the inequality to get
We must have
n ≥ 6.
Make it rain.