calculus need answers

Question

Given f(x) > 0 with f ′(x) < 0, and f ′′(x) > 0 for all x in the interval [0, 2] with f(0) = 1 and f(2) = 0.2, the left, right, trapezoidal, and midpoint rule approximations were used to estimate the integral from 0 to 2 of f of x, dx. The estimates were 0.7811, 0.8675, 0.8650, 0.8632 and 0.9540, and the same number of subintervals were used in each case. Match the rule to its estimate.

MatchTermDefinition
	left endpoint	A) 0.9540
	right endpoint	B) 0.8650
	midpoint	C) 0.8632
	trapezoidal	D) 0.8675
	actual area	E) 0.7811

Answer 1

Hello Joshua!

The first thing to understand with the question is the description of the function f(x). The first derivative is negative over the interval so that indicates its a decreasing function over the interval. The second derivative is also positive meaning that its concave up. If you sketch this function it would look like something similar to f(x) = 1/x. Knowing this information you can proceed to matching up areas with the summation method.

There is no actual calculation required. All that is needed to know is how the different sums error to the actual area. The number for divisions n is the same in all summations. Since the function is decreasing, you would know that the left endpoint summation would have the highest value of area. In contrast, the right summation method would give you the smallest area.

The Midpoint summation method will have the smallest error and the error is slightly higher than the actual value. The Trapezoid summation has the next smallest area and is slightly below the actual error. Therefore we get the following matches:

A .9540 Left Endpoint Summation

E .7811 Right Endpoint Summation

D .8675 Midpoint Summation

B .8650 Actual area

C .8632 Trapezoidal Summation