Question

Calculus Help Needed ASAP. Help appreciated!

Given f(x) > 0 with f ′(x) > 0, and f ′′(x) > 0 for all x in the interval [0, 3] with f(0) = 0.1 and f(3) = 1, the left, right, trapezoidal, and midpoint rule approximations were used to estimate the integral from 0 to 3 of f of x, dx. The estimates were 0.8067, 0.9635, 1.0514, 1.0753 and 1.3439, and the same number of subintervals were used in each case. Match the rule to its estimate.

Answer 1

To think through this problem, I first drew a sketch of the curve. We are told that f(x)>0 which means that the function is positive (above the x-axis). We are also told that f’(x)>0. Since the first derivative is positive the graph is strictly increasing over the interval. Further since the second derivative is positive (f’’(x)>)) the function is concave up (like a smile not a frown).

I divided the interval into 3 (we Are not given how many rectangles/trapezoids are used so this is just an example to help us think through the problem). See the sketches in the attached. 

Right Endpoint
Here we make a rectangle by using the point on the curve that is on the right side of each rectangle. 

Left Endpoint
Here we create a rectangle using a point on the curve that is to the left of each rectangle.

Midpoint
Use the point in the middle of the rectangle’s side (and on the curve) to create the rectangle.

Trapezoidal
Use both the left and right points on the curve (from each interval) and connect them with a segment to get a trapezoid.

The rectangles and trapezoids are used to estimate the area under the curve so I shaded the area of the rectangles (or trapezoids) in each picture. Also the top picture shows the actual area shaded in.

From the pictures it should be clear that the Left endpoint gives an area much less (an underestimate) than the actual. So the left-endpoint goes with .8067

It should also be clear that the right end point produces a large overestimate (way more is shaded than the area under the curve) so that must give the largest answer which is 1.3439

It appears that the trapezoids give the most accurate area and that thisis a little bit more than the actual. The midpoint is harder to tell from pictures but a curve that is increasing and concave up will have an area larger than that given by using the midpoint. That means that the midpoint goes with .9635 (since that is an underestimate) and the trapezoids go with 1.0753 since that is a slight overestimate and the actual area is 1.0514.