Understanding Endpoints & Overestimation/Underestimation of a Function

[phillyolly]

I am just seeking some clarification.

Can one tell me the difference between right and left endpoints when drawing a function and estimating its area with these endpoints? And how are they connected to underestimation and overestimation?

If you can draw graphs and point out which ones are over/underestimated, and which have left/right endpoints, it will be very helpful.

Thanks a lot.

 

[Antiphon]

The left endpoint is on the left.

Draw a strait line to the right endpoint. What do see?

 

[LCKurtz]

You are given a function f(x) on an interval [a,b]. To estimate the value of

$$\int_a^b f(x)\ dx$$

you make a partition of the interval:

$$a = x_0 < x_1, ... ,< x_n = b$$

and pick points c_i in [x_i-1,x_i] for i = 1..n. You then approximate the integral with rectangles of height f(c_i) on each subinterval:

$$\sum_{i=1}^n f(c_i)\Delta x_i$$

If you pick the c_i on the left of its subinterval [x_i-1,x_i] then c_i=x_i-1 and if you pick it on the right of its subinterval, then c_i = x_i.

If f is an increasing function its value will increase as x moves to the right. Using the left end point and drawing the approximating rectangle, you can see that the rectangles always underestimate the area under the curve, but if f is decreasing, just the opposite happens and the area of the rectangle overestimates the area under the curve.

Surely your text has pictures of this.