6.2 Error Bounds for Approximating Sums

Finding constants for error bounds

Theorem 3 (Error bounds for approximating sums), page 387 of the textbook Let , and let , , , and denote left, right, trapezoid, and midpoint sums for , each with equal subdivisions.
  • Left- and right-rule errors  Let be a constant such that for all  in . Then 

  • Midpoint- and trapezoid-rule errors  Let be a constant such that for all in  . Then

Using GeoGebra to find K_1 and K_2 (video instructions)

Using Geogebra to find K_1 and K_2 (written instructions)

1. Enter the function  and the interval  in the Input bar as follows. Then press enter. In this example I will use the function  on the interval .

 f=function[sin(x^2),0,3]  2. Next, plot a new function  that is the derivative of  over the same interval by entering the following into the Input bar. It may take a moment for the graph of  to appear. g=derivative[f]  3. Since we want to find an upper bound for the absolute value of , plot the function . h=abs(g) 4. At this point you can use the graph to estimate a value for . If you want to obtain the optimal bound, try the extremum command. GeoGebra can identify all of the local extrema (the local maxes and local mins) of a function on an interval. To obtain the  coordinates of the local extrema of , type the following into the Input bar. Note: the introduction of the absolute value may cause some discontinuities in the function , which may in turn lead to some errors in the implementation of the extremum function. extremum[h,0,3] In order to find the absolute max of  on the interval , one must also take into account the endpoints of the interval. Compare the -coordinates of the local extrema and the -coordinates of  at and to determine the absolute maximum value of the function on the closed interval  5. To find , we need to differentiate the function  and find the maximum. Follow the steps above with  replaced by. j=derivative[g] k=abs(j) extremum[k,0,3] Helpful Hints

  • You can reset the applet at any time by clicking on the circle made of two arrows in the upper-right hand corner. This will erase all of your functions. 
  • To make a function or a point disappear or reappear on the coordinate plane, click on the radio buttons on the left side of the screen.
  • You can pan the cartesian plane by holding down the mouse, and zoom in and out by scrolling with the mouse.