# 6.2 Error Bounds for Approximating Sums

- Author:
- BatesMath, leeb, Marie Mueller

## Finding constants for error bounds

**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)

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.