TI-83/83 Plus: Finding x-intercepts ofa Second Derivative Graph

The following pages contain some instructions on the usage of the TI-83/83 Plusgraphing calculator.The example used below is taken out of the text titled Calculus Concepts - An InformalApproach to the Mathematics of Change, 1st Edition by LaTorre, Kenelly, Fetta, and etal.

Example#1 Post-secondary Education p. 292: Consider the following model for thepercentage of students graduating from high school in South Carolina from 1982 through1990 who entered post-secondary institutions

( ) 792.50672.3355.11057.0 23 ++= xxxxf percent where x=0 in 1982Find the inflection point of the model.

Your calculator can be very helpful in checking your analytical work when finding theinflection points. When you are not required to obtain an analytical solution usingderivatives or when only an approximation to the exact answer is required, the followingprocedure can be implemented.

Press to go into the function Editor menu. Enter the above equation into yourcalculator by \Y1 =. Be sure to use press for the leading negative sign as opposed tosubtraction, which is . When you have finished entering the equation of the model,your screen should look like the screen on the left given below.

Press to move the cursor down by \Y2 =. Press to go into the MATH menu.Press several times to move the cursor down by 8:nDeriv(. At this point, your screenshould look like the screen in the middle given above. This particular function of thecalculator uses the symmetric difference quotient to calculate the slope of a secant line,which in turn can be used as an approximate value for the slope of the tangent line at agiven point on a function.Press to select 8:nDeriv( and paste it onto the function Editor by \Y2 =. At thispoint, your screen should look like the screen on the right given above.The Nderiv function takes three arguments, separated by commas. The first one is thelocation of the function in the function Editor whose derivative is needed. To get Y1pasted by nDeriv( as the first argument, first press to go into the VARS menu.Press to scroll over to the VARS Y-VARS menu. Press to select 1:Functionfrom the VARS Y-VARS menu. Press to select 1:Y1 from the FUNCTION menu

and paste it by nDeriv( as the first argument. Press . Press for the generic inputvariable x. Press . Press again to obtain an approximation for the derivative atevery x pixel between the Xmin and Xmax values. Press . At this point, your screenshould look like the screen on the left given below.

Press to select the WINDOW menu. The appropriate WINDOW menu settingsused for this model are displayed in the middle given above. Press to see thegraph of the function and the graph of the numerical derivative of the function. At thispoint, your screen should look like the screen on the right given above.Press to go into the function Editor menu. Press three times to move the cursordown by \Y2 =. Press to move the blinking cursor over to the equal sign next to \Y2 inthe function Editor. Press to turn off the function in \Y2 location in the functionEditor.At this point, the derivative of the calculators first derivative needs to be placed intofunction editor by \Y3 =. Press to move the cursor down by \Y3 =. Press . Press to go into the MATH menu. Press several times to move the cursor down by8:nDeriv(. Press to select 8:nDeriv( and paste it onto the function Editor by \Y3 =.To get Y2 pasted by nDeriv( as the first argument, first press to go into the VARSmenu. Press to scroll over to the VARS Y-VARS menu. Press to select1:Function from the VARS Y-VARS menu. Press to move the cursor down by 2:Y2.Press to select 2:Y2 from the FUNCTION menu and paste it by nDeriv( as thefirst argument. Press . Press for the generic input variable x. Press . Press again to obtain an approximation for the derivative of the first derivative at everyx pixel between the Xmin and Xmax values defined by the above WINDOW settings.At this point, your screen should look like the screen on the left given below.

Press to see the graph of the function and the graph of the numerical derivative ofthe calculators first derivative. At this point, your screen should look like the screen onthe right given above. The graph of the calculators numerical second derivative of thefunction takes longer to produce because the calculator approximates the derivative of thederivative at each pixel point before graphing.

An inflection point is a point of greatest or least slope. One can easily find an inflectionpoint of a function by finding where the first derivative of the function has a maximum orminimum value. The first derivative graph seems to reach a maximum at about x=4. Thesecond derivative graph seems to cross the x-axis at about x=4 also. It seems that the x-value of the maximum of the first derivative graph (i.e., the slope graph) is the x-axiscrossing of the second derivative graph.The x-axis crossing of the second derivative graph can be found by using the CALCmenu. To accomplish this task, first press to go into the function Editor menu. Press

to move the blinking cursor over to the equal sign next to \Y1 in the function Editor.Press to turn off the function in \Y1 location in the function Editor. Now, the onlyfunction that remains turned on in the function Editor menu is in \Y3, which is the secondderivative function. Press and to go to the CALC menu. Press to move thecursor down to 2:zero. Press to select this option from the CALCULATE menu.Press to move the cursor anywhere to the left of where the second derivative graphcrosses the x-axis. At this point, your screen should look like the screen on the left givenbelow.

Press to mark the location of the left bound. The calculator will prompt you for theright bound. Press to move the cursor anywhere to the right of where the graph crossesthe x-axis. At this point, your screen should look like the screen in the middle givenabove. Press to mark the location of the right bound. At this point, your screenshould look like the screen on the right given above. Notice that the calculator hasmarked the interval between the left and right bounds with two small triangles at the topof the screen. The calculator is also prompting you for a guess value for the x-axiscrossing of the second derivative graph. Any value between the two small triangles willdo. Press to move the cursor somewhere between the left and right bounds. Press to see the location of the x-axis crossing of the second derivative graph displayedby the calculator as x=4.2730999.At this point, your screen should look like the screen given below.

The second derivative is zero when 27.4x years after 1982. The output from the function is( ) %6.5127.4 f and the rate of change at that point is ( ) 1.227.4 f percentage points per

year.

Copyright 2001 by Mike Turegun