For a recent review lesson, I composed a question which I thought would nicely review using definite integrals to find areas and volumes and to illustrate the significant difference that could occur between the working required for such a question on Paper 1 (no GDC) and one on Paper 2 (GDC allowed).  However, much to my surprise (and slight shock), the question also served to illustrate to students (and to me) that even a very powerful technological device like their TI-84 GDC can make mistakes.  I have come across a few other errors that the TI-84 can make - and have warned students to always make sure that their GDC result makes sense.  But this error (see image at right) was a bit different.

I wrote the question in two versions - one for Paper 1 and one for Paper 2 - as follows:

I had my students first answer the Paper 2 version - and then was going to have them find the exact answers for the area and the volume without their GDC for the Paper 1 version.

Finding the exact area of region R in part (a) is not too challenging because finding the anti-derivative of sin(3x) is quite straightforward.  However, finding the exact volume of the solid of revolution in part (b) requires finding the anti-derivative of which is not straightforward.

So, I was a little suprised when one of my students announced just moments after everyone started working on the Paper 1 version of the question that he had the exact answer for the volume of the solid of revolution.  I asked him about his method and he simply said "my calculator gave me the exact value".  The approximate answer (3 significant figures) for the volume found by evaluating the appropriate definite integral in the Paper 2 version was 1.50 units³.  As shown in the first image above, the result given by the calculator to 10 significant figures is 1.495499938.  I do mention to students that the TI-84 (like most GDCs) have an internal precision of 14 significant figures - so the actual number the TI-84 is using is 1.4954999379353 (but displaying just 10 significant figures).  What this student had done was simply apply the 'convert to fraction' command to the answer he got for the Paper 2 version of the question which produced the exact rational number of .  But the exact volume is units³, clearly an irrational number.  The TI-84 had made a mistake.  Hmmmm ....

The strange thing is that if the TI-84 was converting truncated values for either the 10-digit approximation of 1.495499938 (shown on the screen) or the 14-digit approximation of 1.4954999379353 (stored internally) to a fraction neither one converts to .  Furthermore, the fraction approximated to 14 significant figures is 1.4954999383553 - the last 5 digits don't even match with the 14-digit approximate value of (see results from Wolfram Alpha below).

This error produced by the TI-84 is intriguing and makes me curious to find out more precisely how the internal workings of the GDC produces such an error (watch for future blog entries on GDC errors).  But it did provide an illustration - although unexpected - to students (and to me the teacher) to never completely trust your GDC - and make sure that results from technology need to make sense.  It is not possible for a number to be both irrational and rational.

Two computations made by Wolfram Alpha:

1.   approximated to over 900 digits

2.   The fraction expressed as an exact repeating decimal.  The repeating block contains 811 digits.