don steward
mathematics teaching 10 ~ 16

Sunday, 19 April 2009

greatest value

use a spreadsheet or a calculator or zoom in using graphing software to work out the greatest value of y if:

this work is intended to highlight a need to sometimes look beyond initial appearances....

the first question is relatively straightforward,
with a maximum value that seems to be (for integers) 2
but is actually a maximum y = 2.25 (when x = 2.5, as some students might know)

for the second question, a quartic, the value 24 seems to be the highest value for y

the y-values are (designed to be) 24 for x = 1, 2, 3 and 4:

the intention is that students learn not to accept these initial highest values but to dig deeper

 to find the highest value is actually 25

exploring the shape of the quartic function, it comes up through (0,0) then has two peaks, like a tooth, symmetrical about and with a minimum at x = 2.5

then goes back down to negative values through (5,0)

the first maximum is at x = 1.38196601125... and the second is at 5 minus this value, both giving the same y value = 25

you get to this first highest value of y around x = 1.38966 (using a spreadsheet set to 30 decimal places)

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