Uncle
Banned
Exhibit A:
What you see:
ex
What your calculator really has:
[maths]e^{x} = 1 + x + \frac{1}{2!}x^{2}+\frac{1}{3!}x^{3}[/maths]
[maths]+\frac{1}{4!}x^{4}+\frac{1}{5!}x^{5}+\frac{1}{6!}x^{6}[/maths]
[maths]+\frac{1}{7!}x^{7}+\frac{1}{8!}x^{8}+\frac{1}{9!}x^{9}[/maths]
[maths]+\frac{1}{10!}x^{10} + \frac{1}{11!}x^{11} + \frac{1}{12!}x^{12}[/maths]
[maths]+\frac{1}{13!}x^{13}+\frac{1}{14!}x^{14}+\frac{1}{15!}x^{15}[/maths]
[maths]+\frac{1}{16!}x^{16}+\frac{1}{17!}x^{17}+\frac{1}{18!}x^{18}[/maths]
[maths]+\frac{1}{19!}x^{19}+\frac{1}{20!}x^{20} + \frac{1}{21!}x^{21} [/maths]
[maths]+ \frac{1}{22!}x^{22} + \frac{1}{23!}x^{23} + \frac{1}{24!}x^{24}[/maths]
[maths]+ \frac{1}{25!}x^{25} + \frac{1}{26!}x^{26} + \frac{1}{27!}x^{27}[/maths]
[maths]+ \frac{1}{28!}x^{28} + \frac{1}{29!}x^{29} + \frac{1}{30!}x^{30} + ...[/maths]
The calculator uses a Taylor Polynomial, it requires around a degree of 30 to maintain accuracy to 12 decimal places.
You probably can't enter a 30th degree polynomial into your calculator but why don't you try entering "what your calculator says" and an actual Taylor polynomial representation of degree 6 with some reasonable accuracy:
Enter e2 on your calculator:
Answer:
Enter
[maths]1 + 2 + \frac{1}{2!}(2)^{2}+\frac{1}{3!}(2)^{3}[/maths]
[maths]+\frac{1}{4!}(2)^{4}+\frac{1}{5!}(2)^{5}+\frac{1}{6!}(2)^{6}[/maths]
on your calculator:
Answer:
Sketches of:
ex in red.
[maths]1 + x + \frac{1}{2!}x^{2}+\frac{1}{3!}x^{3}+\frac{1}{4!}x^{4}+\frac{1}{5!}x^{5}+\frac{1}{6!}x^{6}[/maths] in blue.
The exponential function's 6th degree Taylor polynomial representation is almost perfect for a small value of x (at around x = 2).
The moral of the story is, your calculator doesn't know the elementary functions sin x, cos x, log x, it uses Taylor polynomial approximations of very high degree to maintain accuracy on its little 12 digit LCD screen and you probably didn't even know it until you read this thread which is one of the most glorious ones of all time.
EDIT:
For those in first year physics the formula for the length change due to temperature is given by the formula:
L = L0[1 + α(T - T0)]
However it originally came from:
L = L0eα(T - T0)
To make life easier, the approximation ex ≈ 1 + x is only valid if x is much much less than 1.
This is allowed because the coefficient of thermal expansion/contraction (α) for many materials is as small as 10-6.
Also you may encounter problems maybe because, say a bar is moved from your warm bedroom to the cold winter night and the temperature change is very small, compared to shoving, say, simonloo or a metal bar in a high speed furnace from 200C to 55000C like the ones in industrial plants.
Substitute the values and therefore:
L = L0[1 + α(T - T0)]
What you see:
ex
What your calculator really has:
[maths]e^{x} = 1 + x + \frac{1}{2!}x^{2}+\frac{1}{3!}x^{3}[/maths]
[maths]+\frac{1}{4!}x^{4}+\frac{1}{5!}x^{5}+\frac{1}{6!}x^{6}[/maths]
[maths]+\frac{1}{7!}x^{7}+\frac{1}{8!}x^{8}+\frac{1}{9!}x^{9}[/maths]
[maths]+\frac{1}{10!}x^{10} + \frac{1}{11!}x^{11} + \frac{1}{12!}x^{12}[/maths]
[maths]+\frac{1}{13!}x^{13}+\frac{1}{14!}x^{14}+\frac{1}{15!}x^{15}[/maths]
[maths]+\frac{1}{16!}x^{16}+\frac{1}{17!}x^{17}+\frac{1}{18!}x^{18}[/maths]
[maths]+\frac{1}{19!}x^{19}+\frac{1}{20!}x^{20} + \frac{1}{21!}x^{21} [/maths]
[maths]+ \frac{1}{22!}x^{22} + \frac{1}{23!}x^{23} + \frac{1}{24!}x^{24}[/maths]
[maths]+ \frac{1}{25!}x^{25} + \frac{1}{26!}x^{26} + \frac{1}{27!}x^{27}[/maths]
[maths]+ \frac{1}{28!}x^{28} + \frac{1}{29!}x^{29} + \frac{1}{30!}x^{30} + ...[/maths]
The calculator uses a Taylor Polynomial, it requires around a degree of 30 to maintain accuracy to 12 decimal places.
You probably can't enter a 30th degree polynomial into your calculator but why don't you try entering "what your calculator says" and an actual Taylor polynomial representation of degree 6 with some reasonable accuracy:
Enter e2 on your calculator:
Answer:
7.389056099
Enter
[maths]1 + 2 + \frac{1}{2!}(2)^{2}+\frac{1}{3!}(2)^{3}[/maths]
[maths]+\frac{1}{4!}(2)^{4}+\frac{1}{5!}(2)^{5}+\frac{1}{6!}(2)^{6}[/maths]
on your calculator:
Answer:
7.355555556
Sketches of:
ex in red.
[maths]1 + x + \frac{1}{2!}x^{2}+\frac{1}{3!}x^{3}+\frac{1}{4!}x^{4}+\frac{1}{5!}x^{5}+\frac{1}{6!}x^{6}[/maths] in blue.
The exponential function's 6th degree Taylor polynomial representation is almost perfect for a small value of x (at around x = 2).
The moral of the story is, your calculator doesn't know the elementary functions sin x, cos x, log x, it uses Taylor polynomial approximations of very high degree to maintain accuracy on its little 12 digit LCD screen and you probably didn't even know it until you read this thread which is one of the most glorious ones of all time.
EDIT:
For those in first year physics the formula for the length change due to temperature is given by the formula:
L = L0[1 + α(T - T0)]
However it originally came from:
L = L0eα(T - T0)
To make life easier, the approximation ex ≈ 1 + x is only valid if x is much much less than 1.
This is allowed because the coefficient of thermal expansion/contraction (α) for many materials is as small as 10-6.
Also you may encounter problems maybe because, say a bar is moved from your warm bedroom to the cold winter night and the temperature change is very small, compared to shoving, say, simonloo or a metal bar in a high speed furnace from 200C to 55000C like the ones in industrial plants.
Substitute the values and therefore:
L = L0[1 + α(T - T0)]
Last edited:
