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Re: [Phys-L] quickest route



Regarding JD's calculation:

The maximum occurs at θ = atan(2). You can show this via
simple calculus. In the algebra-based introductory physics
course, you can find the maximum by plotting the function
sin(θ) + cos(θ)/2
and eyeballing the maximum, or using the spreadsheet
program's root-finding tool. I didn't evaluate this
arctangent in my head, but my computer tells me it is
63.43°, at which point the direct route wins by about
12 percent.

The function

f(x) = sin(x) + cos(x)/2

can be maximized over the 1st quadrant without calculus or graphical
methods by simply using some trigonometry (including a cosine
difference formula) and rewriting it as:

f(x) = (sqrt(5)/2)*cos(x - atan(2)) .

The maximum can be read off by inspection as occurring at
x = atan(2) with the maximum value being sqrt(5)/2.

In general when extremizing a function of the form:

f(x) = a*cos(x) + b*sin(x)

if it is re-written as

f(x) = (sqrt(a^2 + b^2))*cos(x - atan(b/a))

it makes life a lot easier without using any calculus.

BTW, I switched the argument of the function from [theta] to x
because I noticed on my last post that when I sent Unicode versions
of Greek [theta] and [mu] they came back to me on the list garbled
by being turned into gravely accented e and i respectively. But the
post in the archives seems to have the Greek characters unmolested.
If anyone can tell me how to send Greek characters to the list
without them being molested by the software I would appreciate it.

David Bowman