Expert: Sherry Wallin - 12/1/2008

Question
Rearrange the expression  to make r the subject.

my answer is

s = 4 pi r^2
s/(4 pi) = r^2
sqrt(s/4 pi) = r

is this correct?
and my question
2.
Find the first four terms in the expansion of 1/(1+3x)^2

my answer for this is (which is correct)
f(0)=1; f'(0)=-6, f"(0)=54, and f'"(0)=-648.

then this follow up question comes up I don't understand

State the values of x for which the expansion is valid.

Thank you very much :)

Answer
Your answer could be simplified some more in part 1. When you take the sqrt you need to state that there are two possible values. You always get two when you take a sqrt. So you have r = +- sqrt(s/(4pi)). Note the use of parentheses, these are necessary because otherwise it is unclear if you are dividing by 4 pi or dividing by pi and then multiplying by pi. Also the sqrt of (1/4) is 1/2 so bring it out in front of the radical getting r = +- (1/2)sqrt(s/pi)

To state the values of x for which the expansion is valid...
in the original function in the denominator you have (1+3x)^2, you need to determine what would make the expression undefined, i.e. where the denominator would be zero so you can eliminate that value of the variable, so find 1 + 3x = 0 and solve for x getting x = -1/3 the expression or expansion is invalid BUT you need to worry about all the other terms of the expansion also so you took the derivative for the next terms, see if you can tell what value in any of those would make the expansion undefined. Hint: when you use the quotient rule for the derivative the only thing that changes in the denominator is that you get the denominator squared each time you take the next derivative...