Question

More quadratic approximation... Question 2A-11. Looked at very simply, it is a product of a=bc. So the quadratic approximation (Q) should be Q(a) = Q(b)Q(c). However, the answer for this seems to be closer to Q(a) = bQ(c) and doesn't approximate the first part of the product at all. I was able to solve this question correctly by only approximating to the second part. If I approximated the first part of the equation ("b" above), it got very messy and incorrect.

So my question is: is this the correct way to solve this problem, by only approximating the second half? If so, why?

Answer 1

Sorry about not posting the true equation above (please see below)

\[p=cv \left( 0 \right)^{-k}\left( 1=\frac{ DeltaV }{ v \left( 0 \right) } \right)^{-k}\]

Answer: \[\frac{ c }{ v(0)^{k} }(1-k \frac{ DeltaV }{ v(0) } + \frac{ k(k+1) }{ 2 }\left( \frac{DeltaV }{ v(0) } \right)^{2})\]

It seems to me that the first term (cv(0)^-k) was just copied below and not approximated.

Answer 2

We don’t do anything with the expression on the left because it’s a constant. Where you have this\[v(0)\]you’re supposed to have this\[v_0\]and v_0 represents an unspecified but constant value for volume. The problem is to determine how much change in pressure would occur with a small change in volume, assuming we're starting at v_0. The only variables here are p and Delta v.

Sometimes it isn't obvious whether a letter in a formula represents a variable or a constant. If you can't tell from the context, the use of a subscript is usually a tipoff that it's a constant.

By the way, don't worry if you're puzzled by the last sentence in the statement of the problem, because that statement doesn't make sense and it was a mistake to include it. It says in effect to find the approximation where Delta v is small, but we're *always* looking for an approximation where the change from the known value is small, so the statement is redundant and confusing.

If you haven’t already done so, you may want to review the document on approximation that’s available here: http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/readings/course-reader/. It’s from one of the other MIT OCW courses in single variable calculus, and it provides helpful insight into approximation. It’s also the document referred to in these exercises as “the Notes.”

Answer 3

Didn't mean to repeat myself in referencing the Notes on approximation. I forgot I already mentioned that document.