Editing CommonMathsQuestions
You are currently browsing as guest..
To change this, fill in the following fields:
Username
Password
Click here to reset your password
Who can read this page?
The World
Members
Council
Admin
You have been granted an edit lock on this page
until Thu Mar 28 15:40:08 2024.
Press
to finish editing.
Who can edit this page?
World editing disabled
Members
Council
Admin
Here is a page for explanations of some of the more tricky questions that turn up occasionally. ---- !! Why is EQN:\frac{d}{dx}ln(x)=\frac{1}{x} ? To start with, it's worth looking at the graph and seeing that this is reasonable. * When /x/ is very small, /ln(x)/ is very negative and growing quickly ** so the derivative is large positive. * When /x=1,/ /ln(x)/ is zero and growing gently. ** so the derivative is about 1, although not necessarily exactly so. * As /x/ gets large, /ln(x)/ grows more slowly, but always grows ** so the derivative is positive, but getting close to 0. So it seems plausible. How about an exact calculation? We start with EQN:y=ln(x) and we want to compute EQN:\frac{dy}{dx} * EQN:y=ln(x) * => EQN:x=e^y * => EQN:\frac{d}{dy}(x)=\frac{d}{dy}(e^y) * => EQN:\frac{dx}{dy}=e^y because EQN:\frac{d}{dy}e^y=e^y * => EQN:\frac{dx}{dy}=x because EQN:x=e^y Now the really tricky part is that when /y/ is a one-to-one function of /x,/ which it is in this case if we restrict ourselves to positive /x,/ then EQN:\frac{dx}{dy}=1/\frac{dy}{dx}. To see that properly you can either use graphs and swap the co-ordinates around, or you can do the limiting process for each side. Once you accept that, we have * EQN:\frac{dy}{dx}=1/x * Hence EQN:\frac{d}{dx}ln(x)=1/x ---- !! What is 0.999999... actually equal to? This is a good one. Students seem to think that it can't be one, because it's "obviously" less than one. How do you convince them otherwise? ---- !! What is the value of EQN:0^0 ?? Hmm ... ---- !! Are there any others? ---- CategoryMaths