Quote:
Originally Posted by Brutha  Memorizing math is besides the point. Books are good at remembering.
At university we usually get one or two pages of paper that we can take with us into math exams as remembering that stuff isn't the point. In one exam we could even take as many books as we would like with us.
If you want to understand math the mathematical way you have to ask yourself how you can use the definition for the limit to prove the theorems that you have at your course.
But in math you don't get any advantage from knowing the word-for-word definition.
It doesn't help you to understand what the definition is about.
Either you can use the definition of the limit to prove things or you can't. Knowing the words of a particular expression doesn't help.
Or you just want to use math to solve real world problem but then you don't need definitions at all.
Could you write out a proof of the mean value theorem from hand (you can look up definitions but not the proof itself)? |
I have not gotten to the proof for the mean value theorem from scratch yet, but I do know how to prove limits of both linear and quadratic functions using the epsilon-delta definition of limit.
Here is a proof for the limit of 3x - 2 as x approaches 2, which equals 4. Throughout the proof I will use "[" and "]" to notate absolute value.
lim (3x - 2) = 4.
x->2
We must show that for each epsilon>0, there exists a delta>0 such that [(3x - 2) - 4]<epsilon whenever 0<[x - 2]<delta. Because our choice of delta depends on epsilon, we need to establish a connection between the absolute values of [(3x - 2) - 4] and of [x - 2].
[(3x - 2) - 4] = [3x - 6] = 3[x - 2].
So, for a given epsilon>0 we can choose delta = epsilon/3. This choice works because
0<[x - 2]<delta = epsilon/3
implies that
[(3x - 2) - 4] = 3[x - 2]<3(epsilon/3) = epsilon.
And that is how my textbook taught me to use the epsilon-delta definition of limit to prove the limit of a linear function. I also know how to use the same definition to prove the limit of a quadratic function.