I am in a mood to share some mathematical gems. Exponentials are really cool, because it’s common ground with everybody.
Teachers write lecture notes so that you can learn not them. Try thinking backwards – sometimes it helps to learn to count backwards if you know how to count forwards.
The idea of thinking backwards is an enigma, but the first two ideas are what I think I did to get an A in Calc I. I got an A in Calc I, the funny thing is that I arrived late for the final, I was sweating it, but it was all worth it to see my teacher’s “are you serious?” face.
I enjoyed an experience where the Calculus teacher was using numerical evidence to demonstrate what he was talking about.
You’d think that you aren’t trained to study exponential functions: but if you think about logarithmic growth – 1 and 2 followed by 4… it takes very little talent to see that the next number is 8.
If it will cost $1 for a weasel and $2 for a ferret and $4 for a puppy dog, we can talk about this as being twice as much or 2^2 (four) times as much.
Such comparisons are common ground and many curves in Calculus will be e^x, where e is a number and x is the variable. This is an exponential, it is as rewarding as it is common to study this in Calculus.
When we learn something new, our brain stores this information as compactly as possible in what we call the gist of the lesson. There is a way of taking notes, for example, that in some ways predates the dinosaurs: this is called Cornell Notes.
The notes that your teacher teaches out of may be stored as Cornell Notes in her brain. It doesn’t include all the example problems, elaborations, and anecdotes that it takes to learn the material.
The brain doesn’t focus on etcetera for long periods of time, but you are in luck. Remember that expanding the summation and doing the work pays off in a good grade on the test.
Often, in Linear Algebra or Analysis, the object of study is really compact and your professor just talks about an object with friendly familiarity. So do your teachers teach to you or teach to themselves?
In Physics, you must format your brain to work with any two or three vectors, whether or not you normally process information this way.
It’s been said by a sage that “Physics is all about the conservation of pain: the more pain you put into the homework, the less pain you will have on the test.”
Hopefully you are glad to ask for advice when you are in pain.
I like to think the best of people, so it’s likely that your teacher will be glad to clear up tough points, if you read the book, took notes, and tried the exercises.
Here’s a cool way to do homework: try to solve the equation for x. Give up and look in the back of the book.
Now write that you will have x=62 and try to get to that answer. Sometimes you forget a factor of 4. That is the way life is, 2+2=2*2.
Now, as for the last sentence of the first paragraph: think backwards, assign your own meaning to this idea.
The answers to last week’s sequences are
0,0,0,0,0,0
32,16,8,4,2,1
1,11,21,1211,111221,312211,13112221
1,4,9,16,25,36
1,1,2,3,5,8,13,21
9,28,14,7,22,11,34,17
2,3,5,7,11,13
1,2,3,4,11,12,13,14,21,22,23
2,6,10,14,22,26,30 (hint: divide by 2 and factor: you will see that these numbers are increasing and have a special property in their prime factorization.)
3,7,11,15,19,23,27,31
1,3,7,11,15,23,27,31
2 4 6 8 10 12 14 ….
3 9 15 21 27 33 …
5 25 35 55 65 85 …
7 49 77 91 119 133 …
11 121 143 187 209 …
13 169 221 247 299 …
1,2,3,4,20,69,238,863
1,2,3,4,6,8,11
1,2,1,2,2,3,1,2,3,2,3,5,1
0, 1, 4, 9, 16, 22, 25, 27, 31, 34, 36, 37, 45, 49, 55, 58, 64, 67, 70, 81, 82, 88, 91, 97
0,1,9,36,100,225,441,784
The rest can be found on the online encyclopedia for integer sequences.
Questions about this week’s Calculus Workhop? Write to us at thexavierite@yahoo.com with your questions, comments and concerns regarding the world of math and numbers. It’s a big, mathematical world out there: don’t get lost. Get answers.
Nick Vasilyev
Viewpoints Correspondent