Calculus: Finding the derivative with the CASIO fx-991ES

With exams looming, people might come to tutoring with questions about their calculators:  What does it do (besides the obvious)?  How can it really help me? The tutor offers a handy hint about the CASIO fx-991ES.

Some exams don’t allow the use of just any calculator.  Make sure, as a student, you know which one(s) you can use.  If you need to buy a different one for the exam, pick it up as soon as you can:  familiarity with your calculator is key.

For those allowed to use the CASIO fx-991ES, its derivative function might be useful. As far as I can see (and would expect anyway), it doesn’t seem to do symbolic derivatives, but will find the value of the derivative at a specific x value.

Example:

Find the value of

d/dx (x^0.5-1)/(sinx +2), x=5

Even those who can confidently evaluate this derivative are likely not excited about doing it.

Well, here’s how to have the CASIO fx-991ES do it for you:

  1. I was in MATH mode when I did this. (That’s shift setup 1).
  2. Press shift∫. Enter the derivative expression: (√x-1)/(sin(x)+2) (Note: it autobrackets with sin.) Use the arrows at left and right of the replay to move around as needed;eg, to get out of the radical sign.
  3. After your expression is entered, arrow over to x= and enter your desired value at which to evaluate the derivative. (x=5, in this case).
  4. Press =. The screen may go blank for a few seconds while the calculator computes the answer. Then, you should be greeted with a restatement of the expression and the answer across the bottom: -0.1087192572, in this case.

I don’t have to explain to calculus students how convenient this function can be:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Perl: getting input from the user with STDIN

Tutoring, you often straddle a couple of generations.  The tutor reaches back to an old programming tradition….

When I was a kid in the early 80s, computers were very new to have around.  To make them more appealing to the home market – where they were suddenly available and even affordable – computers were given programs that allowed them to interact with the user.  You’d see such situations as this:

Computer screen:  Hello!  What’s your name?

The user would type in: Gerald

Now the computer would reply:

Hello, Gerald! I’m so glad to meet you!

How did our predecessors manage that feat – and how can we do the same?

In Perl, you can do the following:

#!/usr/bin/perl
print “Hello. What is your name?\n\n”;
$usrname=<STDIN>;
chop $usrname;
print “\nHello, $usrname! I’m so glad to meet you!”;

<STDIN> means for the computer to gather input from the standard source; ie, the console.

Chop removes the newline (the ENTER stroke) Gerald pressed to “enter” his name.

Part of tutoring is continuing traditions:)

Source: Robert’s Perl Tutorial

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Statistics: the normal distribution

Tutoring statistics, you help students who, in many cases, are facing completely new ideas (to them).  The tutor continues his goal of explaining statistics from a “day-to-day” point of view.

In an introductory stats course – which is probably as far as most people ever need to take it – probability is often the first topic.  You might continue with it for a few weeks even, without hearing of the normal distribution.  However, the normal distribution will almost always turn out to be the main focus of the course.

The normal distribution is the “bell curve” some people refer to.  Any large population follows it; therefore, the normal distribution is very useful in big business and for government.  People who are studying stats towards a professional career (rather than just for academic reasons) get schooled in the normal distribution because of its practical application.

A population that follows the normal distribution (virtually all do) is called a normal population.  Such a population follows some predictable traits which are very useful to know:

  1. The mean cuts the population exactly in half:  50% lies below it, the other 50% above.
  2. 68% of the population lies within one standard deviation of the mean. For example: if the mean is 100 and the standard deviation is 20, 68% of the population lies between 80 and 120.
  3. 95% of the population lies within two standard deviations of the mean. To follow along with the population mentioned above, 95% of it will lie between 60 and 140.

I’ll be continuing with implications of the facts above – as well as more attributes of the normal distribution – in coming posts. Cheers.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Programming: subroutines

Tutoring computer science, you deal with constructs that are both ideas and real-life segments of code.  The tutor brings forth the idea of subroutines.

In old-style programming (ie, during the 70s and 80s), the concept of a subroutine was crucial.  Other names for it were “procedure” or “function”.  The subroutine was a self-contained mini-program that was called, did a job for the main program, then was left alone again.  It could be called repeatedly.  A typical program you might write for an assignment could have numerous subroutines; a large program might easily have dozens or more.

As I recall, before the days of object-oriented programming, subroutines were the focus. Students were taught to think and program in terms of subroutines: the main task was meant to be divided into smaller, separate ones, for each of which a subroutine could be written. Not only could each subroutine be written independently; it could also be tested independently.  Finding a bug is much easier in a subroutine than in a larger program.  Moreover, it is easier for another person, who didn’t write the program, to understand how it works when the program’s functionality is divided among subroutines. Each subroutine would, of course, be appropriately commented (see my post here), so that the reader would easily know its purpose.

The approach to programming that centers around subroutines is called procedural. You’ll hear it called the procedural paradigm. While it’s probably been eclipsed by object oriented programming (the object oriented paradigm), it’s still preferred in some circles.

I’ll be further discussing underlying approaches to programming – for instance, the object oriented approach – in future posts. Cheers:)

Source: Wikipedia
Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: proof that 1 ≠ 0

Tutoring math, you often hear the comment that math involves itself in silly pursuits.  The tutor gets the point, and apologizes.

Sometimes, non-math people will point out that math harbours absurdities. The infinite string of numbers with a finite sum (see my post here) is one example. Irrational numbers might also qualify (see my post here), if due to their name alone.

I’ve heard people joking that there’s a proof for the fact that 1 is not equal to 0. Who would need to prove something so obvious? Surely, those math people are just amusing themselves, while everyone else has to put up with the output.

Those sentiments are definitely understandable; I’ve even heard my own math professors say so. However, there are two mollifying considerations:

  1. The ideas that seem absurd are often taken out of context.
  2. Often, the most obvious ideas are the hardest to prove.

The fact that 1 is not equal to 0 is obviously true in our common number system. We all count on its truth. Yet, to discover further ideas about our numbers, we often have to prove what we already know. That’s why math people so often embark on apparently ridiculous errands.

For those who still want to see proof that 1 is not equal to 0, here’s my version:

I’ll use an indirect proof (see my post here about that): I’ll first suppose the opposite of what I want to prove.

Suppose 1 is equal to 0.

In our number system, a number is itself times 1. For example, 10=10(1).

In general terms, the number n is equal to n(1).

Now, if 1=0, then n(1)=n(0). Therefore, n=0.

It follows that if 1=0, then every number is equal to 0. However, we know that not every number is equal to 0. Therefore, 1 cannot equal 0.

No doubt, there are other ways to prove 1 is not equal to 0. Is it worth proving? In some circles, yes – but probably not for most people:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Statistics: the meaning of mean and standard deviation

Tutoring math, you get asked about statistics.  The tutor offers an interpretation of mean and standard deviation.

Back when I was in university, there was a 2nd-year statistics course. It was the introductory stats course and was required for many degrees. Of course, being in math, I had to take it. However, I recall that business students, as well as social science students, commonly took it as well. I rarely had those people in my classes; that stats course was the exception.

I was told that the class had a 50 percent failure/dropout rate – the highest of all the courses at the university. It was even higher than calculus, which had a 40 percent non-completion rate. However, the stats course did have the enrollment of many students who weren’t math-oriented.

A common sentiment expressed by statistics students is that it’s hard to apply the ideas to everyday life. In a way, the feeling is ironic, since the first couple of courses in statistics – much more than most math courses – focus on everyday applications.

Introductory statistics focuses closely on the mean and the standard deviation of a population. The mean is another name for the average; most people understand it as the “expected” value. Consider mean height, for instance. If you imagine a person you can’t see, but want to guess their height, your best guess is the mean height of the population.

Standard deviation is harder to understand for most people. It’s the measure of how far apart the population’s values are – how spread out they are. Thinking of heights again: in a population with low standard deviation, people’s heights would mainly be close to the same. In a population with high standard deviation, the heights would likely be quite different from person to person.

I’ll be discussing more technical aspects of mean and standard deviation in coming posts. This, I hope, might be a good first step:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Auto batteries, part two: cold cranking amps and internal resistance

The tutor continues to explore auto batteries (aka car batteries).  While this topic is rare during tutoring, it’s probably relevant to virtually everyone at some point….

 
I recall, decades ago, a friend of mine bought a new battery for his car. “It’s got 600 cold cranking amps,” he smiled. “No more doubts about starting.”

I soon learned the meaning of 600 cold cranking amps: his car started perfectly from then on. At the same time, he was impressed by the number because of its technical meaning; being a mechanic, it spoke to him in a more precise way.

Years went by: I finished my degree, then went back for (of all things) some mechanics training. Today, when events from the past pop in my head, I try to answer questions that I let go at the time.

So it is with the issue of “cold cranking amps”. The other day I suddenly realized: if a 12-volt battery is pushing 600 amps, its internal resistance must be less than 12/600=0.02Ω. The obvious question: do car batteries really have internal resistance that low?

I started searching the net. The answer was harder to find than I’d expected, but here are some numbers:

car battery resistance link
0.003Ω chiefdelphi.com
0.01Ω tap.iop.org
0.001Ω furryelephant.com

So I guess it’s true: 600 cold cranking amps – or even more – is possible, based on internal resistance alone.

I have yet to define “cold cranking amps”; I will do so in a coming post. Cheers:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Perl: repeated text

Tutoring Perl programming, you are aware of so many little tricks Perl contains for doing specific tasks.  The tutor shares a fun one.

 
I recall, when I was a kid, a common trick to show off was to fill up the computer screen with your name. A kid with a little programming instruction could easily do it; a kid with no knowledge of programming, had no hope of pulling it off.

At the time, I had no knowledge of programming. However, I saw a kid do the trick on a Commodore Vic-20 (I’m really showing my age here). For years I wondered how he did it.

The kid likely did it in BASIC; Perl wasn’t even invented then. However, I’ll show how it’s done in Perl:

#!/usr/bin/perl
$name=”Johnny Smith! ” x100;
# the x100 means repeat the name 100 times
print “$name”;

Of course, you could replace x100 with x500 or whatever. Don’t overdo it; you just want to fill the screen, right?:)

I might start coverage of BASIC; it’s a fun language. If I recall correctly, I think it uses dollar signs in front of variables the way Perl does. It’s been awhile.

Source: McGrath, Mike. Perl in easy steps. Warwickshire, UK: Computer Step, 2004.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Auto batteries: seasonal reflections

Tutoring high school chemistry includes electrochemistry.  The tutor shares some reflections about auto batteries as winter approaches.

 
I recall more than one mechanic talking about how a device might work fine for a long time but have a hidden weakness. When that device is, for whatever reason, put under abnormal stress, it likely fails. The failure is surprising: hasn’t it worked for months (or years) with no problems? Why, then, does it suddenly fail, at a time you really need it to work?

From my experience, auto batteries can give that “false sense of security.” It’s not the battery’s fault, of course; it’s just the life cycle in most of North America (ie, summer to winter).

From purchase, an auto battery of good quality is likely strong and reliable for a few years anyway. Depending on driving habits, it may maintain its vigor much longer than that. However, time is working against the battery: potentially, the chemical process of sulphation, among other factors.

So, the battery likely weakens over time, yet continues starting the car just as expected. Let’s imagine the battery becomes considerably weaker in late May. The driver likely won’t even notice: from late spring through early fall, the weather is warm. In the heat, the car’s oil might be more agreeable to letting the engine turn over. The days are bright, warm, and dry: the driver doesn’t use the headlights, heater, or wipers as much. Life is easy for the battery. Yet, during those carefree months, the battery may already be too weak to start the car in the cold.

When the inevitable “first winter storm” comes, the car may not start. The driver is surprised. (I’ve been there.) Really, though, the driver’s been on borrowed time for weeks or months already. The battery just wouldn’t reveal its weakness until put under stress.

I’ve been reading up on auto batteries lately (just for the pictures, of course:). In coming posts I’ll discuss some of my findings about this fascinating topic we all depend on.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: simplifying fractions with radicals

Tutoring high school math, you work with radicals.  The tutor discusses simplifying fractions with radicals.

I’ve written several articles about working with radicals; you’ll find them by keying “radicals” in the search box. Ones that might be helpful towards this one are here, here, and here.

Example: Simplify √(28)/√(32)

Solution: First, we use the rule

√(a)/√(b)=√(a/b)

which leads, in our case to

√(28/32)

Now, the fraction can be reduced on the inside:

√(28/32)=√(7/8)

Now we can back out again:

√(7/8)=√(7)/√(8)

Now, from my article Simplifying Radicals…part 1, we know that

√(8)=√(4)√(2)=2√(2)

So we have

√(7)/√(8)=√(7)/(2√(2))

Next, the denominator needs to be rationalized (see my article here):

√(7)/(2√(2))*√(2)/√(2)

=(√(7)*√(2))/(2√(2)√(2))

=√(14)/4

Apparently, √(28)/√(32) simplifies to √(14)/4

Radicals are posed in seemingly endless combinations during high school math. They are part of the daily diet in calculus as well. Therefore, I’ll be exploring further examples with radicals in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.