Why Formulas for Derivatives

Posted on September 19, 2023 by Lin McMullin

That’s pretty obvious: Finding derivatives using limits is a pain!

Knowing the derivatives of the common functions makes it easy to find.

No hiding it: you need to memorize these formulas. The best way is to learn them as you get them, a few at a time. I’m sure your teacher will not give them to you all at once. From the first day, memorize them by saying them to yourself as you use them. Don’t wait until the night before the test.

It’s really not too bad: there are only about seventeen formulas for the derivatives of the Elementary Functions. The Elementary Functions as those you’ve learned about already: powers if x including fractional and negative powers (one formula for all), six trigonometric functions, six inverse trigonometric function, exponential functions (one for base e and one for other bases), logarithm functions (natural, and the others).

Formulas are really theorems. The proofs of the formulas are interesting from a mathematical point of view. You should follow along when your teacher shows them to you so that you understand why they work and where they come from. You will not be asked to reproduce the proofs on the AP Calculus Exam.

AP Calculus Course and Exam Description Unit 2 topics 2.5 – 2.7

Why Theorems?

Posted on September 15, 2023 by Lin McMullin

All the important things in mathematics are written as theorems.

Theorems are statements of mathematical facts that have been proven to be true based on axioms, definitions, and previously proved theorems. They summarize information in a general way so that it may be applied to specific new situations.

All the rules, formulas, laws, etc. that you study in mathematics are really theorems.

The form of a theorem is IF one or more things are true, THEN something else is true. For example, IF a function is differentiable at a point, THEN it is continuous at the point.

The IF part is called the hypothesis (the function is differentiable at a point) and the THEN part is called the conclusion (the function is continuous at the point).

The word “implies” can replace the IF and the THEN. So, the theorem above may be shortened to “Differentiability implies continuity.” When this happens be sure you understand what has been omitted.

HINT: It is always a good idea when learning a new theorem to identify the hypothesis and the conclusion for yourself.

Proof

The proof of a theorem is an outline of the reasoning that shows how previous results (axioms, definitions, and previous theorems) lead to the conclusion. They are carefully written to convince mathematicians and other interested people that the theorem is true.

In AP Calculus, you will not prove every theorem. The reason you as beginning calculus students should look at proof is (1) to help you understand why the theorem is true, and (2) to begin learning how to do proof yourself.

Good news / bad news: You will not be asked to prove theorems on the AP Calculus exams. You will be asked to “justify your answer” or “show your reasoning” or the like. To do this you will need to state that hypotheses of the theorem you are using are true for the situation in the question and therefore, you may say that the conclusion applies in this case. To use the theorem in the example, you would have to establish that the function you are given is differentiable, then you may say it is continuous.

for any theorem, you need to know and understand both the hypothesis and the conclusion.

Related Statement: The Contrapositive

The contrapositive of a theorem is a statement that says if the original theorem’s conclusion is false, then its hypothesis is false. This makes sense: When the original hypothesis is true, the conclusion must be true. So, if the conclusion is false something must be wrong with the hypothesis. For any theorem, its contrapositive is always a true theorem.

The contrapositive of the example above is “If a function is not continuous at a point, then it is not differentiable there.” In fact, this particular contrapositive is one you will be using soon.

Related Statement: The Converse

The converse of a theorem is formed by interchanging the hypothesis and the conclusion. The converse of our example is “If a function is continuous at a point, then it is differentiable there.” This statement is false! There are continuous functions that are not differentiable. An example is the absolute value function at the origin.

Converses may or may not be true. They must be proved separately, if possible. It is a mistake to assume the converse is true without first proving it. This mistake is so common it has a name; it is called the fallacy of the converse.

Related Statement: The Inverse

The inverse is (hold on tight) the contrapositive of the converse. It states that if the original hypothesis is false, then the original conclusion is false. for our example: “If a function is not differentiable, then it is not continuous.” (This example is false.)

The inverse is not necessarily true; it is true if the converse is true. Like the theorem / contrapositive pair, the converse / inverse pair are true or false together. Sometimes all four are true, sometimes not.

Finally, any of the four statements may be considered “the theorem” and the other three will change their names accordingly. The theorem states the idea in the form in which it is usually used. The converse, if it is important and true, is given and proved separately at the same time. The contrapositive and the inverse go along for the ride and do not have to be proved separately.

Why Continuity?

Posted on September 5, 2023 by Lin McMullin

We would like to study nice well-behaved functions; functions that are smooth and that don’t do strange things. Yeah, well good luck with that.

One of the things that might be nice is that you could draw the graph of a function from one end of its domain to the other without taking your pen off the paper. And a lot of functions are like that, but not all.

Some functions have holes in them, others jump from one y-value to another without hitting points in between. Some “go off to infinity” and come right back; others go off the top of the graph and come back from the bottom. Some go really crazy around a point.

Functions that you can draw from one end of their domain to the other without lifting your pen are said to be continuous.

More mathematically: A function is continuous at a point (that is, at a single value of x) if, and only if, as you travel along the graph towards the value (from either side of the x-value), the y-values on the graph are approaching the y-value at the point. In symbols: $\displaystyle \underset{{x\to a}}{\mathop{{\lim }}}\,f\left( x \right)=f\left( a \right)$ .

A function is continuous on an interval if, and only if, it is continuous at every value in the interval.

Wait! What?? I have to check all the points??

Technically, yes; practically, no. Most of the time you can easily show that a function is continuous everywhere by looking at its limit in general.

Moreover, you will learn to see where a function is not continuous. This is an important skill: looking at a function and suspecting there is a problem with continuity.

Take a quick look at some of the problems that functions may have at a point. Graph these on your calculator. They all have a “problem” at x = 3. Graph each example and you will see what they look like. Try to figure out why they have a “problem” and what causes it.

$\displaystyle f\left( x \right)=\frac{1}{{x-3}}$ .
$\displaystyle g\left( x \right)=\frac{1}{{{{{\left( {x-3} \right)}}^{2}}}}$
$\displaystyle h\left( x \right)=\frac{{{{x}^{2}}-3x}}{{x-3}}$ . This function has a single hole in the graph at (3, 3); It one may be difficult to see. Try using ZDecimal. A single point is missing because there is no value at x= 3 because the denominator is zero.
$\displaystyle j\left( x \right)=\frac{{{{x}^{2}}\sqrt{{{{x}^{2}}-6x+9}}}}{{2x-6}}$ .

$\displaystyle k\left( x \right)=\cos \left( {\frac{1}{{x-3}}} \right)$ Zoom in several times at (3, 0) where the function has no value.

$\displaystyle m\left( x \right)=\left\{ {\begin{array}{*{20}{c}} {{{x}^{2}}} & {x<3} \\ {4-x} & {x\ge 3} \end{array}} \right.$

Learn to suspect that a function may have a discontinuity. (It’s not always at x = 3) The problem is often a zero denominator.

This is not just a game or some curious functions. One of the main tools of calculus called the derivative, which you will study next, is defined as the limit of a special function which is never continuous at the point you are interested in.

So, let’s continue on to continuity.

AP Calculus Course and Exam Description

Unit 1 topics 1-10 – 1.16, Unit 2 topic 2.4

Why Infinity?

Posted on August 25, 2023 by Lin McMullin

First, right from the start: Infinity is NOT a number.

Lots of folks think of infinity as the largest number possible, greater than anything else. That’s understandable because infinity, denoted by the symbol $\displaystyle \infty$ , is often used that way by those unlucky folks who don’t understand mathematics.

We’ll start with an example: Consider the fraction $\displaystyle \frac{1}{{{{{\left( {x-3} \right)}}^{2}}}}$ . This fraction has no value when x = 3 because there the denominator is zero. And you cannot divide by zero. Nothing personal, no one, no matter how smart, can divide by zero. Ever. Permanently and forever not allowed. Don’t even think about it! (Actually, think about it; just don’t do it.)

What you should say in such cases is that the expression has no value, or is “undefined,” or “the limit does not exist,” abbreviated DNE.

In situations like the example we say, “the limit of the fraction as x approaches 3 equals infinity,” abbreviated $\displaystyle \underset{{x\to 3}}{\mathop{{\lim }}}\,\frac{1}{{{{{\left( {x-3} \right)}}^{2}}}}=\infty$ . This means that the expression gets larger as x gets closer to three. The expression will be greater than any (large) number you want, if you are close enough to three.

You don’t believe me? Okay pick a large number, maybe $\displaystyle {{10}^{8}}$ . I say pick any value for x between 2.9999 and 3.0001 ( $\displaystyle 3-{{10}^{{-4}}}<x<3+{{10}^{4}}$ ) and the expression will be larger than $\displaystyle {{10}^{8}}$ . Try it on your calculator.

How about $\displaystyle {{10}^{{20}}}$ ? Try a number between 2.9999999999 and 3.0000000001. I can play this game all day.

Try graphing the $\displaystyle y=\frac{1}{{{{{\left( {x-3} \right)}}^{2}}}}$ on your calculator. (Hint: Whenever you come across something like this, it is a great idea to graph the expression on your graphing calculator. Graphs can help you see what’s going on. Keep that in mind for the future.)

That’s the way to think about infinity: Infinity is what you say when you’re working with an expression that grows greater than any number you choose.

You may also use infinity to say what happens all the way to the left or right of the graph, its end behavior. The variable, x, may “approach infinity,” that is x moves further to the right (or is greater than any number you choose) the fraction above gets closer to zero: $\displaystyle \underset{{x\to \infty }}{\mathop{{\lim }}}\,\frac{1}{{{{{\left( {x-3} \right)}}^{2}}}}=0$ .

You may not do arithmetic with infinity.

$\displaystyle \infty +\infty \ne 2\infty$

$\displaystyle \infty -\infty \ne 0$

Arithmetic is for numbers.

You will see a number of expressions whose limit is equal to infinity, like $\displaystyle \underset{{x\to 3}}{\mathop{{\lim }}}\,\frac{1}{{{{{\left( {x-3} \right)}}^{2}}}}=\infty$ . Which really means, just what we saw above: that as you (not “you” but x) get closer to 3, the value of the expression will be greater than any number you pick. The $\displaystyle \infty$ symbol is a shorthand way of saying this.

The opposite of infinity, $\displaystyle -\infty$ , sometimes called “negative infinity,” means that the expression gets less than (i.e. more negative), than any negative number you choose.

Even though the expression has no limit, you are allowed to say the limit equals infinity. That’s funny when you think about it. It might be better if everyone said “undefined” or DNE, but they don’t. What can I say?

A word of warning: You may only say “equals infinity” is situations like the example above.

There are other similar expressions that have no limit where it is incorrect to say the limit equals infinity. For example,

$\displaystyle \frac{{\left| x \right|}}{x}$ has no value, is “undefined,” when x = 0, but $\displaystyle \underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{{\left| x \right|}}{x}\ne \infty$ . (Hint: this is where you should look at a graph on your graphing calculator to see why.)
$\displaystyle \underset{{x\to 3}}{\mathop{{\lim }}}\,\frac{1}{{\left( {x-3} \right)}}$ does not exist. This is very similar to the first example but look at the graph and you’ll see a big difference.