Tag Archives: The “Why …” Series

Why Existence Theorems?

Posted on by
Reply

An existence theorem is a theorem that says, if the hypotheses are met, that something, usually a number, must exist.

For example, the Mean Value Theorem is an existence theorem: If a function f is defined on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in the open interval (a, b) such that \displaystyle {f}'\left( c \right)\left( {b-a} \right)=f\left( b \right)-f\left( a \right).

The phrase “there exists” also means “there is” and “there is at least one.” In fact, it is a good idea when seeing an existence theorem to reword it using each of these other phrases. “There is at least one” reminds you that there may be more than one number that satisfies the condition. The mathematical shorthand for these phrases is an upper-case E written backwards: \displaystyle \exists .

  • …then there is a number c in the open interval (a, b) such that…
  • …then there is at least one number c in the open interval (a, b) such that…
  • …then \displaystyle \exists c in the open interval (a, b) such that…

Textbooks, after presenting an existence theorem, usually follow-up with some exercises asking you to actually find the value that exists: “Find the value of c guaranteed by the Mean Value Theorem for the function … on the interval ….” These exercises may help you remember the formula involved.

But, the important thing about most existence theorems is that the number exists, not what the number is.

Other important existence theorems in calculus

The Intermediate Value Theorem

If f is continuous on the interval [a, b] and M is any number between f(a) and f(b), then there exists a number c in the open interval (a, b) such that f(c) = M.

Another wording of the IVT: If f is continuous on an interval and f changes sign in the interval, then there must be at least one number c in the interval such that f(c) = 0

Extreme Value Theorem

If f is continuous on the closed interval [a, b], then there exists a number c in [a, b] such that \displaystyle f\left( c \right)\ge f\left( x \right) for all x in the interval.

Another wording: Every function continuous on a closed interval has (i.e. there exists) a maximum value in the interval.

If f is continuous on the closed interval [a, b], then there exists a number c in [a, b] such that \displaystyle f\left( c \right)\le f\left( x \right) for all x in the interval. Or: Every function continuous on a closed interval has (i.e. there exists) a minimum value in the interval.

Critical Points

If f is differentiable on a closed interval and \displaystyle {f}'\left( x \right) changes sign in the interval, then there exists a critical point in the interval.

Rolle’s theorem

If a function f is defined on the closed interval [a, b] and differentiable on the open interval (a, b) and f(a) = f(b), then there must exist a number c in the open interval (a, b) such that \displaystyle {f}'\left( c \right)=0.

Mean Value Theorem – Other forms

If I drive a car continuously for 150 miles in three hours, then there is a time when my speed was exactly 50 mph.

If a function f is defined on the closed interval [a, b] and differentiable on the open interval (a, b), then there is a point on the graph of f where the tangent line is parallel to the segment between the endpoints.

Cogito, ergo sum

And finally, we have Descartes’ famous “theorem:” Cogito, ergo sum (in Latin) or in the original French, Je pense, donc je suis, translated as “I think, therefore I am” proving his own existence.

Why Analytical Applications?

Posted on by
Reply

The last unit showed you some ways the derivative may be used to solve problems in the context of realistic situations. This unit looks at analytical applications of the derivative – that is applications apparently unrelated to any kind of real situation. This is a bit misleading since the things you will learn are meant to be extended to practical problems. It’s just that for now we will study the ideas and techniques in general, not in any context.

The unit begins with two important theorems. The Mean Value Theorem that relates the average rate of change of a function to the instantaneous rate of change (the derivative), The MVT, as it is called, helps prove other important ideas especially the Fundamental Theorem of Calculus at the beginning of the integration.

The other theorem is the Extreme Value Theorem. The EVT tells you about the existence of maximum and/or minimum values of a function on a closed interval.

Both are existence theorems, theorems that tell you that something important or useful exists and what conditions are required for it to exist. More on existence theorems in my next post.

As with all theorems, learn the hypothesis and conclusion. The graphical interpretation makes these easy to understand.

You will learn how to determine where a function is increasing and decreasing. This leads to finding the maximum or minimum points – where the function changes from increasing to decreasing or vice versa: You will learn three “tests” – theorems really – to justify the extreme value.

Along with that you will learn some more about the second derivative and concavity.

These ideas and theorems will help you accurately draw the graph of a function and nail down the precise location of the important points and tell what is happening between them. Yes, your graphing calculator can do that, but you’re taking this course to learn why.

You will be asked to determine information about the function from its derivatives – plural.  The derivative may be given as a function, a graph, or even a table of values.

You will also be asked to justify your reasoning – tell how you can be sure what you say is correct. You do that by citing the theorem that applies and check its hypotheses, not by Paige’s method:

These concepts are tested on the AP Calculus exams and often produce the lowest scores of the six free-response questions. Yet, if you learn these concepts, that question can be the easiest.

P.S. Some books use the Latin words extremum (singular) or extrema (plural). They mean the extreme value(s). Maybe they have hung around so that the uninitiated will think calculus is difficult and confusing. I don’t know. Use them if you like: impress your (uninitiated) friends.

Course and Exam Description Unit 5 Topics 5.1 through 5.9

Why L’Hospital’s Rule?

Posted on by
2

Why L’Hospital’s Rule?

We are now at the point where we can look at a special technique for finding some limits. Graph on your calculator y = sin(x) and y = x near the origin. Zoom in a little bit. The line is tangent to sin(x) at the origin and their values are almost the same. Look at the two graphs near the origin and see if you can guess the limit of their ratio at the origin:   \displaystyle \underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{{\sin \left( x \right)}}{x}?.

In this example, if you substitute zero into the expression you get zero divided by zero and there is no way to divide out the zero in the dominator as you could with rational expressions.  

This kind of thing is called an indeterminate form. The limit of an indeterminate form may have a value, but in its current form you cannot determine what it is. When you studied limits, you were often able to factor and divide out the denominator and find the limit for what was left. With \displaystyle \frac{{\sin \left( x \right)}}{x} you can’t do that.  

But by replacing the expressions with their local linear approximations, the offending factor will divide out leaving you with the ratio of the derivatives (slopes). This limit may be easier to find.

The technique is called L’Hospital’s Rule, after Guillaume de l’Hospital (1661 – 1704) whose idea it wasn’t! He sort of “borrowed’ it from Johann Bernoulli (1667 – 1784).

L’Hospital’s Rule gives you a way of finding limits of indeterminate forms. You will look at indeterminate forms of the types \displaystyle \tfrac{0}{0} and \displaystyle \tfrac{\infty }{\infty }. The technique may be expanded to other indeterminate forms like  \displaystyle {{1}^{\infty }},\ 0\cdot \infty ,\ \infty -\infty ,\ {{0}^{0}},\text{and }{{\infty }^{0}}, which are not tested on the AP Calculus exams.

Like other “rules” in math, L’Hospital’s Rule is really a theorem. Before you use it, you must check that the hypotheses are true. And on the AP Calculus exams you must show in writing that you have checked.

Course and Exam Description Unit 4 Topic 7

Why Related Rates?

Posted on by
Reply

There are situations where a dependent variable is dependent on more than one independent variable. For example, the volume of a rectangular box depends on its length, width, and height, \displaystyle V=lwh.

Think of a box shaped balloon being blown up.  The volume and all three dimensions are all changing at the same time. Their rates of change are related to each other.

Since rates of change are derivatives, all the derivatives are related. Given several of the rates, you can find the others.

In these problems you use implicit differentiation to find the relationship between the variables and their derivatives. That means that you differentiate with respect to time. And time is usually not one of the variables in the equations. \displaystyle V=lwh – see no t anywhere. Really, the length, width and height are all functions of time; you just don’t see the t.

Sometimes the substitutions required to work your way through these problems are the tricky part. So, be careful with your algebra.

Course and Exam Description Unit 4 sections 4.3 to 4.5.

Why Linear Motion?

Posted on by
Reply

Now that you know how to compute derivatives it is time to use them. The next few topics and my next few posts will discuss some of the applications of derivatives, and some of the things you can use them for.

The first is linear motion or motion along a straight line.

Derivatives give the rate of change of something that is changing. Linear motion problems concern the change in the position of something moving in a straight line. It may be someone riding a bike, driving a car, swimming, or walking or just a “particle” moving on a number line.

The function gives the position of whatever is moving as a function of time. This position is the distance from a known point often the origin. The time is the time the object is at the point. The units are distance units like feet, meters, or miles.

The derivative position is velocity, the rate of change of position with respect to time.  Velocity is a vector; it has both magnitude and direction. While the derivative appears to be just a number its sign counts: a positive velocity indicates movement to the right or up, and negative to the left or down. Units are things like miles per hour, meters per second, etc.

The absolute value of velocity is the speed which has the same units as velocity but no direction.

The second derivative of position is acceleration. This is the rate of change of velocity. Acceleration is also a vector whose sign indicates how the velocity is changing (increasing or decreasing). The units are feet per minute per minute or meters per second per second. Units are often given as meters per second squared (m/s2) which is correct but meters per second per second helps you understand that the velocity in meters per second is changing so much per second.   

Of these four, velocity may be the most useful. You will learn how to use the velocity (the first derivative) and its graph to determine how the particle moves over intervals of time: when it is moving left or right, when it stops, when it changes direction, whether it is speeding up or slowing down, how far the object moves, and so on. You can also find the position from the velocity if you also know the starting position.

You will work with equations and with graphs without equations. Reading the graphs of velocity and acceleration is an important skill to learn.

The reasoning used in linear motion problems is the same as in other applications. What you do is the same; what it means depends on the context.

Not to scare anyone, but linear motion problems appear as one of the six free-response questions on the AP Calculus exams almost every year as well as in several the multiple-choice questions.

So, let’s get moving!

Course and Exam Description Unit 4 Sections 4.1 and 4.2

Why Definitions?

Posted on by
Reply

Definitions name things; mathematical definitions name things very precisely.

A good definition (in mathematics or anywhere) names the thing defined in a sentence that,

  1. Puts the thing into the nearest class of similar objects.
  2. Gives its distinguishing characteristic (not all its attributes, only those that set it apart).
  3. Uses simpler (previously defined) terms.
  4. Is reversible.

An example from geometry: A rectangle is a parallelogram with one right angle.

THING DEFINED: rectangle.

NEAREST CLASS OF SIMILAR OBJECTS: parallelogram.

DISTINGUISHING CHARACTERISTIC: one right angle. The other characteristics – the other three right angles, opposite sides parallel, opposite sides congruent, etc. – can all be proven as theorems based on the properties of a parallelogram and the one right angle. No need to mention them in the definition. This also helps keep the definition as short as possible.

PREVIOUSLY DEFINED TERMS: parallelogram, right angle. These you are assumed to know already; they have been previously defined.

IS REVERSIBLE: This means that, if someone gives you a rectangle, then without looking at it you know you have a parallelogram and it has a right angle, AND if someone gives you a parallelogram with a right angle, you may be absolutely sure it is a rectangle. In fact, you could write the definition the other way around: A parallelogram with one right angle, is a rectangle. Either way is okay.

Definitions often use the phrase … if, and only if.... For example: A parallelogram is a rectangle if, and only if, it has a right angle. The phrase indicates reversibility: the statement and its converse (and therefore, its contrapositive and inverse) are true.

When you get a new term or concept defined in calculus (or anywhere else), take a minute to learn it. Look for the nearest class of similar things, its distinguishing characteristics, and be sure you understand the previously defined terms. Try reversing it; say it the other way around. At that point you’ll pretty much have it memorized.

Definitions are never proved. There is nothing to prove; they just name something. Statements in mathematics that need to be proved are called theorems.

Finally, you may take the words in bold above as the definition of a definition!

Why Techniques for Differentiation?

Posted on by
Reply

You have learned and used formulas for finding the derivatives of the Elementary Functions. These can be applied to functions made up of the Elementary Functions and extended to other expressions.

Many functions are made by combining the Elementary Functions. For example, polynomials are the sum and differences of a constants and powers of x multiplied by constants (their coefficients).

The functions you will look at next are the sums, differences, products, quotients, and/or composites of the Elementary Function. You will learn five techniques for handling these.

Sums and differences are found by differentiating the individual terms. Then there are the Product Rule for products of functions, the Quotient Rule for quotients and the Chain Rule for compositions. The techniques are often used in combinations.

Learn to see the patterns in the functions and learn what procedure to use for each.

HINT: Memorize the techniques as you learn them. After all these years, I still say the formulas and techniques as I use them. So, for products I repeat in my mind “the first times the derivative of the second plus the second times the derivative of the first” as I do the computation. Forget about mnemonics – just say the technique as you use it, and you’ll memorize it easily. 

Implicit relations and inverses have their own techniques; they use the basic formulas and techniques in different ways.

Since derivatives are functions, they have their own derivatives. The derivative of the first derivative is called the second derivative. Then there is the third derivative, the fourth derivative and so on. Mostly, you will use only the first three. No new rules to learn; higher order derivatives are computed the same way as the first derivative, and in fact, they often get simpler.

The proofs of the formulas and techniques (they are really theorems) 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 3 Sections 2.8 – 2.10 and Unit 3 all