Category Archives: The “Why …” Series

Why Antiderivatives?

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Antiderivatives are needed to evaluate definite integrals.

The next thing to consider is how to find antiderivatives.

Each of the formulas you learned for finding a derivative may be reversed to find antiderivatives. For example, since \displaystyle \frac{d}{{dx}}\sin \left( x \right)=\cos \left( x \right), it follows that.\displaystyle \int{{\cos \left( x \right)dx}}=\sin \left( x \right)

I wish it were all that simple.

There are three concerns.

First, if the original function included a constant, this constant will disappear when you differentiate. Think about it: adding a constant translates the graph up or down but does not change the shape; the slope (derivative) remains the same.

This means that a function has an infinite number of antiderivatives. The good news is they are all the same except for the constant of integration.

The \displaystyle \cos \left( x \right)is the derivative of \displaystyle \sin \left( x \right)+3,\sin \left( x \right)-8,\pi +\sin \left( x \right) and all kinds of similar things.

To remind you of this you should write \displaystyle \int{{\cos \left( x \right)dx}}=\sin \left( x \right)+C where C is a constant, a number, called the constant of integration.

Next, very similar looking functions have very different antiderivatives found in very different ways. I won’t scare you with examples, you’ll see them soon enough.

Finally, there are many simple looking functions, that you can easily differentiate that do not have an antiderivative that is any function you’ve seen.

In the last parts of Unit 6, you will learn some methods integration. BC students will learn a few additional methods. You’ve only scratched the surface: there are many more, but these can wait until you get to university (or maybe Mathematica knows them – I wouldn’t be surprised).

As you learn these methods of integration you will have to decide when to use each. Learn which method is appropriate in each situation.

Course and Exam Description Unit 6.8 thru 6.14

Why the Fundamental Theorem of Calculus?

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The Fundamental Theorem of Calculus is, well, fundamental. It relates the derivative and the integral.

Writing a Riemann sum with all that fancy notation is tedious. To speed things up a special notation is used to replace it. The limit of the Riemann sum for a function on an interval [a, b] is written as its definite integral:

\displaystyle \underset{{\left| {\Delta x} \right|\to 0}}{\mathop{{\lim }}}\,\sum\limits_{{i=1}}^{\infty }{{f\left( {{{x}_{i}}} \right)\Delta x}} \displaystyle =\int_{a}^{b}{{f\left( x \right)dx}}

The \displaystyle f\left( x \right) (called the integrand) is the function with no fancy notation and the dx, called differential x replaces the \displaystyle \Delta x. The a and b, called the lower and upper limit of integration respectively, show you the interval the Riemann sum was formed on (which the Riemann sum does not).

Keep in mind that behind every definite integral is a Riemann sum. Therefore, all the properties of limits apply to definite integrals. They can be added and subtracted, a constant may be factored out, and so on.

The Fundamental Theorem of Calculus, the FTC, tells you how to evaluate a definite integral (and therefore its Riemann sum): Simply evaluate the function of which \displaystyle f\left( x \right) is the derivative at the endpoints of the interval and subtract.

To keep this in mind you can write the FTC like this considering the integrand as the derivative (of something):

\displaystyle \int_{a}^{b}{{{f}'\left( x \right)dx}}=f\left( b \right)-f\left( a \right).

For example, since  \displaystyle d\sin \left( x \right)=\cos \left( x \right)dx,

\displaystyle \int_{0}^{{\pi /2}}{{\cos \left( x \right)dx=\sin }}\left( {\tfrac{\pi }{2}} \right)-\sin \left( 0 \right)=1-0=1

That’s all there is to it!

But wait! There’s more! This reveals another important idea: Since derivatives are rates of change, the FTC says that the integral of a rate of change is the net amount of change over the interval. Also called the accumulated change.

Well, okay, there is the problem of finding the function whose derivative is the integrand which is not always easy. This function is called the antiderivative of the integrand; another name is the indefinite integral. (The notation for an antiderivative or indefinite integral is the same as for a definite integral without the limits of integration). The truth is that finding the antiderivative is not as straightforward as finding the derivative. We will tackle that soon.

Course and Exam Description Unit 6.3 thru 6.7

Why Riemann Sums?

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You are now ready to move into the study of integration, the other “half” of calculus. To integrate is defined as “to bring together or incorporate parts into a whole” (Dictionary.com).

The initial problem in integral calculus is to find the area of a region between the graph of a function and the x-axis with vertical sides. This is done by lining up very thin rectangles, finding their individual areas and incorporating them into a whole by adding their areas.

The way the rectangles areas are found and added is to use a Riemann sum. The width of each rectangle is a small distance along the x-axis and the length is the distance from the x-axis to the curve. As you use more rectangles over the same interval, their width decreases, and the approximation of the area becomes better.

Yes, that’s limits again. As the number of rectangles increases (\displaystyle n\to \infty ), their width decreases (\displaystyle \Delta x\to 0) and the (Riemann) sum approaches the area.

You will start by setting up some of these Riemann sums with a small number of rectangles to help you get the idea of what’s happening. (Lots of arithmetic here.)

Written in mathematical notation, a Riemann sum looks like this \displaystyle \sum\limits_{{n=1}}^{\infty }{{f\left( {{{x}_{n}}} \right)\Delta x}}. The interval on the x-axis is divided into subintervals of width \displaystyle \Delta x; these do not have to be the same, but almost always are. The \displaystyle f\left( {{{x}_{n}}} \right)  is the function’s value at some point, \displaystyle {{x}_{n}}, in each interval. So, \displaystyle f\left( {{{x}_{n}}} \right)\Delta x is the area of the rectangle for that subinterval. The sigma sign sums them up.

And the \displaystyle \underset{{\Delta x\to 0}}{\mathop{{\lim }}}\,f\left( {{{x}_{n}}} \right)\Delta x gives the area.

Most of the time the limit will not be easy to find, so you’ll avoid it! Soon you will learn a quick and efficient way to find the limits.

Riemann sums can be used in many other applications as you will soon learn.

Course and Exam Description Units 6. 1 and 6.2

Why Optimization?

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Unit 5 ends with a return to a realistic context. To optimize something means to find the best way to do it. “Best” or “optimum” may mean the quickest, the cheapest, the most profitable, or the easiest way to do something.

For example, you may be asked to build a box of a given volume with the least, and therefore cheapest, amount of material. Thus, these are really problems where you need to find the maximum or minimum of the function that models the situation.  There are applications to engineering, finance, science, medicen, and economics among others.

The most difficult part of these problems is often writing the equation to be optimized; not the calculus involved. Once you have the model, finding the extreme value is easy.

The last part of this unit extends the ideas of this unit to implicit relations, those whose graph may not be a function. These too, increase, decrease, and have extreme values. The same techniques help you to find them.  

Course and Exam Description Unit 5 Sections 11 and 12

A note for teachers: You are not behind scheduel. Please remember that I am posing this series ahead, probably well ahead, of where you are. This is so that they will be here when you get here.

Why Existence Theorems?

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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?

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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?

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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