Calculus unlimited pdf

No calculus, as such, was used in this definition. This suggested that the same concept could be used to define the tangent line and thus serve as a limit-free foundation for the differential calculus. We introduced this new definition of the derivative into our class notes and developed it in our calculus classes for several years. As far as we know, the definition has not appeared elsewhere. If our presumption of originality is ill-founded, we welcome your comments.

Repository Staff Only: item control page. A Caltech Library Service. Calculus Unlimited. Once we know how to recognize rapidly vanishing functions, Theorem 2 will provide a useful test for differentiability and a tool for computing derivatives.

Notice that Theorem 2, like Theorem 1, is of the ''if and only if' type. Thus, it has two independent parts. Each of the two parts of the theorem is called the converse of the other. In the proof which follows, we prove the second part first, since it is convenient to do so. Recall that m was any number greater than mo.

Prove that x - XO 2 vanishes rapidly at xo. Fill in the details of the last two paragraphs of the proof of Theorem 2. By evaluating these polynomials for values of x very near 1, on a calculator, try to guess which of the polynomials vanish rapidly at 1. Factoring the polynomials may help you to understand what is happening.

By Theorem 2, each of the expressions in square brackets represents a function which vanishes rapidly at xo, so we need to show that the sum of two rapidly vanishing functions is rapidly vanishing. The proof of this rests upon the fact that a constant multiple of a rapidly vanishing function is again rapidly vanishing.

Theorem 3 proves these two basic properties of rapidly vanishing functions. Theorem 3 1. If rl x vanishes rapidly at Xo, and a is any real number, then arl x vanishes rapidly at Xo.

Proof 1. Adding these inequalities gives You should now be able to see why we used eI2. Theorem 4 1. Sum Rule. Constant Multiple Rule. If f x is differentiable at xo, and a is any real number, then 'the function af x is differentiable at xo, and its derilJatilJ2 there is af' x o.

See Solved Exercise 5. Prove that, if rl x and r:l x vanish rapidly at Xo, then so does rl x - ':lex. Prove part 2 of Theorem 4 the constant multiple rule. Let a be a nonzero constant, and assume that af overtakes g at xo. This means that Theorem 3 is a special case of the sum and constant multiple rules. Our proof of Theorem 4 pro- ceeded from this special case to the general case. Can just one of them be nondifferentiable at xo? What does the graph look like in this case? The Product Rule The sum rule depended on the fact that the sum of two rapidly vanishing func- tions is again rapidly vanishing.

For the product rule, we need a similar result for products, where only one factor is known to be rapidly vanishing. Theorem 5 If rex vanishes rapidly at Xo and f x is differentiable at xo, thenf x r x vanishes rapidly at xC. Proof Note that part 2 of Theorem 3 is a special case of this theorem, where [ x is constant. We prove Theorem 5 in two steps, the first of which shows that [ x can be "sandwiched" between two constant values.

The number B is called a bound for If x I near xo. We could just as well have used slopes! On a sufficiently small interval II. The reader may fill in the algebra required to determine a possible choice for B. See Solved Exercise 6. Step 2. Clearly f x r x vanishes at xo. We now apply Theorem 1 just as we did in proving part li of Theorem 3.

Thus, by Theorem 1,itx r x vanishes rapidly at xo. We can now deduce the product rule from Theorem 5 by a computation. Theorem 6 Product Rule. Note that the correct formula for the derivative of a product appeared as the coefficient of x - xo in our computation; there was no need to know it in advance. Now that we have proven the product rule, we may use one of its impor- tant consequences: the derivative of x n is nx n -1 see Solved Exercise 7.

Find a function which vanishes rapidly at both 3 and 7. Find a function which vanishes rapidly at 1,2, and 3. Sketch a graph of this function. Show that x - a n vanishes rapidly at a if n is any positive integer greater than or equal to 2.

Prove that a is a multiple root of g x if and only if g x vanishes rapidly ata. If f x and g x are defined in an interval about xo, andf' xo and fg ' xo both exist, does g' xo necessarily exist? Compare Exercise 9. Then a bow-tie argument like that on p. In Theorem 5, we actually proved that the product of a rapidly vanishing function and a function bounded between two values is again rapidly vanishing. Now the constant multiple. Corollary Quotient Rule.

With these basic results in hand we can now readily differentiate any rational function. See your regular calculus text. Using Fig. Could such a g be differentiable at 01 Exercises Let a and b be real numbers. Suppose that f x vanishes rapidly at a and that g x is differentiable at a.

Could fig be differentiable at Xo without f and g themselves being differen- tiable there? Let if x ;;. Sketch the graphs of g x and x 2 g x. Prove that if r is rapidly vanishing at x0 and f is locally bounded at x 0, then fr vanishes rapid- ly at xo. Let if x is irrational if x is rational Show that [ x is not differentiable at. Isf x dif- ferentiable at O?

Prove your answer. Suppose that f x and g x are differentiable at Xo and that they both vanish there. Prove that their product vanishes rapidly at xo.

Find a function which vanishes rapidly at every integer. To such questions as, "how do we know that there is a number whose square is 21" and "how is rr constructed? How does one define the circumference of a circle? One way to settle them, without recourse to geometric intuition, is to write down a list of unambiguous rules or axioms which enable us to prove all we want. A set of axioms for the real numbers was developed in the middle part of the 19th Century.

In more advanced courses one has to face the question of showing that there exists a system of numbers obeying these axioms, but we shall merely assume this here. Mathematics that is useful in applications to science is rarely discovered by means of axiom systems. Axiornatics is more frequently the fmal product of a piece of mathematics created for some need. The aXioms for real numbers were agreed on only after centuries of trial and error, and only after the basic theorems were already discovered.

It is not our intent to show that all the usual manipulative rules follow from the axioms, since that job is too long and is done in algebra courses. Our aim is merely to set out our assumptions in a clear fashion and to give a few illustrations of how to use them. Addition and Multiplication Axioms Our first axioms pertain to the operation of addition.

There is a number 0 "zero'. The next axioms pertain to multiplication and its relation with addition. Multiplication Axioms There is a multiplication operation ". For all x, y, and z, x 6'. In principle, one could prove all the usual rules of algebraic manipulation from the axioms above, but we will content ourselves with the few samples given in the exercises below.

As usual, we write x 2 for x x, x 3 for x 2 x, etc. Prove that x. Exercises Prove the following identities. Order Axioms From now on we will use, without further justification, the usual rules for alge- braic manipulations. The previous exercises were intended to convince the reader that these rules can all be derived from the addition and multiplication axioms. We tum now to the order axioms. Order Axiums There is a relation such that, for certain pairs x and y of real numbers the statement "x read "x is less than or equal to y" is true.

This relation has the following properties: 1. If x and z is any number. Also, we write y ;;;.. Again, one can prove all the usual properties of the inequality signs from the axioms above. As before, we limit ourselves to a few instances. Solved Exercises 6. Prove: if x and c 0, then ex ey. Prove that a b. Prove: x 2 ;;;.. Prove that there are no infinitesimal real numbers.

In fact, the rational numbers i. If Xl and X2 are elements of Sand y is a number between them, i. Thus, the set S has "no holes" in the sense that it contains every number between any two of its members.

Our intuition tells us that S ought to be an interval. This is done in Solved Exercise We would then have found the square root of 2. The completeness axiom makes our intuitive notion into a property of real numbers, taking its place alongside the addition, multiplication, and order axioms.

We need one definition before stating the axiom. The theory is called "nonstandard analysis. Any interval is a convex set. See Solved Exercise The completeness axiom asserts the converse. Completeness Axiom Every convex set of real numbers is an interval.

The force of the completeness axiom lies in the fact that intervals have endpoints. Thus, whenever we can prove a set to be convex, the completeness axiom implies the existence of certain real numbers. Here is some further motivation for the completeness axiom. Suppose that S is a convex set of real numbers which does not extend infinitely in either direc- tion on the number line. IfYI andY2are still not endpoints, we can imagine spreading the calipers more and more until no more spreading is possible.

The points beyond which the caliper tips cannot spread must be endpoints of S; they mayor may not belong to S. Solved Exercises Prove that [a, b is convex. Let c be the right-hand endpoint of S which exists by the completeness axiom. Prove that. J2is irrational; Le. Prove that any open interval a, b contains both rational and irrational numbers. Let S be the set consisting of those numbers x for which x E [0,1 or x E 1,2]. Prove that Sis not convex. What are the endpOints of the convex set?

Let A and B be sets of real numbers such that every element of A is less than every element of B, and such that every real number belongs to either A or B. Using the completeness axiom, show that there is exactly one real num- ber c such that every number less than c is in A and every number greater than c is inB. Problems tor Chapter 4 1. Using the addition and multiplication axioms as stated, prove the following identities.

Prove the following identities. Prove the following statements. Prove that every real number is less than some positive integer. This result is often referred to as the "Archimedian property.

Which of the following sets are convex? Show that S is convex and describe S as an interval. Prove the following inequality using the order axioms: a 2 -tb 2. Suppose that A and B are convex sets such that every element of A is less than every element of B. Calculus, Publish or Perish, Inc. The Definition of ContinuitY Naively, we think of a curve as being continuous if we can draw it ''without re- moving the pencil from the paper.

If the curve is continuous, at least a "piece" of the curve on each side of xo,Yo should be between these lines. Compare this with the behavior of the discontinuous curve in Fig. The following defmition is a precise fonnulation, for functions, of this idea. CIt X o Fig. Definition If Xo is an element of the domain D of a function f. The property by which continuity is defmed might be called the "principle of persistence of inequalities": fis continuous atxo when every strict inequality which is satisfied by f xQ continues to be satisfied by f x for x in some open interval about xo.

The intervals I and J in the definition may depend upon the value of Cl and C2. The following example illustrates this idea. Worked Example 1 The mass y in grams of a silver plate which is deposited on a wire during a plating process is given by a function I x , where x is the length of time in seconds during which the plating apparatus is allowed to operate.

Being realistic, you know that you cannot control the time prectsely, but you are willing to accept the result if the mass is less than 0. Show that if f is continuous, there is a certain tolerance r such that, if the time is within r of 3 seconds, the resulting mass of silver plate will be acceptable. Solution We wish to restrict x so that f x will satisfy the inequalities 1.

From condition 1 of the definition, there is an open interval I containing 3 such that 1. Of course, to get a specific value of r which works, we must know more about the function f. Continuity tells us only that such a tolerance '[ exists. Theorem 1, which appears later in this chapter, gives an easy way to verify that many functions are continuous.

First, though, we try out the definition on a few simple cases in the follOWing exercises. Solved Exercises. Let [ x be the absolute value function. Decide whether each of the functions whose graphs appear in Fig. Explain your answers. Show that, for any constants a and b. A geometric argument will suffice.

How should you define f l to make the resulting function continuous? Is there any way to definef O so that the resulting function will be continuous? Our intuition snggests that if a curve is smooth enough to have a tangent line then the curve should have no breaks-that is, a differentiable function is continuous. The following theorem says just that. Theorem 1 If the function f is differentiable at xo, then f is continuous at Xo. Proof We need to verify that conditions 1 and 2 of the definition of con- tinuity hold, under the assumption that the definition of differentiability is met.

Referring to Fig. Condition 1 is verified in an analogous manner. We leave the details to the reader. This method is certainly much easier than attempting to verify directly the conditions in the defInition of continuity.

The argument used in this example leads to the follOWing general result. Any polynomial P x is continuous. Let P x and Q x be polynomials, with Q x not identically zero. In Chapter 3, we proved theorems concerning sums, products, and quo- tients of differentiable functions.

One can do the same for continuous functions: the sum product and quotient where the denominator is nonzero of contin- uous functions is continuous. Using such theorems one can. Interested readers can try to work these theorems -out for themselves see Exer- cise 11 below or else wait until Chapter 13, where they will be discussed in connection with the theory of limits.

Solved Exercises 5. Prove that x 2 - 1! Is the converse of Theorem 1 true; i. Prove or give an example. Let [be. Exercises 7. Find a function which is continuous on the whole real line, and which is differentiable for all x except 1,2, and 3. A sketch will do. Let fbe defmed in an open interval about Xo. Our intuitive notions of continuity suggest that every continuous function has the intermediate value property, and indeed we will prove that this is true.

Unfortunately, the intermediate value property is not suitable as a defini- tion of continuity; in Fig. Before, proving the main intermediate value theorem, it is convenient to begin by proving an alternative version.

Suppose that f is continuous on [a, b and that f a is less [greater] than some number d. The completeness axiom states that every convex set is an interval. Thus, we begin by proving that Sis convex; Le. We have proven that yES, so S is convex. By the completeness axtom, S is an interval. Nothing less than a can be in S, so a is the left- hand endpoint of S. Since S is contained in [a, b] , it cannot extend infinitely far to the. Thus, case 2 cannot occur. We can now deduce the usual form of the intermediate value theorem.

Proof If there were no such c, then the alternative version, which we have just proven, would enable us to conclude that [ a andf b lie on the same side of d. Find a formula for a function like that shown in Fig. You may use trigonometric functions. Let T be the set of values of a function fwhich is continuous on [a, b] ; Le. Prove that T is convex. Prove that f must have a critical point a point where [' vanishes somewhere on the interval -1,1.

Prove that f has at least two real roots. Study Fig. Similarly f is decreasing if and only if [ x - [ xo changes sign from positive to negative at xo. If we substitute the definition of "overtake" in the definition of in- creasing, we obtain the following equivalent reformulation. Definition' Let f be a function whose domain contains an open interval aboutxo. The opposite happens if the function is decreasing at xo. Sohltion Let [be 1,3. We have verified i and li of part 1 of Dermition', sof is increasing at 2.

The transition defmition of the derivative of f at Xo tells us which lines overtake and are overtaken by the graph of f at xo. This leads to the next theorem. Theorem 3 Let f be differentiable at x o.

Proof We shall prove parts 1 and 3; the proof of part 2 is similar to 1. The follOWing is another proof of Theorem 3 directly using the original defmition of the derivative in Chapter 1.

By Theorem 3 part 3, X S - x 3 - 2x 2 is increasing at Theorem 3 can be interpreted geometrically: if the linear approximation to [at Xo that is, the tangent line is an increasing or decreasing function, then [ itself is increasing or decreasing at Xo- If the tangent line is horizontal, the behavior of fatxo is not determined by the tangent line.

Combined with the techniques for differentiation in Chapter 3, Theorem 3 provides an effective means for deciding where a function is increasing or decreasing. A ball is thrown upward with an initial velocity of 30 meters per second. When is it falling? We would expect f b to be larger thanf a.

In fact, we have the following useful result. The statement of Theorem 4 may appear to be tautological-that is, "triv- ially true"-but in fact it requires a proof which will be given shortly. Like the intermediate value theorem, Theorem 4 connects a local property of func- tions increasing at each point of an interval with a global property relation between values of the function at endpoints.

We do not insist thatfbe increas- ing or decreasing at a or b because we wish the theorem to apply in cases of the type illustrated in Fig. Also, we note that if f is not continuous, the result is not valid see Fig. Proof of Theorem 4 We proceed in several steps. By the same kind of argument as we used iIi the proof of the intermediate value theorem alternative ver- sion it is easy to ihow that S is convex.

By the completeness axiom, S is an interval. Since I is increasing at x, S contains all the points sufficiently near to x and to the right of x, so x is the left-hand endpoint of S. Thus S contains points to the right of c, con- tradicting the fact that c is the righthand endpoint of S.

Notice that we have not yet used the continuity off at a and b. Choose any y in a,b. We shall now rephrase Theorem 4. The following terminology will be convenient. Definition Let [be a function defined on an interval I. Theorem 4' Let f be continuous on [a, b] and increasing [decreasing] at all points of a, b. Then fis increasing [decreasing] on [a, b]. For example, the function in Fig. Draw a sketch to con- vince yourself. Combining Theorems 3 and 4' with the intermediate value theorem gives a result which is useful for graphing.

I' is'never zero on a,b. Then f is monotonic on [a, b. To check whether f is increasing or decreasing on [a. If I' took values with both signs, it would have to be zero somewhere in between. Iff' is positive everywhere, f is increasing at each point of Ca, b by Theorem 3. By Theorem 4', f is increasing on [a, b. If f' is negative everywhere, f is decreasing on [a, b. Proof By the intermediate value theorem, i' is: everywhere positive or everywhere negative on a, b.

Suppose it to be everywhere positive. Thu:sf j:s iucrca:sjng on a, b. Similarly, if I' is everywhere negative on a. Similar statements hold for half-open intervals [a, b or a, b. Applications of these results to the shape of graphs will be given in the next chapter. What is wrong here?

Prove the analogue of Theorem 4 for decreasing functions. Show by example that the conclusion in Solved Exercise 17 may not be valid if f is discontinuous at b. The Extreme Value Theorem The last theorem in this chapter asserts that a continuous function on a closed interval has maximum and minimum values.

Again, the proof uses the complete- ness axiom. We begin with a lemma which gets us part way to the theorem. Boundedness Lemma Let f be continuous on [a, b]. Then there is a num- ber B such that f x '" Bfor all x in [a, b].

We say that f is bounded above by B on [a, b] see Fig. Proof If P is any subset of [a, b] , we will say that f is bounded above on P if there is some number m which may depend on P such that f x '" m for all x in P.

Let S be the set consisting ofthosey in [a,b] such thatfis bounded above on [a,y]. Also, y must belong to [a, b if y 1 and y 2 do, so S is. By the completeness axiom, S is an interval which is contained in [a,b]. Let c be the right-hand endpoint of S.

Thus, S is of the form [a, c]. Then f has both a maximum and minimum value on [a, b],. Proof We prove that there is a maximum value, leaving the case of a mini- mum to the reader.

Consider the set T of values off, i. We saw in Solved Exercise 12 that T is convex. By the completeness axiom, T is an interval. By the boundedness lemma, T cannot extend infmitely far in the positive direction, so it has an upper endpoint, which we call M. Prove that, if fis continuous on [a,b], then the set Tofvalues of [ see Solved Exercise 12 is a closed interval.

Mustf a andf b be the endpoints of this interval? Find a specific number M which works in Solved Exercise Give an example of a continuous function I on [0, 1] such that neither 1 0 nor f l is an endpoint of the set of values of Ion [0,1].

Is the boundedness lemma true if the closed interval [a, b is replaced by an open interval a, b? We can define all the notions of this section, including continuity, differen- tiability, maximum and minimum values, etc. It is possible to prove that I is continuous on [0,2] ; you may assume this now.

Thus, it is really necessary to work with the real numbers. Prove that, if I is a continuous function on an interval I not necessarily closed , then the set of values of I onl is an interval. Could the set of values be a closed interval even if I is not? Prove that the volume of a cube is a continuous function of the length of its edges. Did you ever weigh 15 ktlograms? Write a direct proof of the "minimum" part of the extreme value theorem.

Prove that, if I and g are continuous on [a. Show by example that the sum of two functions with the intermediate value property need not have this property. Prove that the intermediate value theorem implies the completeness axiom. For extensive examples, consult your regular calculus text. We will give an example not usually found in calculus texts: how to graph a general cubic or quartic. If I' is continuous at Xo, then a turning point is also a critical point why? To investigate the behavior of the graph ofI at a turning point, it is useful to treat separately the two possible kinds of sign change.

Suppose first that I' changes sign from negative to positive at xo. Then there is. Applying the remark following the corollary to Theorem 5 of Chapter 5, we find that f is decreasing on a,xol and increasing on [xo,b. For this reason, we call Xo a local mini- mum point for f In case t' changes from positive to negative at xo, a similar argument shows that f behaves as shown in Fig. I x Fig. We can summarize the results we have obtained as follows. Theorem 1 Suppose that f' is continuous in an open interval containing xo 1.

If the sign change of f' is from negative to positive, then Xo is a local minimum point. If the sign change oft' is from positive to negative, then Xo is a local maximum point. If t' is continuous, then every turning point is a critical point, but a criticial point is not necessarily a turning point.

Solved Exercises 1. Are they local maximum or minimum points? For each of the functions in Fig. Can a function be increasing at a turning point? Proof For parts 1 and 2, we use the definition of turning point as a point where the derivative changes sign. By Theorem 2, zero is a local maximum point and 4 is a local minimum point. We may still use the first derivative test to analyze the critical points, however.

Concavity The sign of I" xo has a useful interpretation, even ifI' xo is not zero, in terms of the way in which the graph of f is "bending" at xo. We first make a prelimi- nary defInition. Definition If I and g are functions defmed on'a set D containing xo, we say that the graph lies above the graph olgnear Xo if there is an open interval!

The concepts of continuity, local maximum, and local minimum can be expressed in terms of the notion just dermed, where one of the graphs is a horizontal line, as follows: 1. Definition Let I be defined in an open interval containing Xo. The curve "hold:s water" where it is concave upward and "sheds water" where it is concave downward. It appears from Fig. Home page url. Elementary Calculus by Michael Corral - mecmath. Yet Another Calculus Text by Dan Sloughter Introduction to calculus based on the hyperreal number system for readers who are already familiar with calculus basics.

It covers hyperreals, continuous functions, derivatives, geometric interpretation, optimization, integrals, applications, etc. The Calculus for Beginners by John William Mercer - Cambridge University Press The author has been guided by the conviction that it is much more important for the beginner to understand clearly what the processes of the Calculus mean, and what it can do for him, than to acquire facility in performing its operations.