Most of the questions conform to the examination-style universities of Indian. For some of us, the word conjures up memories of ten-pound textbooks and visions of tedious abstract equations. And yet, in reality, calculus is fun and accessible, and surrounds us everywhere we go.
In Everyday Calculus, Oscar Fernandez demonstrates that calculus can be used to explore practically any aspect of our lives, including the most effective number of hours to sleep and the fastest route to get to work. He also shows that calculus can be both useful—determining which seat at the theater leads to the best viewing experience, for instance—and fascinating—exploring topics such as time travel and the age of the universe.
Throughout, Fernandez presents straightforward concepts, and no prior mathematical knowledge is required. For advanced math fans, the mathematical derivations are included in the appendixes. The book features a new preface that alerts readers to new interactive online content, including demonstrations linked to specific figures in the book as well as an online supplement.
Whether you're new to mathematics or already a curious math enthusiast, Everyday Calculus will convince even die-hard skeptics to view this area of math in a whole new way. This undergraduate text develops two types of quantum calculi, the q-calculus and the h-calculus. As this book develops quantum calculus along the lines of traditional calculus, the reader discovers, with a remarkable inevitability, many important notions and results of classical mathematics.
This book is written at the level of a first course in calculus and linear algebra and is aimed at undergraduate and beginning graduate students in mathematics, computer science, and physics. By breaking down differentiation and integration into digestible concepts, this guide helps you build a stronger foundation with a solid understanding of the big ideas at work.
This user-friendly math book leads you step-by-step through each concept, operation, and solution, explaining the "how" and "why" in plain English instead of math-speak. Through relevant instruction and practical examples, you'll soon learn that real-life calculus isn't nearly the monster it's made out to be. Calculus is a required course for many college majors, and for students without a strong math foundation, it can be a real barrier to graduation.
Breaking that barrier down means recognizing calculus for what it is—simply a tool for studying the ways in which variables interact. It's the logical extension of the algebra, geometry, and trigonometry you've already taken, and Calculus For Dummies, 2nd Edition proves that if you can master those classes, you can tackle calculus and win. Includes foundations in algebra, trigonometry, and pre-calculus concepts Explores sequences, series, and graphing common functions Instructs you how to approximate area with integration Features things to remember, things to forget, and things you can't get away with Stop fearing calculus, and learn to embrace the challenge.
With this comprehensive study guide, you'll gain the skills and confidence that make all the difference. Calculus For Dummies, 2nd Edition provides a roadmap for success, and the backup you need to get there. Now, Gonick, a Harvard-trained mathematician, offers a comprehensive and up-to-date illustrated course in first-year calculus that demystifies the world of functions, limits, derivatives, and integrals.
Using clear and helpful graphics—and delightful humor to lighten what is frequently a tough subject—he teaches all of the essentials, with numerous examples and problem sets. For the curious and confused alike, The Cartoon Guide to Calculus is the perfect combination of entertainment and education—a valuable supplement for any student, teacher, parent, or professional.
Calculus II For Dummies offers expert instruction, advice, and tips to help second semester calculus students get a handle on the subject and ace their exams. It covers intermediate calculus topics in plain English, featuring in-depth coverage of integration, including substitution, integration techniques and when to use them, approximate integration, and improper integrals.
Grove C. Foundations of Hyperbolic Manifolds, John G. Fourier Analysis and Its Applications, S. Axler F. Gehring K.
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Partial Differential Equations, Jurgen Jost. Permutation Groups, John D. Dixon Brian Mortimer. Principles of Random Walk, Frank Spitzer.
Probability Theory I, M. Probability, A. Problems in Analytic Number Theory, M. Ram Murty. Quantum Groups, Christian Kassel. Random Processes, M. Real and Functional Analysis, Serge Lang. Riemann Surfaces, Hershel M. Farkas Irwin Kra. Riemannian Geometry, Peter Petersen. Riemannian Manifolds, John M. Rings and Categories of Modules, Frank W. Anderson Kent R. Several Complex Variables, H. Grauert K. Sheaf Theory, Glen E. Singular Homology Theory, William S. Smooth Manifolds and Observables, Jet Nestruev.
The Symmetric Group, Bruce E. Theory of Complex Functions, Reinhold Remmert. Topological Vector Spaces, H. Schaefer M. Topological Vector Spaces, Helmut H. Topology and Geometry, Glen E. Weakly Differentiable Functions, William P. Protter Charles B. Morrey Jr. Protter C. Algebra, L. Applied Partial Differential Equations, J. David Logan. Aspects of Calculus, Gabriel Klambauer.
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Linear Algebra, Serge Lang. Linear Algebra, Robert J. Linear Algebra, Klaus Janich. Mathematical Analysis, Andrew Browder. Mathematical Logic, H. Ebbinghaus J. Flum W. Mathematics and Its History, John Stillwell.
Methods of Mathematical Economics, Joel Franklin. Naive Set Theory, Paul R. Notes on Set Theory, Yiannis N. Numbers and Geometry, John Stillwell. Optimization Techniques, L. Primer of Modern Analysis, Kennan T. Silverman John Tate. Second Year Calculus, David M. Short Calculus, Serge Lang. Hence 1 holds. Show that E is compact if and only if every sequence in E has a subsequence converging to a point in E. Show sup E and inf E belong to E.
Show that distinct x n cannot belong to the same member of U. We now do so. Absolutely convergent series are convergent, as we shall see in Corollary These are the easiest series to sum. Example 2 Formula 2 of Example 1 and the next result are very important and both should be used whenever possible, even though we will not prove 1 below until the next section. Series 97 number p. Here are some remarkable formulas that can be shown by techniques [Fourier series or complex variables, to name two possibilities] that will not be covered in this text.
It is worth emphasizing that it is often easier to prove limits exist or series converge than to determine their exact values. In particu- lar, 3 in We next give several tests to assist us in determining whether a series converges. Absolutely convergent series are convergent. We next state the Ratio Test which is popular because it is often easy to use.
Moreover, an important result concerning the radius of convergence of a power series uses the Root Test. We give the proof after the proof of the Root Test. Then clearly an converges; see Exercise Inequality 1 in the proof of the Ratio Test shows that the Root Test is superior to the Ratio Test in the following sense: Whenever the Root Test gives no information [i.
Series which converges by the Root Test. Nevertheless, the tests usually fail together as the next remark shows. We have three tests for convergence of a series [Comparison, Ra- tio, Root], and we will obtain two more in the next section. Of course, none of these tests will give us the exact value of the series 1. As noted in Before trying the Comparison Test we need to decide whether we believe the series converges or not.
Since n2 converges, the series 1 converges by the Comparison Test. Hence the series 1 converges by the Ratio Test. It is also possible to show 1 converges by comparing it with a suitable geometric series. Therefore the series 1 diverges by Corollary Since the terms of the series 1 are not all nonnegative, we will not be able to use the Comparison Test It turns out that this series converges by the Alternating Series Test Justify your answers.
Hint : Use Theorem Then the series both converge or else they both diverge. Prove this. Hint : This is almost obvious from Theorem Compare Example 8. Show an is a geometric series. Prove there is Here are some sums that can be handled. We illustrate. The series diverges very slowly.
However, an integral test is useful to establish the next result. Actually, we have already mentioned 2 [without proof! The techniques just illustrated can be used to prove the following theorem. Since n diverges, 1 we see that np diverges by the Comparison Test.
The interested reader may formulate and prove the general result [Exercise Proof We need to show that the sequence sn converges. See Exercise There we considered a decimal expansion K. We will prove the converse after we formalize the process of long division. The development here is based on some suggestions by Karl Stromberg. Fig- ure If we name the digits d1 , d2 , d3 ,. Now Eq. Thus by induction, 5 holds for all n. Every nonnegative real number x has at least one decimal expansion.
The proof will be similar to that for results in As noted in Discussion Similarly, 2. The next theorem shows this is essentially the only way a number can have distinct decimal expansions.
If x has decimal expansion K. The reader can easily check these claims [Exercise Now suppose x has two distinct decimal expansions K. Sequences a contradiction. An expression of the form K. We call such an expansion a repeating decimal. Example 1 Every integer is a repeating decimal. We evaluate this as follows: 3. By the usual long division process in If the details seem too complicated to you, move on to Examples 4—7. A real number x is rational if and only if its decimal expansion is repeating.
As we saw in Since a and b are integers, each rk is an integer. Example 4 An expansion such as. These facts and many others are proved in a fascinating book by Ivan Niven [49]. Sequences must be a positive integer. Claim 1. First, we obtain a recursive relation for In ; see 3. We use integration by parts Theorem Claim 2.
Sequences As noted in Exercise 9. So, for large n, the integer bn In lies in the interval 0, 1 , a contradiction. Zhou and Markov use a similar technique to prove tan r is irrational for nonzero ratio- nal r and cos r is irrational if r 2 is a nonzero rational. Example 7 There is a famous number introduced by Euler over years ago that arises in the study of the gamma function. The remark in Ex- ercise Exercises 11 Note the interesting pattern. Each sn has a decimal n n n n expansion 0.
Remark : This shows the elements of 0, 1 cannot be listed as a sequence. Most of the calculus involves the study of continuous functions. In this chapter we study continuous and uniformly continuous functions. Properly speaking, the symbol f represents the function while f x represents the value of the function at x.
However, a func- tion is often given by specifying its values and without mentioning its domain. Continuity real-valued function. Let f be a real-valued function whose domain is a subset of R. The function f is said to be continuous if it is continuous on dom f.
The next theorem says this in another way. Now assume f is continuous at x0 , but 1 fails. This shows f cannot be continuous at x0 , contrary to our assumption. Hence f is continuous at each x0 in R. The graph of f in Fig. Prove f is continuous at 0. The function f in Example 2 is also continuous at the other points of R; see Example 4.
Show f is discontinuous, i. Let f be a real-valued function. Here is an easy theorem. Continuity Proof Consider a sequence xn in dom f converging to x0.
Theorem 9. This proves kf is continuous at x0. These new functions are continuous if f and g are continuous. Let f and g be real-valued functions that are continuous at x0 in R.
Likewise, Theorem 9. Then Theorem 9. Example 4 For this example, let us accept as known that polynomial functions and the functions sin x, cos x and ex are continuous on R.
Then 4ex and sin x are continuous on R by Theorem The function x4 sin x is continuous on R by ii of Theorem Several applications of Theorems Example 5 Let f and g be continuous at x0 in R. Prove max f, g is continuous at x0. However, we illustrate a useful technique by reducing the problem to results we have already established. By The- orem Be sure to specify their domains. Use these facts and theorems in this section to prove the following functions are also continuous.
Prove every rational function is continuous. Hint : Use part a. Show f is discontinuous at every x in R. Properties of Continuous Functions Show P is discontinuous at every positive integer. Because postage rates tend to increase over time, A and B are actually functions. Let f be a continuous real-valued function on a closed interval [a, b].
Then f is a bounded function. Proof Assume f is not bounded on [a, b]. By the Bolzano- Weierstrass Theorem The number x0 also must be- long to the closed interval [a, b], as noted in Exercise 8. It follows that f is bounded. By the Bolzano-Weierstrass theorem, there is a subsequence ynk of yn converging to a limit y0 in [a, b].
Thus f assumes its maximum at y0. It follows easily that f assumes its minimum at x0 ; see Exercise If the domain is not a closed interval, one needs to be careful; see Exercise Example 1 Let f be a continuous function mapping [0, 1] into [0, 1].
A rigorous proof involves a little trick. Properties of Continuous Functions Therefore f is one-to-one and each nonnegative y has exactly one nonnegative mth root.
Let f be a continuous strictly increasing function on some interval I. Then f I is an interval J by Corollary Let g be a strictly increasing function on an interval J such that g J is an interval I. Then g is continuous on J. Proof Consider x0 in J. We assume x0 is not an endpoint of J ; tiny changes in the proof are needed otherwise.
However, Exercise Let f be a one-to-one continuous function on an interval I. By the Intermediate Value Theorem This contradicts the one-to-one property of f. We will show f is strictly increasing on I. Where does it break down? Uniform Continuity Show there exists an unbounded continuous function on S.
The- orem Such functions are said to be uniformly continuous on S. We will say f is uniformly continuous if f is uniformly continuous on dom f. It makes no sense to speak of a function being uniformly continuous at each point. It is now straightforward to verify 1. The squeamish reader may skip this demonstration and wait for the easy proof in Example 6.
Uniform Continuity We were lucky! How- ever, these results are not accidents as the next important theorem shows. If f is continuous on a closed interval [a, b], then f is uniformly continuous on [a, b]. Proof Assume f is not uniformly continuous on [a, b]. We conclude f is uniformly continuous on [a, b]. Continuity The preceding proof used only two properties of [a, b]: a Boundedness, so the Bolzano-Weierstrass theorem applies; b A convergent sequence in [a, b] converges to an element in [a, b].
As noted prior to Theorem Hence Theorem If f is continuous on a closed and bounded set S, then f is uniformly continuous on S. See also Theorems Example 5 In view of Theorem Example 5 illustrates the power of Theorem One of the important applications of uniform continuity concerns the integrabil- ity of continuous functions on closed intervals.
To see the relevance of uniform continuity, consider a continuous nonnegative real-valued function f on [0, 1]. Then the sum of the areas of the rectangles in Fig.
Relation 1 may appear obvious from Fig. The next two theorems show uniformly continuous functions have nice properties. If f is uniformly continuous on a set S and sn is a Cauchy sequence in S, then f sn is a Cauchy sequence. This proves f sn is also a Cauchy sequence. Then sn is obviously a Cauchy sequence in 0, 1. Therefore f cannot be uniformly continuous on 0, 1 by Theorem The next theorem involves extensions of functions. The function g in Example 8 does not extend to a continuous func- tion on the closed interval, and it turns out that g is not uniformly continuous.
These examples illustrate the next theorem. Suppose now that f is uniformly continuous on a, b. To prove 1 , note that sn is a Cauchy sequence, so f sn is also a Cauchy sequence by Theorem Hence f sn converges by Theorem To prove 2 we create a third sequence un such that sn and tn are both subsequences of un. Here is another useful criterion that implies uniform continuity. Proof For this proof we need the Mean Value theorem, which can be found in most calculus texts or later in this book [Theorem This proves the uniform continuity of f on I.
For a direct proof of this fact, see Example 2. Use any theorems you wish. Hint : Assume not. Use Theorems Justify your answers, using appropriate theorems or Exercise Compare with Theorem Limits of Functions In this section we formalize this notion.
This section is needed for our careful study of derivatives in Chap. Continuity b Observe that limits, when they exist, are unique. Exercise 9. This establishes 1. It follows [Exercise 9. Of course, a direct proof of 2 also can be given. First we prove some limit theorems in considerable generality.
Consider a sequence xn in S with limit a. Theorems 9. Likewise iii follows by an application of Theorem 9. The next theorem is less general than might have been expected; Example 7 shows why. This follows immediately from Theorem Example 7 We give an example to show continuity of g is needed in Theo- rem Limits of Functions As in Theorem First we state and prove a typi- cal result of this sort.
Then, after Corollary Proof We imitate our proof of Theorem Then 1 in Corol- lary Exercises x Also indicate when they do not exist. This is called the squeeze lemma. Warning: This is not immediate from Exercise Justify all claims. Hint : Use Exercise 9. Hint : Use Theorem 9. More thorough treatments appear in [33,53] and [62].
In particular, for this brief introduction we avoid the technical and somewhat confusing matter of relative topolo- gies that is not, and should not be, avoided in the more thorough treatments. We say f is continuous on a subset E of S if f is continuous at each point of E. We will not develop the theory, but gener- ally speaking, functions that look continuous will be. Example 3 Functions with domain R and values in R2 or R3 , or generally Rk , are also studied in several variable calculus.
This func- tion maps R onto the circle in R2 about 0, 0 with radius 1. We did not prove that any of the paths above are continuous, because we can easily prove the following general fact. Proof Both implications follow from formula 1 in the proof of Lemma So fj is continuous on [a, b]. The next theorem shows continuity is a topological property; see Discussion Proof Suppose f is continuous on S.
Thus f is continuous at s0. Continuity at a point is also a topological property; see Exer- cise The next theorem and corollary illustrate the power of compactness. Let E be a compact subset of S. Proof To prove i , let U be an open cover of f E. Hence there exist U1 , U2 ,. This proves i. Assertion ii in Theorem The next corollary should be compared with Theorem Let f be a continuous real-valued function on a metric space S, d.
If E is a compact subset of S, then i f is bounded on E, ii f assumes its maximum and minimum on E. This implies i. Since f E is compact, it contains sup f E by Exercise This tells us f assumes its maximum value on E at the point s0. Similarly, f assumes its minimum on E. Example 4 All the functions f in Example 2 are bounded on any compact subset of R2 , i.
Likewise, all the functions g in Example 2 are bounded on each closed and bounded set in R3. Note Corollary The proof of Proposition Now assume G is compact, but f is not continuous at t0 in E.
The remainder of this section will be devoted to the Baire Cate- gory Theorem and some interesting consequences. A point s is in the closure of a set E if every open set con- taining s also contains an element of E. For example, Q is dense in R, since every nonempty open interval in R contains rationals; see 4. Continuity Each part of the next theorem is a variant of what is called the Baire Category Theorem. Let S, d be a complete metric space. Proof All the hard work will be in proving: Part a.
Part b. Thus its complement Fm contains a nonempty open set. Part c. In other words, we may assume each An is closed and nowhere dense. Part b implies that A contains no nonempty open set. Part d. For this reason, it is common to divide the subsets of S into two categories. Category 1 consists of sets that are unions of sequences of nowhere dense subsets of S, and Category 2 consists of the other subsets of S.
Here is part d of Theorem A complete metric space S, d is of the second category in itself. Each of the spaces Rk is of second category in itself. This and Corollary This proposition also shows closed subsets of complete metric spaces are themselves complete metric spaces. Therefore, nonempty perfect subsets of complete metric spaces are uncountable. The Cantor set in Example 5 on page 89 is an interesting compact perfect set. Recall Exercise Example 8 a Consider any function f on R. No nondegenerate interval I in R can be written as the disjoint union of two or more nondegenerate closed intervals.
Assume I is the disjoint union of two or more nondegenerate closed intervals. It is also clear that the family can be listed as a sequence i. By Discussion Then K is the set of endpoints of the removed intervals, so K is nonempty and countable. We will show K is perfect, and this will be a contradiction by Discussion This shows K is perfect, completing the proof.
A close examination of Example 8 b and Theorem Other ap- plications of this theorem appear in Theorem Hint : It is somewhat easier to use Theorem Hint: Assume a nondegener- ate interval I in R is the union of two or more disjoint nondegenerate closed intervals.
Show [x, y] is also the union of two or more disjoint nondegenerate closed intervals, contradicting the case covered in the proof of Theorem As noted in the proof of Corollary The set E is disconnected if one of the following two equivalent conditions holds.
We show that conditions a and b above are equivalent. To check b , we need only verify 3. Thus 3 holds, and condition b holds. Now suppose that b holds. Conversely, intervals in R are connected. Each possibility leads to a contradiction, so [a, b] and the original interval I are connected. Thus E is not connected, a contradiction. The next corollary generalizes Theorem If E is a connected subset of S, then f E is an interval in R.
In particular, f has the intermediate value property. Example 2 Curves are connected. If E in S, d is path-connected, then E is connected. Then 1 and 2 hold with F in place of E. Thus F is disconnected, but F is connected by Theorem Continuity are convex. Convex sets E in Rk are always path-connected.
For more details, see any book on several variable calculus. Example 4 This example is adapted from [15]. Then f is continuous if and only if G is path-connected. For the converse, assume G is path-connected. By Proposition Now assume f is not continuous on I. The points in these spaces are actually functions themselves. Let S be a subset of R. In this metric space, a sequence fn converges to a point [function! Continuity We will study this important concept in the next chapter, but without using metric space terminology.
In our metric space terminology, Theorems Incidentally, the empty set is connected. Show F is a uniformly continuous real-valued function on the metric space C S.
Show F is a uniformly continuous function from R into C S. The set E is open if every function in E is interior to E. In fact, this topology does not come from any metric at all! Show that the graph of f is connected but not path-connected; compare Exercise This is from [11].
Use Theorem Sequences and Series of
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