MAT137 Lecture Notes

1 MAT137 Lecture NotesTyler Holden,c 2014-2015 Contents1 Logic and Sets and notation .. Fundamental Logic .. , Then .. and the Contrapositive .. Definitions and Theorems .. and Proofs .. Induction .. 162 A Quick Primer on Absolute Values .. Definition .. Measure of Distance .. Limited Intuition .. Waving Arguments .. Of epsilons and deltas .. of Limits .. Limits at Infinity .. Continuity .. The Squeeze Theorem .. The Value Theorems .. Value Theorem .. Value Theorem .. 5013 First Principles .. of Differentiability.

2 Some Derivative Formulas .. Product Rule .. Derivatives .. Derivatives .. Rates of Change .. Physics .. The Chain Rule .. The Inverse Trigonometric Functions .. Inverse of What? .. Inverse Trigonometric Functions .. Derivatives .. Exponentials and Logarithms .. Functions .. Derivatives .. Differentiation .. 804 Applications Of Implicit Differentiation .. Idea of Implicit Functions .. Implicit Differentiation Works .. Related Rates .. The Mean Value Theorem .. and Rolle s Theorem .. Theorem Proper .. Maxima and Minima of Functions.

3 L H opital s Rule .. Indeterminate Types .. o -notation .. Curve Sketching .. Curve Sketching .. with Parameters .. 1135 Infima and Suprema .. Upper Bound .. and Results .. Sigma Notation .. The Definite Integral .. Upper and Lower Integrals .. Riemann Sums .. Anti-Derivatives .. The Fundamental Theorem of Calculus .. Computing Areas with Integrals .. Indefinite Integrals .. 1346 Integration Integration by Substitution .. Integration by Parts .. Integrating Trigonometric Functions .. Trigonometric Substitution.

4 Partial Fractions .. 1497 Applications of the Volumes .. 1538 Improper First Principles .. Intervals .. Functions .. Comparison Tests .. Basic Comparison Test .. Limit Comparison Test .. 1669 Sequences and The Basics .. Limits of Sequences .. Functions .. Infinite Series .. Special Series .. Convergence Tests .. Tests .. Tests .. Kinds of Convergence .. 18410 Power Taylor Series .. Taylor Remainder .. Power Series .. Differentiation and Integration of Taylor Series .. Applications of Taylor Series .. 19741 Logic and Proofs1 Logic and ProofsMany of you have already waded through the quagmire that is high-school calculus, endlesslyberated with salvos of mindless computational questions asking you to find numbers with whichyou associate no meaning.

5 This is what I call recipe mathematics, wherein the student is providedwith a recipe and the necessary ingredients, and is told to prepare a mathematical is far removed from our goal in this class: our goal is to show the students how realmathematics is done. We are going to be problem solving and proving theorems, and this is amurky world which is hard to teach, and harder to is often obsessed withrigour: the process of removing ambiguity and attaining thehighest possible level of absolute, infallible deduction. To do this, the student must first understandthe rigid framework in which mathematics is Sets and notationBefore we can begin to speak complicated sentences, we must first learn the words of a any collection of well-defined and distinct objects.

6 By this we mean that you can put asmany things as you like into a set, so long as they are concrete and all different. We often surroundthe elements of a set by curly braces{,}, for example{1,2,3,..},{cat,dog,bird},{ ^, _},{ , , , }.We can put anything we want into a set1so long as the object is a well-defined thing (for example,we cannot consider the set of all objects which I think are interesting. What objects are in thisset? It is ambiguous), and all the elements of the set are distinct (so the object{1,1,2}is not aset, because the element 1 appears multiple times).We use the symbol (read as in ) to talk about when an element is in a set; for example,1 {1,2,3}but _ / {dog,cat}.

7 We can also talk about subsets, which are collections of items ina set and indicated with a sign. For example,{2,4,6} {1,2,3,4,5,6}since every element on the left-hand-side is also present in the sets can have many objects within them, it is often impractical to list them all , we might useset-buildernotation, which allows to say the set of all things which satisfysome property. For example,{x:x >0}is read as the set of allxsuch thatxis greater than 0, while{month : month ends in ber }={September, October, November, December},1 This is actually a complete lie, but the reason why it is a lie is rather subtle. The quintessential example of this issomething known asRussell s paradox.

8 LetSbe the set whose elements are the sets which do not contain themselvesas subsets. IsSan element of itself? This is a self-referential 2015 Tyler Holden51 Logic and Sets and notationis the set of all months for which the end of the name of the month ends in ber . Another way wecould say this, is to letMbe the set of all months, and consider the set{x M:xends in ber }.More interesting than months are sets of numbers, since they tend to be quite big. Some setsthat we will be involved with a lot are as follows: Thenaturals2N={0,1,2,3,..}, TheintegersZ={.., 2, 1,0,1,2,..}, TherationalsQ={p/q:p,q Z,q6= 0}, TherealsR(the set of all infinite decimal expansion).

9 I have been somewhat sloppy in defining the real numbers here since their construction is actuallyrather involved. Nonetheless, I believe the average student is familiar with some of the IntervalsIn addition to the universes listed above, we will often find ourselves concerned with subsets ofR, called intervals. Almost certainly the student has seen these in one form or another, but were-introduce them here to ensure that everyone is on the same page. Effectively, intervals represent connected subsets of real numbers, such as1 x 5, 2< x <4, 10< x 1, x > can be a pain to write down though, so for the sake of compactness we introduce a newnotation: Ifa,bare real numbers witha < b, the following hold4(a,b) ={x R:a < x < b}[a,b) ={x R:a x < b}(a,b] ={x R:a < x b}[a,b] ={x R:a x b}Intervals of the form (a,b) are said to beopen intervals, while [a,b] are said to beclosed other two [a,b) and (a,b] arehalf-open intervals.

10 With a slight abuse of notation, we can alsochoose to let our sets be unbounded by writing infinities:( ,a) ={x R:x < a}(a, ) ={x R:x > a}2 Some mathematicians do not believe that 0 is a natural things can happen if we are not careful with defining the real numbers. For example, consider the number0. 9, where our bar indicates that the number 9 is repeated infinitely often after the decimal place. It turns out that0. 9 = 1, demonstrating that decimal expansions of real numbers are not people prefer to use the notation ]a,b[ for the interval (a,b), but we will avoid this relatively uncommonpractice in this 2015 Tyler Fundamental Logic1 Logic and ProofsWe must always use open brackets when enclosing the infinity, since to use closed brackets [,] wouldimply a point at infinity, and no such point exists in5R.