MAT137 Lecture Notes - home.tykenho.com

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.

2 The Value Theorems .. Value Theorem .. Value Theorem .. 5013 First Principles .. of Differentiability .. 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.

3 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 .. L H opital s Rule .. Indeterminate Types .. o -notation .. Curve Sketching .. Curve Sketching .. with Parameters .. 1135 Infima and Suprema .. Upper Bound.

4 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 .. Partial Fractions .. 1497 Applications of the Volumes.

5 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.

6 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. 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.

7 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.

8 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).

9 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}. 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.

10 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. 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.