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Calculus with Proofs
University of Toronto St. George
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Browse the full collection of course materials, past exams, study guides and class notes for MAT137Y1 - Calculus with Proofs at University of Toronto St. George verified by our …

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Boris Khesin

Gracia-Saz A.

Thaddeus Janisse

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Class Notes

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MAT137Y1 Lecture Notes - Lecture 3: Distributive Property

The lower limit of summation does not have to be 1; it can be any integer. If ak is a function of k, and m and n are non-negative integers with m n, th

MAT137Y1 Lecture Notes - Lecture 4: Rational Number

Let a = {2,3,4} for all x in a, x > 0: For all x that exists in integer numbers, x > 0. This is false as if we take x = -6, then -6 is a integer,

MAT137Y1 Lecture Notes - Lecture 5: Injective Function

A function f : (a, b) r is injective on (a, b) if x (a, b), Y (a, b), x 6= y f(x) 6= f(y). If c does not= 0, show that the function f(x) = cx is inject

MAT137Y1 Lecture Notes - Lecture 6: Natural Number

A proof used to show that a result holds for every natural number n. If p(1) is true, and p(k) p(k + 1), then p(n) is true for all n n. 4. know that p(

MAT137Y1 Lecture Notes - Lecture 7: Asymptote

MAT137Y1 Lecture 8: Definition of a Limit

Date: january 22, 2020 assume that f(x) is defined on a punctured interval (c , c) (c, c + ), where > 0 represents a small distance to the left and

MAT137Y1 Lecture 9: Proofs with Limits

The limit lim f(x) = l means that for all > 0, there exists > 0 such that if 0 < |x c| < , then. The two definitions of limit in definition

MAT137Y1 Lecture Notes - Lecture 10: Product Rule

MAT137Y1 Lecture Notes - Lecture 11: Classification Of Discontinuities, Inverse Trigonometric Functions, Algebraic Function

A function f is continuous at x = c if lim f(x) = f(c). x c. A function f is left continuous at x = c if lim f (x) = f (c) x c- is right continuous at

MAT137Y1 Lecture Notes - Lecture 12: Intermediate Value Theorem

MAT137Y1 Lecture Notes - Lecture 13: Maxima And Minima

Given a function f and interval [a, \, b][a,b], the local extrema may be points of discontinuity, points of non- differentiability, or points at which

MAT137Y1 Lecture Notes - Lecture 14: Mean Value Theorem

MAT137Y1 Lecture 15: Applications of MVT

MAT137Y1 Lecture Notes - Lecture 16: Infimum And Supremum, Empty Set

Similarly, a is bounded from below if there exists m r, called a lower bound of a, such that x m for every x a. Lower bounds: suppose that a r is a set

MAT137Y1 Lecture Notes - Lecture 17: Monotonic Function, Bounded Function

MAT137Y1 Lecture Notes - Lecture 22: Riemann Sum

Let"s use four rectangles of equal width of 1. 1. This partitions the interval [0,4][0,4] into four subintervals, [0,1],[0,1], [1,2],[1,2], [2,3] On ea

MAT137Y1 Lecture 23: Antiderivatives and Indefinite integrals

MAT137Y1 Lecture Notes - Lecture 24: Antiderivative

MAT137Y1 Lecture Notes - Lecture 25: Antiderivative

MAT137Y1 Lecture Notes - Lecture 26: Arithmetic Progression, Geometric Progression, Royal Institute Of Technology

Sequences converging to zero: we say that the sequence sn converges to 0 whenever the following hold: for all (cid:0) > 0, there exists a real numbe

MAT137Y1 Lecture Notes - Lecture 27: Monotonic Function, Bounded Function, Ratio Test

MAT137Y1 Lecture Notes - Lecture 28: Linear Map, Invertible Matrix

Big theorm: the big theorem only applies when m = n. Theorem 1: suppose t : r m r n a linear transformation and a is an n m matrix with t(x) = ax, and

MAT137Y1 Lecture 29: Series Definition

MAT137Y1 Lecture 30: Properties of Series

MAT137Y1 Lecture 31: Comparison Tests

MAT137Y1 Lecture Notes - Lecture 32: Alternating Series, Absolute Convergence

MAT137Y1 Lecture Notes - Lecture 33: Ratio Test

MAT137Y1 Lecture 34: Power Series

MAT137Y1 Lecture 35: Taylor Polynomials

MAT137Y1 Lecture Notes - Lecture 36: Taylor Series

MAT137Y1 Lecture Notes - Lecture 37: Analytic Function

MAT137Y1 Lecture 38: New Power Series

MAT137Y1 Lecture 39: Applications