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All Educational Materials for MAT136H1 at University of Toronto St. George (UTSG)

UTSGMAT136H1Emile Le BlancWinter

MAT136H1 Study Guide - Comprehensive Final Exam Guide - Joule, Fish Measurement, Scion Tc

OC2241881107 Page
20 Nov 2018
0
5 -i ~e 0. \ t d~ta. nw t x / \j j t t t (tan 0 \ l j to l. 11 "f) at<-,. { un c1lr" k fllidbolgt =-x 1 fn, o -to 1. L " fttut rl(!q,n~lt h~ l. () ctt~
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UTSGMAT136H1Natalia Cristina Garcia FritzWinter

[MAT136H1] - Final Exam Guide - Comprehensive Notes fot the exam (38 pages long!)

OC114616638 Page
27 Mar 2017
0
Mat136h1 s - lecture 1 - calculus 1(b) Email using mail. utoronto. ca email account and include mat136h in subject line. Office: ba6290g; office hours:
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UTSGMAT136H1P.DesrochersFall

[MAT136H1] - Final Exam Guide - Everything you need to know! (77 pages long)

OC115262677 Page
30 Mar 2017
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1- Final Exam Guide - Comprehensive Notes for the exam ( 38 pages long!)

OC53748838 Page
29 Mar 2018
0
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UTSGMAT136H1Emile Le BlancSpring

[MAT136H1] - Final Exam Guide - Everything you need to know! (50 pages long)

OC113064350 Page
30 Mar 2017
0
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UTSGMAT136H1Anthony LamWinter

MAT136H1 Study Guide - Comprehensive Final Exam Guide -

OC113065774 Page
20 Nov 2018
0
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UTSGMAT136H1Asif ZamanSummer

[MAT136H1] - Final Exam Guide - Everything you need to know! (127 pages long)

OC1141343127 Page
28 Nov 2016
33
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UTSGMAT136H1Emile Le BlancWinter

MAT136H1- Final Exam Guide - Comprehensive Notes for the exam ( 107 pages long!)

Yeasmin Sultana Begum107 Page
28 Mar 2018
0
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UTSGMAT136H1WolckuzFall

MAT136H1- Final Exam Guide - Comprehensive Notes for the exam ( 32 pages long!)

OC141235332 Page
29 Mar 2018
0
Bases of subspaces, dot product, cross product, projections. 4. 3, 5. 1, 5. 2 change of coordinates, matrix inverse, elementary matrices. 5. 4, 5. 5, 6
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UTSGMAT136H1Natalia Cristina Garcia FritzWinter

[MAT136H1] - Midterm Exam Guide - Everything you need to know! (32 pages long)

OC114616632 Page
7 Feb 2017
28
Mat136h1 s - lecture 1 - calculus 1(b) Email using mail. utoronto. ca email account and include mat136h in subject line. Office: ba6290g; office hours:
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Frequently-seen exam questions from 2014 - 2018.
UTSGMAT136H1AllFall

MAT136H1 Quiz: MAT136H1 Quiz 3 2015 Fall Solutions

OC25402945 Page
25 Oct 2018
0
We integrate both sides of the equation, and we get: (cid:90) (cid:90) y2dy = (1 + ex)dx y2dy = (1 + ex)dx. 3 y3 = x + ex + c. Solve the initial-value
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UTSGMAT136H1AllFall

MAT136H1 Study Guide - Midterm Guide: Minor Places In Arda, Scilab, Even And Odd Functions

OC25402947 Page
25 Oct 2018
0
Lec 0101 lec 5101 tut 0101 tut 0201 tut 5101 tut 5201. This exam contains 7 pages (including this cover page) and 5 problems. Check to see if any pages
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UTSGMAT136H1AllFall

MAT136H1 Study Guide - Quiz Guide: Noh, Axa

OC25402944 Page
25 Oct 2018
0
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UTSGMAT136H1AllFall

MAT136H1 Quiz: MAT136H1 Long Quiz 1 2015 Fall Solutions

OC254029410 Page
25 Oct 2018
0
1: write the integral cos(x) dx as a limit of riemann sums. Solution: notice that a = 0 and b = 0. Therefore, x = b(cid:0)a n and xi = a + i x = 1: in
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UTSGMAT136H1AllFall

MAT136H1 Study Guide - Quiz Guide: Improper Integral, Asymptote

OC25402944 Page
25 Oct 2018
0
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UTSGMAT136H1AllFall

MAT136H1 Quiz: MAT136H1 Quiz 2 2014 Fall

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25 Oct 2018
0
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UTSGMAT136H1AllFall

MAT136H1 Final: MAT136H1 Final Exam 2013 Summer Solutions

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25 Oct 2018
0
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UTSGMAT136H1AllFall

MAT136H1 Final: MAT136H1 Final Exam 2012 Fall Solutions

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25 Oct 2018
0
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UTSGMAT136H1AllFall

MAT136H1 Study Guide - Quiz Guide: Improper Integral, Alternating Series, Ratio Test

OC254029410 Page
25 Oct 2018
0
2 an improper of type 1 and we have. 1 x 1 is continuous on [2, ). Therefore, this is (let u = x 1) (cid:90) . 1 x 1 dx = lim t . = lim t dx (cid:90) t
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UTSGMAT136H1AllFall

MAT136H1 Study Guide - Quiz Guide: Squeeze Theorem

OC25402945 Page
25 Oct 2018
0
Calculate the sum of the series(cid:80) n=1 an whose partial sums are given by the following expression: n3 1. 2n3 + 3n 1 n3 1. Determine whether the s
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 23: Sequences & Series

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11 Mar 2019
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 22: The Logistic Model

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11 Mar 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 24: Properties & Convergence of Series

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11 Mar 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 19: Separation of Variables

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4 Mar 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 24: Growth and Decay

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4 Mar 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 20: Growth and Decay

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4 Mar 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 18: Euler's Method

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15 Feb 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 17: Differential Equations & Slope Fields

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13 Feb 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture Notes - Lecture 16: Spose

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11 Feb 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 16: Midterm Test Review Part 2

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11 Feb 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 15: Comparison of Improper Integrals

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11 Feb 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 14: Improper Integrals

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8 Feb 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 13: Numerical Methods for Definite Integrals

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5 Feb 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 12: Algebraic Identities and Trigonometric Substitutions

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1 Feb 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 11: In-Class Review (Top Hat)

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30 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 10: Tables of Integrals & CAS

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28 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 3: Area Between Curves

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25 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 5: Differential Equations and Motion

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25 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 8: Integration by Parts

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26 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 2: The Fundamental Theorem and Interpretations

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25 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 4: Constructing Antiderivatives

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25 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 6: Second Fundamental Theorem of Calculus

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26 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 9: Integration Practice Problems

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26 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 7: Integration by Substitution

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26 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 1: Key Concept: The Definite Integral 5.1 and 5.2

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23 Jan 2019
0
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UTSGMAT136H1Park, SFall

MAT136H1 Lecture 17: Integration (VOLUME OF REVOLUTION)

@hanny2 Page
12 Oct 2019
0
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UTSGMAT136H1Park, SFall

MAT136H1 Lecture 15: Integration (AREA & VOLUME)

@hanny2 Page
9 Oct 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 26: Final Review

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5 Apr 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture Notes - Lecture 25: Fax

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3 Apr 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 24: 8.4 Variable Density

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29 Mar 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 23: 8.1 Areas and Volumes & 8.2 Applications to Geometry

OC25564852 Page
27 Mar 2019
0
8iareasfvolg. com ft pute thevolume of a square base pyramid base 756 ft 756ft ht by h. Step l createhorizontal slicesof thickness oh step 2 bottom lay
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 22: Review lecture

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22 Mar 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 21: 10.2 Taylor Series 10.3 Finding and Using Taylor Polynomials

OC25564853 Page
19 Mar 2019
0
10. 2tglorflxzfldtfkacx astfcazq a. l. gwsaiesi. hn n th yo polynomialof fix about thepoint t i. tt n g _go seriesthat you should be familiar with memo
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 20: 10.1 Taylor Polynomials

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15 Mar 2019
0
Approximateby a degree 2 polynomial fix 12s c gx c i e a 0 approximate about thepoint 0. Pdx g t c x g 2 fix. R g 12 2 x f pill f oc. iof"co g p o. Pic
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 19: 9.9 Test for convergence and divergence 9.5 Power Series differential Intervel of convergence

OC25564854 Page
13 Mar 2019
0
Should use it whenyouknow howto evaluatethe indetlerent. nl integral qq. I comparisontest i o e anebn n n. Converge converge diverge diverge no. It tt
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 18: 9.3 Comparing Series Via Integrals

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9 Mar 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 17: 9.1 Sequence and Series 9.2 Geometric Series

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6 Mar 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 16: 11.7 Logistic Modelling

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28 Feb 2019
0
Ph birthrate gu deathrate b f k constant. Logistic model population growing butalternating a limit p ipo negatiegnwthnh_e f. cn logistic model k l nega
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture Notes - Lecture 15: Thx

OC25564852 Page
27 Feb 2019
0
Jim are having a cupofcoffee j cools hiscoffeewith 3 tsb of cream theyboth wait tominutes m thencools her otkewith 3 tbsptcreamp. inno drinks the. Is m
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 12: 11.4 separable equations

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15 Feb 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture Notes - Lecture 11: Encyclopedia Of Indo-European Culture

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12 Feb 2019
0
1 y 1 theslope i y canbeany number y dy dx y . 2xy gcxj. tt g fix g 9 f x zx. fi i. 2x ig f g that contradicts our assumptions that g x fix are differe
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 10: Review For the Midterm

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8 Feb 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 9: 7.7 Comparing Improper Integrals

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6 Feb 2019
0
77comparingimproperintldf. in ding an exact value of an improper integral is often hard so we can often apure it with aknownintegral tosay ifgiven inte
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 8: 7.6 improper integral

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1 Feb 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 7: 7.4&7.5 algebraic identities and trigonometric substitutions & numerical methods for definite integral

OC25564853 Page
29 Jan 2019
0
Hx2eif. dxhl ex hxe f tuixexijfuz. du parametriceacrde parametric partof a hyperbole s0 sno lcoshx. snhx. I d. tt dx 1 dtil. it lpnhessxj l. is. 1snow
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 6: 7.2 Integration by Parts

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25 Jan 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 5: 6.4 Fundamental Theorem of Calculus & 7.1 Ingtegration by Substitution

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23 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 7: Graph of the Integral Function

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22 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesSpring

MAT136H1 Lecture 6: Antiderivatives

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19 Jan 2019
0
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UTSGMAT136H1Sarah Mayes-TangWinter

MAT136H1 Lecture 6: Jan 18 Lec06

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18 Jan 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 4: Differential Equation and Motions and Differentrial Equations

OC25564853 Page
18 Jan 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 3: 6.1& 6.2 Antiderivatives Graphically & Numerically& Constructing Antiderivatives Analytically

OC25564853 Page
17 Jan 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 2: 5.3 The Fundamental Therom+Interpretations

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17 Jan 2019
0
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UTSGMAT136H1Debanjana KunduWinter

MAT136H1 Lecture 1: 5.1&5.2 Left/Right Hand Sums & Definite Intergrals

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17 Jan 2019
0
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UTSGMAT136H1Sarah Mayes-TangWinter

MAT136H1 Lecture 5: Jan 16 Lec05

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17 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesWinter

MAT136H1 Lecture 5: Families of antiderivatives

OC26419253 Page
17 Jan 2019
0
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UTSGMAT136H1Sarah Mayes-TangWinter

MAT136H1 Lecture Notes - Lecture 4: Antiderivative

OC25087572 Page
16 Jan 2019
0
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UTSGMAT136H1Kathlyn DykesWinter

MAT136H1 Lecture 4: Using area to compute antiderivatives

OC26419253 Page
14 Jan 2019
0
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UTSGMAT136H1Sarah Mayes-TangWinter

MAT136H1 Lecture 1: MAT136 Lec1

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12 Jan 2019
0
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UTSGMAT136H1Sarah Mayes-TangWinter

MAT136H1 Lecture 2: Jan 09 Lec02

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12 Jan 2019
0
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UTSGMAT136H1Sarah Mayes-TangWinter

MAT136H1 Lecture 3: Jan 11 Lec03

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12 Jan 2019
0
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UTSGMAT136H1AllWinter

MAT136H1 Midterm: MAT136H1 - Term Test 1

OC53748854 Page
15 Sep 2018
0
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UTSGMAT136H1AllWinter

MAT136H1 Midterm: MAT136H1 - Term Test 1

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15 Sep 2018
0
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UTSGMAT136H1AllWinter

MAT136H1 Final: MAT136H1 - Final Exam - Mayes-Tang, S

OC53748868 Page
15 Sep 2018
0
= 1 2 2 + u 4 u du u= ( 3. = 5 u 5 7 u 7 + c. = 5 sin x5 7 sin x7. Use double angle formula: cos2x=1-2sin 2 x sin 2 x= (1-cos2x)/2 cos2x=2cos 2 x-1 cos
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UTSGMAT136H1AllWinter

MAT136H1 Final: MAT136H1 - Final Exam - Kapovitch, V

OC53748868 Page
15 Sep 2018
0
= 1 2 2 + u 4 u du u= ( 3. = 5 u 5 7 u 7 + c. = 5 sin x5 7 sin x7. Use double angle formula: cos2x=1-2sin 2 x sin 2 x= (1-cos2x)/2 cos2x=2cos 2 x-1 cos
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 32: Taylor polynomials

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11 Apr 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 33: Taylor series convergence

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11 Apr 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 27: Sequence and series

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11 Apr 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 30: Convergence Test

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11 Apr 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 29: Series

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11 Apr 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 23: Review

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11 Apr 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1- Final Exam Guide - Comprehensive Notes for the exam ( 38 pages long!)

OC53748838 Page
29 Mar 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 7: Applications of Integrals

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12 Feb 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 14: Integration by Parts

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12 Feb 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 16: Partial Fractions and Integration

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12 Feb 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 12: Work, Hooke_s Law, Variable Density

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12 Feb 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 8: How to Find the Area & Volume of a Given Function

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12 Feb 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 15: Integration of Rational Functions

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12 Feb 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 11: Work

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12 Feb 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 10: Volume and Rotation

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12 Feb 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 13: Average Values

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12 Feb 2018
0
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UTSGMAT136H1Mayes-Tang, SWinter

MAT136H1 Lecture 9: Volume of domain in 3-space

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12 Feb 2018
0
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Lecture Notes - Lecture 35: Arc Length

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13 Apr 2016
35
Arc length parameterization (t)r (t ) r 0 to (t)r du (t) || d u (t ) r 0 (t)dt v speed. Arc length, s, from reference point, from t s = t0 s = . 2 + 1
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Study Guide - Final Guide: Telephone Numbers In The United Kingdom, If And Only If, Hit106.9 Newcastle

OC537488104 Page
13 Apr 2016
85
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Lecture 35: Review of Integrals

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11 Apr 2016
32
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Lecture Notes - Lecture 38: Hit106.9 Newcastle

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11 Apr 2016
19
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Study Guide - Final Guide: Scilab, If And Only If, Selenium

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8 Apr 2016
87
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Lecture Notes - Lecture 34: Ibm System P

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7 Apr 2016
20
= l lim n f(x) such that f lim x : ex. (x) = a n x 1/x = 1: monotonicity and boundedness, must be monotonically increasing or decreasing, must be bound
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UTSGMAT136H1Anthony LamSpring

MAT187H1 Lecture 37: 12.9 continued and Ch 16 Second Order Differential Equations

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7 Apr 2016
15
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Lecture 33: Review

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5 Apr 2016
23
C n a n = c 0 + c 1 a + c 2 a 2 + . (x. 1 ( (x+1)) = 1 ( + 1 + ( + 1 2 ( + 1 3. Suppose f(x) has a series expansion on (a r,a+r) f f (x) f f (a) (x) f
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UTSGMAT136H1Anthony LamSpring

MAT187H1 Lecture 36: 12.9 continued

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5 Apr 2016
13
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Lecture 32: Revision + Taylor Series

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4 Apr 2016
17
C n a n = c 0 + c 1 a + c 2 a 2 + . (x (x (x n=1. 1 x = 1 + x + x 2 + x 3 + . 1 ( (x+4) = 1 ( + 4 + ( + 4 2 ( + 4 3 x x x ( + 4 | < 1. 1 < x| + 4| < 1.
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Lecture 31: Infinite Geometric Series

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1 Apr 2016
19
1 + x + x 2 + x 3 + x 4 n=0 x n = 1. Let y = 1 + x + x 2 + x 3 x. 1 (1 x) 2 = dx when 1<x<1 x dy = 1 + 2 + 3 2 + 4 3 x. 1 x = 1 (1 x) 2 y = 1. C n a n
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UTSGMAT136H1Anthony LamSpring

MAT187H1 Lecture 34: 12.7 continued & 12.8 Length of Curves

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1 Apr 2016
30
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UTSGMAT136H1Anthony LamSpring

MAT187H1 Lecture 33: 12.6 continued & 12.7 Motion In Space

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30 Mar 2016
26
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Lecture 30: Power Series

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30 Mar 2016
14
C n n = c 0 + c 1 + c 2 x n=1. 0 + c 1 a + c 2 a 2 (x (x n=1. It converges on a ( r a + r. R = radius of convergence interval diverges on ( a r ( + r a
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UTSGMAT136H1Anthony LamSpring

MAT187H1 Lecture 32: 11.3 Calculus in Polar Coordinates & 12.6 Calculus of Vector Valued Functions

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24 Mar 2016
15
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Lecture 29: Alternating Series Cont.

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24 Mar 2016
12
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Lecture 28: Alternating Series

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22 Mar 2016
17
S = 1 2 n = n. 1 n 1 n n=1 n 1 b n n = 0 lim n . + ( 1 n 1 n + ( 1 n b. S = b1 b 2 + b 3 b 4 l sn l l rn l. S = n=1 n| b n 1 ( 1) n 1 n 4. 0 n| < b n+1
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UTSGMAT136H1Anthony LamSpring

MAT187H1 Lecture 31: 11.2 Polar Coordinates: Plotting & 11.3 Calculus in Polar Coordinates

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22 Mar 2016
30
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UTSGMAT136H1Anthony LamSpring

MAT187H1 Lecture 30: 11.2 Polar Coordinates

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21 Mar 2016
39
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Lecture 27: Comparison Test/Limit Comparison Test

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21 Mar 2016
82
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UTSGMAT136H1Anthony LamSpring

MAT187H1 Lecture 29: 11.1 Parametric Equations: Tangents & 11.2 Polar Coordinates

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17 Mar 2016
19
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Lecture 26: Integral Test

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17 Mar 2016
15
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UTSGMAT136H1Anthony LamSpring

MAT136H1 Lecture 25: Series

OC5374883 Page
15 Mar 2016
9
A n = a 1 + a 2 + a 3 + . 1 + 2 + 3 + 4 + 5 n=1 a i = a 1 + a 2 + a 3 + . + a n a n n=1 is convergent a n n=1 is divergent r. R = a r 2 n(1. S (1 a(1 r
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UTSGMAT136H1Anthony LamSpring

MAT187H1 Lecture 28: 11.1 Parametric Curves & Tangent Lines

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15 Mar 2016
12
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UTSGMAT136H1Anthony LamSpring

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= { 1 n+2: fibonacci sequences: a 1 = 1. 1 n 1 n=1 n = a n 1 + a n 2 for all n>2. { n n=1 well defined/exists is said to converge if the limit, lim n a
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=1 y+y 2 y 3 +y 4 = ( )1 k 2k x n k=0. Set x=y 2 : ey2 = 1 + y2 + 2! y4 3! Set x= y 2 : ey2 = 1 y2 + 2! x6 3! x2 x2 = x2 x4 + 2! x8 + . e x5 + . inx s
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+ c dx c x ex dx dx ln sinx anxdx t. | + c x| osxdx c x n+1 + c. = x + c inxdx osx s. X + c ecxdx s otxdx c sinx. X 2 dx u = 9 x2 u xdx, d = 2 du = x.
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Examine behaviour y(t) = ce kt b/k dy = k + b > 0. Y dy = 0 b y = k dt dy = k + b = kk. , ky y < k dy = b k > 0 dt. K > b b > 0 k + b < 0 y > k. , ky d
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4 s in x s (sin x) (1. 1 2 2 + u 4 du u 5. )du u 3 + c sin x3 + c. 2 + 2 sinx + c in xcos xdx s. 1 cos2x 2 in x(1 s os xdx c u 4 u 2 (1 u 7 + c u 5 7.
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Review of exponential models y(x) = ce kx ce (ce ) kx = k kx = k y. Then n(t) = n(0)e kx (from y = ce kx ) dy = k y dx dy = k y dx dn = k dt. 2 k = k =
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> w q (x)dx inconclusive g q (x) g a) b) c) d) > f (x), x (x)dx inconclusive g (x) g. = 2 x 12 x 12 x. Is wrong! b a f (x)dx is defined as q q b lim a
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/n) b c n i=1 c a x = 1/n f (x)dx f (x)dx f (x)dx f (x)dx f (x)dx f (x) x f (yi) y) If a<b then (x)dx (x)dx b a b a. Rules: b a b (x)dx a b for an inpu
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/n b i 1 x i f (x ) x i provided that this limit exists and gives x i = + i a. )/2 b definite integral of f on [a,b] is a (b a f (x)dx. If f has only a
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