# Main topics for the First Midterm

Size: px
Start display at page:

Transcription

1 Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections , nd Sections (essentilly ll of the mteril covered in clss). Be sure to know the results of the problems on HWs 1-6 nd understnd their proofs so tht you cn reproduce them. Mke sure to void these common mistkes: Remember tht cx = c x only works if c 0. When writing n ε, δ-proof, remember tht you need to consider every possible vlue of ε > 0, not just specific vlue. Remember tht δ cn NEVER be negtive. f(x) If lim f(x) = lim g(x) = 0, it does NOT men tht lim x c x c x c g(x) is 0 or undefined, it simply 0 mens you need to find some other wy to compute the limit. Remember tht criticl point is vlue for x, while n extrem is vlue for y = f(x). Remember tht criticl point might not be either locl mximum or minimum. If c is criticl point with f (c) = 0, then the second derivtive test tells you nothing. c might be locl mximum, locl minimum or neither. The first derivtive test my be useful in situtions like this. 1

2 Mke sure you re comfortble with the following: Proofs: Understnd wht it mens to write mthemticl proof. Understnd proof by contrdiction nd induction. In prticulr, understnd wht proof by induction consists of. Remember to include ll prts of proof by induction (bsic cse, inductive hypothesis, inductive step, conclusion) Specify which cse you re considering (e.g. n = 1, n = k, n = k + 1) Specify wht you re ssuming nd wht you wnt to show Know wht is needed to prove sttements like If A, then B or A iff B. Inequlities: Know how to solve liner nd polynomil inequlities (mke sure to check the boundry vlues!). Know how to solve inequlities involving more complicted functions (bsolute vlues, squre roots, trig functions, etc.). Understnd how to express solutions in intervl nottion, or s sets on number line. Understnd wht the quntity b mens geometriclly. Trigonometry: Be fmilir with the functions sin x, cos x nd tn x. Know how to drw their grphs nd compute their vlues t certin points. Know the bsic trig identities (e.g. sin x = sin x, sin 2 x + cos 2 x = 1, etc.). Understnd their reltion to the unit circle, nd know how to use the unit circle to solve problems involving trig functions. Definition of limit: Understnd the intuitive mening of limit, nd the ε, δ definition (nd understnd why they re relted!). Mke sure tht you cn remember the definition of limit without help, nd cn explin wht it mens (i.e. don t just memorize for every ε > 0 there is δ > 0... understnd wht this mens, nd why we defined it this wy). Also mke sure you understnd the definition of one-sided limits. Know how to modify the definition of lim x c f(x) = L in the cse where one or both of c nd L re ±. ε, δ-proofs: Know wht is needed to prove sttement like lim x c f(x) = L, both when f is specific function (like f(x) = x 2 + 1) nd when f is generl function. 2

3 Know how to give n ε, δ-proof when f is liner function, or when f is (simple) non-liner function (such s x 2 +, 1 x, x, x + b where, b re constnts. ) Remember tht you must prove tht your given vlue of δ works, its not enough to simply give the vlue. Remember tht your vlue for δ should NEVER be negtive! Computing limits: Know how to compute limits of most simple functions. In prticulr: polynomils, rtionl functions, nd functions involving bsolute vlues, squre roots nd trig functions. Be fmilir with the theorems bout limits which we proved in clss (or you proved on your homework), nd know how to use them. In prticulr, know how to use the definition of limit to prove sttements for the limits of f + g nd αg. Understnd how to compute the limits of functions like f + g, f g, αf, f g f(x) Understnd why lim x c g(x) might still exist if lim f(x) = lim g(x) = 0. x c x c nd f g. Know how to use the Pinching Theorem to compute limits of more complicted functions. ( ) ( ) 1 1 Remember the formuls lim f(x) = lim f nd lim f(x) = lim x h 0 + h f x h 0 h Know how to compute lim x P (x) nd lim x P (x) for polynomil, P, by looking t the first term. Continuity: Know wht it mens for function to be continuous (both the intuitive explntion, nd the rigorous definition). Understnd the difference between being continuous t c; continuous on (, b); nd continuous on [, b]. Know how you need to redefine the function to mke it continuous in the cse of removble discontinuity. Know how to show using the definition of continuity tht functions like f + g nd αg re continuous. Know how { to check if given function is continuous. In prticulr, when is the function f(x) x < h(x) = g(x) x continuous? Intermedite Vlue Theorem: Know the sttement of the theorem (in prticulr, under wht conditions is it true?). Know how to use it to estimte solutions to equtions. Understnd how it cn be used in proofs. Extereme Vlue Theorem: 3

4 Know the sttement of the theorem (in prticulr, under wht conditions is it true?). Understnd how it cn be used in proofs. Derivtives: Understnd the intuitive mening of derivtive (both s the slope of the tngent line to grph, nd s the liner pproximtion to function). Know how to find the eqution for the tngent to the grph of function f(x) t x = Know the f (x) nottion nd the dy dx nottion for derivtive. In prticulr, understnd why f(c + h) f(c) + hf (c). Know how to tell if function is differentible (nd when it isn t). In prticulr, when { f(x) x < is the function h(x) = g(x) x differentible? Remember tht differentible function is lwys continuous (but not the other wy round). In prticulr, remember tht f(x) = x is n exmple of everywhere continuous function f(x) is not differentible t x = 0 nd know how to show it. Know the rigorous definition of derivtive, i.e. s the limit of f(c+h) f(c) h. Know the lterntive expression for derivtive s limit, i.e. the limit of f(x) f(c) x c Know the difference between being differentible t point, nd being differentible in n intervl. Know how to directly use the definition to compute derivtive (by just tking limit). Know how to use the definition of derivtive to show functions like f + g nd αg re differentible Computing derivtives: In generl, you should mke sure you know how to compute the derivtive of ny function you know how to write down. Know how to compute the derivtives of simple functions (like polynomils). Know how to compute the derivtives of sums nd products of function. ( ) Know how to compute f. g (If you hve trouble remembering the quotient rule, remember tht you cn compute this just by knowing nd using the product ) rule.) Know how to use the chin rule to compute (f g). When using the chin rule, mke sure to keep trck of which vrible you re differentiting with respect to. Remember tht d dx f(y) f (y). Understnd how you cn use the chin rule multiple times to compute the derivtive of something like f(g(h(x))). Remember tht d dx xr = rx r 1 for ny rtionl r. Know the derivtives of sin x nd cos x, nd know how to use these to compute the derivtives of more complicted trig functions. Know how to use implicit differentition to compute dy dx when you do not necessrily hve formul for y in terms of x. ( 1 g 4

5 Word Problems Know how to trnslte rel world problem into clculus problem. Know how to find the mximum or minimum vlue of some quntity. Do not forget to consider the closed intervl of possible vlues for the rgument when mximzing or minimizing some quntity.in prticulr, be sure to check wht hppens t the endpoints. Higher order derivties: Know wht it mens to tke higher order derivtive of function. Understnd the nottions f (x), f (n) (x) nd dn y dx n. Know how to compute the second derivtive of function (i.e. by tking the derivtive twice). Understnd how you cn use induction to prove formul for dn y dx n. Minim nd mxim of functions: Know (nd understnd) the definitions of locl nd bsolute mxim/minim. Know wht criticl point of function is, nd know tht ny locl extrem re criticl points. Remember tht NOT ll criticl points need to be locl extrem. Remember tht criticl points need to be inner points for the domin of the functions. I.e. in order for x = c to be criticl point you need c (, b) such tht f is defined on ll of the points of (, b) including c. Know wht it mens for function to be incresing or decresing, nd know how to determine this by looking t the sign of the derivtive. Distinguish between the definition of n inresing function nd theorems helping you determine when function is incresing. Know how to use the first nd second derivtive tests to determine whether criticl point is mximum, minimum or neither, nd know why these tests work. Understnd why the first derivtive test cn be used in more situtions thn the second. Understnd wht endpoint extrem re, nd how to find them. Understnd how to find the bsolute extrem of function (both function defined on [, b] or n open intervl (, b) or n infinite intervl). Do not forger to consider the endpoints (if they exist) when finding the bsolute extrem of function. Know wht to do if f is not continuous t criticl point c. Men Vlue Theorem: Know the sttement (but not the proof). In prticulr, know the necessry conditions for the function. 5

6 Know how to use this theorem to get informtion bout function from its derivtive. Know how to find the point x = c stisfying the conditions of the theorem. Or if there is no such point, to show tht there is no such number x = c nd explin why. Properties of functions relted to grphing: Mke sure tht you know the following properties of functions nd their grphics. Tht is, you should know the definition, how to chech if given function hs these properties using the definition or n uxiliry theorem, nd wht the grph of function with these properties look like. Concve up/concve down functions. Inflection points. (Remember tht point where f (c) = 0 or f (c) DNE does NOT need to be n inflection point.) Verticl nd horizontl symptotes. (Remember which nd how mny limits you need to check in ech cse.) Verticl tngents nd cusps. (Remember these properties re bout the derivtive nd know how to distinguish between them.) Know how properties of f (x) relte to properties of f(x). Tht is, if you know tht f (x) hs some property (e.g. positive, incresing, hs verticl symptote) wht does this tell you bout the function f(x) nd its grph. Mke sure to distinguish between the definition of concve up function nd theorems helping you determine when function is concve up. Know how to drw the grph of function, either if you re given n eqution for the function, or simply told properties tht the function must stisfy. Integrtion: Understnd intuitively wht n integrl is. Tht is, tht of infinitely smll quntities f(t)dt. f(t)dt is n infinite sum Understnd the reltion between integrls nd re. In prticulr, understnd wht hppens when f(x) is not positive everwhere. Also, how to clculte the integrl vi the res of geometric figures. Mke sure you understnd the nottion f(t)dt. In prticulr understnd tht this is number, which depends only on nd b, nd NOT on t. The vrible t only dummy vrible nd the letter t cn be substituted with ny other unused letter. If the dummy vrible t ever ppers outside of the integrl sign, then you hve done something wrong. Estimting Integrls: Know wht prition of n intervl [, b] is. Know wht the norm, P of prtition P is. Understnd why prtitions with smller vlues of P re considered better. 6

7 Know wht the lower nd upper sums, L f (P ) nd U f (P ) re. Tht is, know the definitions, nd how to compute them for given or n rbitrry prtition. Note tht they involve finding miniml nd mximl vlues of function, so you my need to find the criticl points. Know how to define f(t)dt in terms of L f (P ) nd U f (P ). Know how we cn use upper nd lower sums to compute exct vlues of some bsic integrls (e.g. cdt, tdt) Know wht the Riemnn sum Sf (P ) is, nd know how to compute it. b Understnd the definition lim P 0 S f (P ) = Properties of Integrls: Know bsic properties of integrls: b f(t) + g(t)dt = b kf(t)dt = k Understnd how holds for ll, b, c f(t)dt f(t)dt + g(t)dt f(t)dt f(t)dt is defined for = b nd > b so tht f(t)dt = Fundmentl Theorem of Clculus: Understnd the function F (x) = x c c f(t)dt. f(t)dt + c f(t)dt Tht is, understnd wht this function reprsents (i.e. it is n re under the curve.) Know on which intervl F (x) is continous nd on which differentible (i.e.intervls of the type [, b] nd (, b) respectively) nd the vlues of F (x) t the endpoints of the intervl. Understnd tht the function F (x) is function of x, NOT function of t. (It would be useful to keep trck of which vribles re dummy nd which rel vribles. ) Know tht F (x) = f(x) nd understnd the intuition for it. (This reltionship is relly importnt throughout the rest of the qurter.) 7

8 In prticulr, remember tht F (x) should not depend in ny wy on the choice of c. If c ppers in your fomul for F (x), you hve done something wrong. Know how to compute the derivtive F (x) when composed with nother function g(x) i.e. know how to use the Chin rule to find the derivtive of F (x) = f(t)dt. Know how to use tht F (x) = f(x) in order to find the criticl points of F (x), locl extrem vlues of F (x), inflection points, nd etc. Miscellneous Problems from Homework Know wht is the difference between functions f(x) nd g(x) continuous on n intervl I when f = g. Know wht is the reltionship between the limits of f nd g if f(x) g(x) on some intervl I. Know how to prove tht the limit of the bsolute vlue of function is the bsolute vlue of the limit i.e lim f = L if lim f = L. Know how to prove tht if lim x c f(x) = L, then f(x) is bounded ner but different from c i.e. there re constnts m nd M such tht m < f(x) < M. Know how to prove by definition tht if lim x c f(x) = L > 0, then f(x) > 0 for x ner but different from c. Know how to show tht f(x) is liner function if f = 0 for ll x in some intervl I. Know how to prove by induction problems similr to the ones on Problem set 5. Know the simplifying expressions for the sums n nd n 2. h(x) 8

### Main topics for the Second Midterm

Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the

### Overview of Calculus I

Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,

### Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

### Review of Calculus, cont d

Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

### MATH 144: Business Calculus Final Review

MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives

### MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

### The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

### Riemann Sums and Riemann Integrals

Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties

### 7.2 Riemann Integrable Functions

7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous

### Calculus I-II Review Sheet

Clculus I-II Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing

### Stuff You Need to Know From Calculus

Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you

### Riemann Sums and Riemann Integrals

Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct

### The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

### The Regulated and Riemann Integrals

Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

### A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

### Review of basic calculus

Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

### 1 The fundamental theorems of calculus.

The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection

### AP Calculus Multiple Choice: BC Edition Solutions

AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this

### MATH , Calculus 2, Fall 2018

MATH 36-2, 36-3 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly

### Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl

### 1 The fundamental theorems of calculus.

The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous

### Topics Covered AP Calculus AB

Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.

### Riemann Integrals and the Fundamental Theorem of Calculus

Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums

### MA 124 January 18, Derivatives are. Integrals are.

MA 124 Jnury 18, 2018 Prof PB s one-minute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,

### f(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all

3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the

### 7.2 The Definite Integral

7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

### Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

### MAT137 Calculus! Lecture 20

officil website http://uoft.me/mat137 MAT137 Clculus! Lecture 20 Tody: 4.6 Concvity 4.7 Asypmtotes Net: 4.8 Curve Sketching 4.5 More Optimiztion Problems MVT Applictions Emple 1 Let f () = 3 27 20. 1 Find

### Lecture 1: Introduction to integration theory and bounded variation

Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. You

### Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

### DERIVATIVES NOTES HARRIS MATH CAMP Introduction

f DERIVATIVES NOTES HARRIS MATH CAMP 208. Introduction Reding: Section 2. The derivtive of function t point is the slope of the tngent line to the function t tht point. Wht does this men, nd how do we

### Math 360: A primitive integral and elementary functions

Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

### Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED

Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type

### Final Exam - Review MATH Spring 2017

Finl Exm - Review MATH 5 - Spring 7 Chpter, 3, nd Sections 5.-5.5, 5.7 Finl Exm: Tuesdy 5/9, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.

### Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004

Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when

### Calculus II: Integrations and Series

Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]

### Recitation 3: More Applications of the Derivative

Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech

### How can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?

Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those

### IMPORTANT THEOREMS CHEAT SHEET

IMPORTANT THEOREMS CHEAT SHEET BY DOUGLAS DANE Howdy, I m Bronson s dog Dougls. Bronson is still complining bout the textbook so I thought if I kept list of the importnt results for you, he might stop.

### 38 Riemann sums and existence of the definite integral.

38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

### 0.1 Chapters 1: Limits and continuity

1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97

### f a L Most reasonable functions are continuous, as seen in the following theorem:

Limits Suppose f : R R. To sy lim f(x) = L x mens tht s x gets closer n closer to, then f(x) gets closer n closer to L. This suggests tht the grph of f looks like one of the following three pictures: f

### . Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.

Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos( - 1 2 ) = rcsin( 1 2 ) = rcsin( - 1 2 ) = Cn you do similr problems? Review of Bsic Concepts

### MATH SS124 Sec 39 Concepts summary with examples

This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples

### Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

### Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

### f(a+h) f(a) x a h 0. This is the rate at which

M408S Concept Inventory smple nswers These questions re open-ended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnk-out-n-nswer problems! (There re plenty of those in the

### Integrals - Motivation

Integrls - Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is non-liner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but

### First midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009

Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm-6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No

### Chapter One: Calculus Revisited

Chpter One: Clculus Revisited 1 Clculus of Single Vrible Question in your mind: How do you understnd the essentil concepts nd theorems in Clculus? Two bsic concepts in Clculus re differentition nd integrtion

### W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying

Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)

### . Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =

Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos( - 1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin( - 1 ) = -π 2 6 2 6 Cn you do similr problems?

### B Veitch. Calculus I Study Guide

Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some

### Taylor Polynomial Inequalities

Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil

### n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1

The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the

### The Riemann Integral

Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function

### A sequence is a list of numbers in a specific order. A series is a sum of the terms of a sequence.

Core Module Revision Sheet The C exm is hour 30 minutes long nd is in two sections. Section A (36 mrks) 8 0 short questions worth no more thn 5 mrks ech. Section B (36 mrks) 3 questions worth mrks ech.

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### INTRODUCTION TO INTEGRATION

INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide

### Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

### Review of Riemann Integral

1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.

### Chapters 4 & 5 Integrals & Applications

Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions

### Math 1B, lecture 4: Error bounds for numerical methods

Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

### Chapter 0. What is the Lebesgue integral about?

Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous

### Big idea in Calculus: approximation

Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:

### Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1

Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution

### UNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3

UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,

### Lecture 3. Limits of Functions and Continuity

Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live

### Objectives. Materials

Techer Notes Activity 17 Fundmentl Theorem of Clculus Objectives Explore the connections between n ccumultion function, one defined by definite integrl, nd the integrnd Discover tht the derivtive of the

### FINALTERM EXAMINATION 2011 Calculus &. Analytical Geometry-I

FINALTERM EXAMINATION 011 Clculus &. Anlyticl Geometry-I Question No: 1 { Mrks: 1 ) - Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...

### Math 113 Exam 1-Review

Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between

### 63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

### Math 1431 Section 6.1. f x dx, find f. Question 22: If. a. 5 b. π c. π-5 d. 0 e. -5. Question 33: Choose the correct statement given that

Mth 43 Section 6 Question : If f d nd f d, find f 4 d π c π- d e - Question 33: Choose the correct sttement given tht 7 f d 8 nd 7 f d3 7 c d f d3 f d f d f d e None of these Mth 43 Section 6 Are Under

### P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)

1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

### Math 8 Winter 2015 Applications of Integration

Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

### MAA 4212 Improper Integrals

Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which

### LECTURE. INTEGRATION AND ANTIDERIVATIVE.

ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE. INTEGRATION AND ANTIDERIVATIVE. ROTHSCHILD CAESARIA COURSE, 2015/6 1. Integrtion Historiclly, it ws the problem of computing res nd volumes, tht triggered development

### Numerical Analysis: Trapezoidal and Simpson s Rule

nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =

### The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

### MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further

### SYDE 112, LECTURES 3 & 4: The Fundamental Theorem of Calculus

SYDE 112, LECTURES & 4: The Fundmentl Theorem of Clculus So fr we hve introduced two new concepts in this course: ntidifferentition nd Riemnn sums. It turns out tht these quntities re relted, but it is

### Fundamental Theorem of Calculus

Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under

### THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem

### f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral

Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one

### Chapter 8.2: The Integral

Chpter 8.: The Integrl You cn think of Clculus s doule-wide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in

### 4.4 Areas, Integrals and Antiderivatives

. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

### Functions of One Real Variable. A Survival Guide. Arindama Singh Department of Mathematics Indian Institute of Technology Madras

Functions of One Rel Vrible A Survivl Guide Arindm Singh Deprtment of Mthemtics Indin Institute of Technology Mdrs Contents Limit nd Continuity 3. Preliminries..................................... 3.2

Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

FINALTERM EXAMINATION 9 (Session - ) Clculus & Anlyticl Geometry-I Question No: ( Mrs: ) - Plese choose one f ( x) x According to Power-Rule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+

### Topics for final

Topics for 161.01 finl.1 The tngent nd velocity problems. Estimting limits from tbles. Instntneous velocity is limit of verge velocity. Slope of tngent line is limit of slope of secnt lines.. The limit

### Math 554 Integration

Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we

### Advanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015

Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n

### Section 6.1 Definite Integral

Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

### Definite integral. Mathematics FRDIS MENDELU

Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the

### Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

### Properties of the Riemann Integral

Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University Februry 15, 2018 Outline 1 Some Infimum nd Supremum Properties 2

### Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.

Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must

### Lecture 1. Functional series. Pointwise and uniform convergence.

1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is