# Math Calculus with Analytic Geometry II

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0

2 orem of definite If < b then b f (x) dx = ( under f bove x-xis) ( bove f under x-xis) Exmple x dx = π 3 4 = 9π 4

3 orem of definite Problem 9 Compute 6 x 5 dx by finding s of regions between the grph of f nd the x-xis. Solution to Problem 9 x-intercept 0 = x 5 x = 5 ( under f bove x-xis) A = (6 5 ) ( 6 5) = 49 4

4 orem of definite Solution to Problem 9 (continued) ( bove f under x-xis) 6 A = ( 5 ( )) ( ( ) + 5) = ( 9 ) (9) = 8 4 x 5 dx = A A = = 8 Exmple 0 (Are formuls seldom work) Evlute x dx. No formul for this portion of circle

5 orem of definite orem ( orem of I) If f is continuous on the intervl [, b] nd F (t) = f (t) then b f (t) dt = F (b) F (). Exmple ( ) sin 8x + x 4 = 8 cos 8x + 4x 3 so by FTOC d dx 8 cos 8x + 4x 3 dx = (sin(8 ) + 4 ) (sin(8 ) + 4 ) = sin 6 sin 8 + 5

6 orem of definite Nottion g(x) b = g(b) g() x= or if x is clerly the vrible to plug nd b into cn write g(x) b = g(b) g() With this nottion FTOC I ( orem of I) sys If f is continuous on the intervl [, b] nd F (t) = f (t) then b f (t) dt = F (t) b t=

7 orem of definite FTOC sys roughly Computing under f Suprising! Why should nd slopes be relted? Finding with derivtive f Could prove FTOC from definition of integrl (Definition 9) nd definition of the derivtive, but we won t. Mkes some sense visully

8 orem of definite Note on units for If x mesured in units u nd f (x) mesured in units u then hs units u u Exmple 3 b f (x) dx If t is the time mesured in hours P(t) number of people working in fctory t time t. b P(t) dt is people hours worked between time nd b

9 orem of definite Integrl of rte of chnge If F (t) is the rte of chnge of some quntity F (t) then FTOC I sy tht b F (t) dt = F (b) F () This is net chnge in F from time to b Exmple 4 As in Exmple 4 from lst week t time (in h) v(t) velocity (rte of chnge of position) t time t (in km/h) then 3 0 v(t) dt is distnce trveled (net chnge in position) in units (km/h) h = km

10 orem of definite Exmple 5 t time (in yers) g(t) growth rte (in m/yer) of person t ge t then 6 9 g(t) dt is chnge in height (in m) of person from ge 9 to 6.

11 orem of definite Definition 6 ( ) verge vlue of f on the intervl [, b] is f ve = b Where does this formul come from? Let s derive it using b f (t) dt. Philosophy of Estimte quntity Figure out how to improve estimte Tke limit

12 orem of definite Averge temperture Let f (t) be the temperture t time t. Let s estimte the verge temperture between m nd 0pm. (First estimte with intervls) totl time = 0 ech intervl is 0 f ve f ( + 0 (Better estimte with 3 intervls) totl time = 0 ech intervl is 0 3 f ve f ( ) ( + f ) ( ) + f + 0 ) ( ) + f

13 orem of definite Averge temperture (continued) (Better estimte with n intervls) totl time = 0 ech intervl is 0 n f ve f ( + 0 n = n f i= = 0 ( + i 0 n n f i= ) ( ) + + f + n 0 n n ) n ( + i 0 n ) 0 n

14 orem of definite Averge temperture (continued) (Tke limit to get exct verge) f ve = lim n 0 = 0 lim n = 0 0 n f i= n f i= f (t) dt ( + i 0 n ( + i 0 n ) 0 n ) 0 n We ve derived the formul in Definition 6.

15 orem of definite Problem 7 Find the verge vlue of the funtion on the intervl [, ] Solution to Problem 7 h ve = b = h(x) = 4 x b ( ) = 4 π h(x) dx 4 x dx = π

16 orem of definite Problem 8 t is time mesured in dys since Jn., 003 R(t) is the distnce from the erth to the sun t time t Wht does represent? Solution to Problem R(t) dt Averge distnce from erth to the sun in 003

17 orem of definite orem 9 If f is integrble then orem 30 If f is integrble then definite b f (x) dx = b f (x) dx b c c f (x) dx + f (x) dx = f (x) dx b

18 orem of definite orem 3 If f is integrble then Proof. b b cf (x) dx = c cf (x) dx = lim n = lim n c b f (x) dx n cf ( + i x) x i= = c lim n = c b n f ( + i x) x i= n f ( + i x) x i= f (x) dx

19 orem of definite orem 3 If f nd g re integrble then Proof. b b f (x) + g(x) dx = f (x) + g(x) dx = lim n = lim n b f (x) dx + b g(x) dx n (f ( + i x) + g( + i x)) x i= n (f ( + i x) x + g( + i x) x) i=

20 orem of definite Proof of orem 3 (continued). n = lim (f ( + i x) x + g( + i x) x) n i= ( n ) n = lim f ( + i x) x + g( + i x) x n i= i= ( ) ( ) n n = f ( + i x) x + g( + i x) x = lim n b i= f (x) dx + b g(x) dx lim n i=

21 orem of definite Problem 33 Compute 6 7 t(v) 7u(v) dv given 6 7 u(v) dv = 6 7 t(v) dv = Solution to Problem t(v) 7u(v) dv = = t(v) dv t(v) dv 7 = 7( ) = u(v) dv u(v) dv

22 Wrning orem of definite Exmple 34 f (x) = b f (x) g(x) dx b {, 0 x 0, < x b f (x) g(x) dx f (x) dx b b f (x) dx g(x) dx nd g(x) = b g(x) dx { 0, 0 x, < x f (x)g(x) dx = dx = 0 but f (x) dx 0 0 g(x) dx = =

23 orem of definite Definition 35 f is even if f ( x) = f (x) f is odd if f ( x) = f (x) Note: Most functions re neither. One function is both. orem 36 If f is even then If f is odd then nd 0 f (x) dx = 0 0 f (x) dx = 0 f (x) dx = f (x) dx f (x) dx = 0 f (x) dx

24 orem of definite Problem 37 Suppose 4 Q(s) ds = 3 0 Q(s) ds = 9 4 Evlute 0 Solution to Problem 37 0 Q(s) ds = 4 Q(s) ds Q(s) ds + = 3 + ( 9) = Q(s) ds

25 orem of definite Problem 38 Compute 8 y(u) du given y is n odd function 8 y(u) du = 7 Solution to Problem y(u) du = y(u) du y(u) du = 7 0 = 7

26 orem of definite Problem 39 Compute 7 h(r) dr given 3 h is n even function 7 h(r) dr = h(r) dr = 0 3 Solution to Problem h(r) dr = = = = 3 7 h(r) dr h(r) dr ( 7 h(r) dr 3 h(r) dr + h(r) dr h(r) dr h(r) dr 3 7 = ( 0) + (4) = 8 0 h(r) dr 7 0 ) h(r) dr

27 orem of definite orem 40 If m f (x) M for x [, b] then m(b ) orem 4 b If f (x) g(x) for x [, b] then f (x) dx M(b ) b f (x) dx b g(x) dx

28 orem of definite Problem 4 Show tht cos(x) sin(x 3 ) dx 99 Solution to Problem 4 For ll x [, 00] cos(x) sin(x 3 ) = cos(x) sin(x 3 ) so cos(x) sin(x 3 ) By orem 40 (00 ) cos(x) sin(x 3 ) dx (00 ) cos(x) sin(x 3 ) dx 99

29 orem of definite Problem 43 For ech pirs of decide which is the lrger π 4 0 cos(x) dx nd π 4 sin(x) dx 0 π π 4 cos(x) dx nd π π 4 Solution to Problem 43 sin(x) dx For ll x [0, π 4 ] cos(x) sin(x) so π 4 0 cos(x) dx π 4 0 sin(x) dx For ll x [ π 4, π ] cos(x) sin(x) so π π 4 cos(x) dx π π 4 sin(x) dx

30 orem of definite Definition 44 (An ntiderivtive) F (x) is n ntiderivtive of f (x) if F (x) = f (x). Exmple 45 so ( d dx x cos(4x + 3) ) = x cos(4x + 3) + 4x sin(4x + 3) x cos(4x + 3) is n ntiderivtive of x cos(4x + 3) + 4x sin(4x + 3) x cos(4x + 3) + 30 is nother ntiderivtive.

31 orem of definite Motivtion Why do we cre bout finding? FTOC I sys tht computing b f (t) dt is esy if we hve n ntiderivtive F of f. Exmple 46 From Exmple 45 bove 7 x cos(4x + 3) + 4x sin(4x + 3) dx = x cos(4x + 3) = (7 cos( ) + 30) ( cos(4 + 3) + 30) = 49 cos(3) cos(7)

32 orem of definite Notice in Exmple 45 we could hve dded ny constnt to x cos(4x + 3) nd we would hve hd nother ntiderivtive of x cos(4x + 3) + 4x sin(4x + 3) We usully dd n unspecified constnt to remind us tht there re mny. Definition 47 ( ntiderivtive) ntiderivtive of f (x) is the set of ll of f (x). orem 48 If f is continuous nd F (x) = f (x) then every ntiderivtive of f is of the form F (x) + C for some constnt C.

33 orem of definite Wht if f is not continuous? ntiderivtive of noncontinuous function Let n F (x) = { ln x + 4, x > 0 ln( x) + 8, x < 0 { F (x) = x, x > 0 { x, x < 0 = x, x > 0 x, x < 0 = x So F (x) is n ntiderivtive of x. Any choice of constnts (4 nd 8 weren t specil) gives sme result. Thus the ntiderivtive of x is F (x) = { ln x + C, x > 0 ln( x) + C, x < 0 = { ln x + C, x > 0 ln x + C, x < 0

34 orem of definite On the other hnd Min reson we cre bout is the FTOC. FTOC only pplies if f is integrble on [, b] x is not integrble on intervls contining 0 so in pplictions we only use one of the two constnts t time Exmple 49 3 x dx = ln x +C Exmple x dx 3 = (ln +C ) (ln 3 +C ) = ln ln 3 cnnot be evluted using FTOC Nottionl wrning By convention we sy tht F (x) + C is the ntiderivtive of f (x) whenever F (x) = f (x) even when this is techniclly incorrect.

35 orem of definite Nottion f (x) dx = F (x) + C mens F (x) + C is the ntiderivtive of f (x) Terminology Since FTOC links ntidifferentition nd integrtion we lso cll (indefinite). following sttements ll men the sme thing: f (x) = d dx F (x) f (x) dx = F (x) + C f (x) is the derivtive of F (x) F (x) + C is the ntiderivtive of f (x) F (x) + C is the indefinite integrl of f (x) F (x) + C is the integrl of f (x)

36 orem of definite Problem 5 Check the following (6x + 3e x ) cos(3x + 3e x ) dx = sin(3x + 3e x ) + C sec x dx = ln sec x + tn x + C Solution to Problem 5 d dx sin(3x + 3e x ) = (6x + 3e x ) cos(3x + 3e x ) d dx ln sec x + tn x = sec x tn x + sec x sec x + tn x tn x + sec x = (sec x) sec x + tn x = sec x Note: Similr to x ntiderivtive of sec x should hve different C for ech intervl [ (n )π, (n+)π ] but nobody does this.

37 orem of definite (See bckbord)

38 orem of definite Ech rule for differentition gives us rule for integrtion From ( ) c d dx F (x) = d dx cf (x) we get orem 5 (Constnt rule for integrtion) cf (x) dx = c f (x) dx

39 orem of definite Proof of orem 5. Suppose d dx F (x) = f (x). We hve the derivtive rule c d dx F (x) = ( ) d dx cf (x) Reinterpreting this rule s n ntiderivtive gives c d dx F (x) dx = cf (x) + C. Thus we my conclude cf (x) dx = c d dx F (x) dx = cf (x) + C = c(f (x) + C ) = c f (x) dx.

40 orem of definite From we get ( ) d dx F (x) + d dx G(x) = d dx F (x) + G(x) orem 53 (Sum rule for integrtion) f (x) + g(x) dx = f (x) dx + g(x) dx

41 orem of definite Proof of orem 53. Suppose d d dx F (x) = f (x) nd dx G(x) = g(x). We hve the derivtive rule ( ) d dx F (x) + d dx G(x) = d dx F (x) + G(x) Reinterpreting this rule s n ntiderivtive gives d dx F (x) + d dx G(x) dx = F (x) + G(x) + C. Thus we my conclude f (x) + g(x) dx = d dx F (x) + d dx G(x) dx = F (x) + G(x) + C = f (x) dx + g(x) dx Note: We drop constnts when we hve on both sides of n eqution.

42 orem of definite Bsic Integrls Ech bsic derivtive gives us bsic integrl differentition rule Tble : Bsic to memorize integrtion rule d dx x r+ = (r + )x r x r dx = r+ x r+ + C if r d dx ln x = x d dx cos x = sin x d dx sin x = cos x d dx ex = e x d dx rctn x = +x d dx rcsin x = x x dx = ln x + C sin x dx = cos x + C cos x dx = sin x + C e x dx = e x + C +x dx = rctn x + C x dx = rcsin x + C

43 orem of definite More bsic You lso know few more derivtive Tble : More bsic to memorize differentition rule d dx tn x = sec x d dx cot x = csc x d dx sec x = sec x tn x d dx csc x = csc x cot x d dx x = (ln ) x integrtion rule sec x dx = tn x + C csc x dx = cot x + C sec x tn x dx = sec x + C csc x cot x dx = csc x + C x dx = x ln + C

44 orem of definite Techniques of integrtion Advnced derivtive give us techniques of integrtion differentition technique of rule integrtion chin rule u-substitution ( 7.) product rule integrtion by prts ( 7.) We will return to these integrtion techniques lter.

45 orem of definite Problem 54 Find formul for 0e x + 7 sin x dx Solution to Problem 54 0e x + 7 sin x dx = Check your nswer! = 0 0e x dx + e x dx sin x dx (Sum rule) sin x dx (Constnt rule) = 0e x 7 cos x + C (Tble ) d dx (0ex 7 cos x) = 0e x + 7 sin x

46 orem of definite Problem 55 Find formul for 8 t 8t dt Solution to Problem 55 8 t 8t dt = 8 = 8 dt t dt 8 t 8 t dt (Sum rule) t dt (Const. rule) = 8 rcsin t 8 t3 3 + C (Tble ) Check your nswer! ( d dt 8 rcsin t 8 3 t3) = 8 t 8 3 3t = 8 t 8t

47 orem of definite Problem 56 Find formul for rcsin(3 π ) Solution to Problem 56 rcsin(3 π ) = + cos u du + sec u du rcsin(3 π ) du + sec u du (Sum rule) = rcsin(3 π ) u + tn u + C (Tbles nd ) Check your nswer! ( d rcsin(3 π ) dt ) u + tn u = rcsin(3 π ) + sec u

48 orem of definite Problem 57 Compute 3 y( 6 6 y) 3 y dy Solution to Problem 57 3 y( 6 6 y) dy = y 3 = = = 3y ( y y 6 ) dy y 3 3y 36y y 5 6 ) dy y 3 3y 6 36y y 3 6 dy 3y 6 36y y dy = y y = 8 7 y y 3 + 7y = 3 y 3 ( ) ( )

49 orem of definite Problem 58 Find n ntiderivtive G(x) of g(x) = sin x + 7 stisfying G(π) = 0. Solution to Problem 58 G(x) = sin x + 7 dx = cos x + 7x + C Use fct tht G(π) = 0 to solve for C. 0 = G(π) = cos π + C So C = 0 + cos π = 0 + ( ) = G(x) = cos x + 7x

50 orem of definite Problem 59 verge vlue of h(x) = x 3 3x on [, ] is 8 solve for. Solution to Problem 59 h ve = x 3 3x dx ( ) = ( 4 x 4 x 3 ) = ( ) [ 4 ( )4 ( ) 3 ] ( ) = ( ) = ( 3) = Use fct tht h ve = 8 to solve for. 8 = h ve = so = ± 8

### 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

### 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:

### 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

### 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

### 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.

### Test 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).

Test 3 Review Jiwen He Test 3 Test 3: Dec. 4-6 in CASA Mteril - Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 14-17 in CASA You Might Be Interested

### Math 113 Exam 2 Practice

Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

### Sections 5.2: The Definite Integral

Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)

### Week 10: Riemann integral and its properties

Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the

### 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

### 5.5 The Substitution Rule

5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in

### Calculus and linear algebra for biomedical engineering Week 11: The Riemann integral and its properties

Clculus nd liner lgebr for biomedicl engineering Week 11: The Riemnn integrl nd its properties Hrtmut Führ fuehr@mth.rwth-chen.de Lehrstuhl A für Mthemtik, RWTH Achen Jnury 9, 2009 Overview 1 Motivtion:

### Section 14.3 Arc Length and Curvature

Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

### Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

### Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

### 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

### Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

### Distance And Velocity

Unit #8 - The Integrl Some problems nd solutions selected or dpted from Hughes-Hllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl

### 2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

### 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 100 Review Sheet

Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s

### Chapter 6. Riemann Integral

Introduction to Riemnn integrl Chpter 6. Riemnn Integrl Won-Kwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl

### 5: The Definite Integral

5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

### MTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008

MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul

### x ) dx dx x sec x over the interval (, ).

Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor

### PDE Notes. Paul Carnig. January ODE s vs PDE s 1

PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................

### Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus

Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = -x + 8x )Use

### We know that if f is a continuous nonnegative function on the interval [a, b], then b

1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going

### 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

### STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

### Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

### 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

### Numerical integration

2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter

### 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

### A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.

A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c

### Midpoint Approximation

Midpoint Approximtion Sometimes, we need to pproximte n integrl of the form R b f (x)dx nd we cnnot find n ntiderivtive in order to evlute the integrl. Also we my need to evlute R b f (x)dx where we do

### 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...

### NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

### Line Integrals. Partitioning the Curve. Estimating the Mass

Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to

### If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du

Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find nti-derivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible

### CS667 Lecture 6: Monte Carlo Integration 02/10/05

CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of

### The Wave Equation I. MA 436 Kurt Bryan

1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

### New Expansion and Infinite Series

Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

### Chapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1

Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the -coordinte of ech criticl vlue of g. Show

### different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).

Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different

### (0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35

7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know

### 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 =

### l 2 p2 n 4n 2, the total surface area of the

Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n n-sided regulr polygon of perimeter p n with vertices on C. Form cone

### RAM RAJYA MORE, SIWAN. XI th, XII th, TARGET IIT-JEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII (PQRS) INDEFINITE INTERATION & Their Properties

M.Sc. (Mths), B.Ed, M.Phil (Mths) MATHEMATICS Mob. : 947084408 9546359990 M.Sc. (Mths), B.Ed, M.Phil (Mths) RAM RAJYA MORE, SIWAN XI th, XII th, TARGET IIT-JEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII

### Topic 1 Notes Jeremy Orloff

Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble

### F is n ntiderivtive èor èindeæniteè integrlè off if F 0 èxè =fèxè. Nottion: F èxè = ; it mens F 0 èxè=fèxè ëthe integrl of f of x dee x" Bsic list: xn

Mth 70 Topics for third exm Chpter 3: Applictions of Derivtives x7: Liner pproximtion nd diæerentils Ide: The tngent line to grph of function mkes good pproximtion to the function, ner the point of tngency.

### SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.

### Notes on Calculus II Integral Calculus. Miguel A. Lerma

Notes on Clculus II Integrl Clculus Miguel A. Lerm November 22, 22 Contents Introduction 5 Chpter. Integrls 6.. Ares nd Distnces. The Definite Integrl 6.2. The Evlution Theorem.3. The Fundmentl Theorem

### Mapping the delta function and other Radon measures

Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support

### Not for reproduction

AREA OF A SURFACE OF REVOLUTION cut h FIGURE FIGURE πr r r l h FIGURE A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundry of solid of revolution of the type

### 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

### Continuous Random Variables

STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

### UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence

### Anonymous Math 361: Homework 5. x i = 1 (1 u i )

Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define

### Czechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction

Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued

Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

### x 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx

. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute

### 1 Error Analysis of Simple Rules for Numerical Integration

cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion

### 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

### KOÇ UNIVERSITY MATH 106 FINAL EXAM JANUARY 6, 2013

KOÇ UNIVERSITY MATH 6 FINAL EXAM JANUARY 6, 23 Durtion of Exm: 2 minutes INSTRUCTIONS: No clcultors nd no cell phones my be used on the test. No questions, nd tlking llowed. You must lwys explin your nswers

### Power Series, Taylor Series

CHAPTER 5 Power Series, Tylor Series In Chpter 4, we evluted complex integrls directly by using Cuchy s integrl formul, which ws derived from the fmous Cuchy integrl theorem. We now shift from the pproch

### MATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals.

MATH 409 Advned Clulus I Leture 22: Improper Riemnn integrls. Improper Riemnn integrl If funtion f : [,b] R is integrble on [,b], then the funtion F(x) = x f(t)dt is well defined nd ontinuous on [,b].

### Math 113 Exam 2 Practice

Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.-3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This

### approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

### Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

### Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

### MA Handout 2: Notation and Background Concepts from Analysis

MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,

### Line and Surface Integrals: An Intuitive Understanding

Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of

### APPM 1360 Exam 2 Spring 2016

APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the

### Calculus MATH 172-Fall 2017 Lecture Notes

Clculus MATH 172-Fll 2017 Lecture Notes These notes re concise summry of wht hs been covered so fr during the lectures. All the definitions must be memorized nd understood. Sttements of importnt theorems

### Integrals along Curves.

Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the

### The Dirac distribution

A DIRAC DISTRIBUTION A The Dirc distribution A Definition of the Dirc distribution The Dirc distribution δx cn be introduced by three equivlent wys Dirc [] defined it by reltions δx dx, δx if x The distribution

### The Definite Integral

CHAPTER 3 The Definite Integrl Key Words nd Concepts: Definite Integrl Questions to Consider: How do we use slicing to turn problem sttement into definite integrl? How re definite nd indefinite integrls

### The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

### Week 12 Notes. Aim: How do we use differentiation to maximize/minimize certain values (e.g. profit, cost,

Week 2 Notes ) Optimiztion Problems: Aim: How o we use ifferentition to mximize/minimize certin vlues (e.g. profit, cost, volume, ) Exmple: Suppose you own tour bus n you book groups of 20 to 70 people

### Summer MTH142 College Calculus 2. Section J. Lecture Notes. Yin Su University at Buffalo

Summer 6 MTH4 College Clculus Section J Lecture Notes Yin Su University t Bufflo yinsu@bufflo.edu Contents Bsic techniques of integrtion 3. Antiderivtive nd indefinite integrls..............................................

### For a continuous function f : [a; b]! R we wish to define the Riemann integral

Supplementry Notes for MM509 Topology II 2. The Riemnn Integrl Andrew Swnn For continuous function f : [; b]! R we wish to define the Riemnn integrl R b f (x) dx nd estblish some of its properties. This

### 11.1 Exponential Functions

. Eponentil Functions In this chpter we wnt to look t specific type of function tht hs mny very useful pplictions, the eponentil function. Definition: Eponentil Function An eponentil function is function

### MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs

MATHS NOTES The Institute of Eduction 06 SUBJECT: Mths LEVEL: Higher TEACHER: Aidn Rontree Topics Covered: Powers nd Logs About Aidn: Aidn is our senior Mths techer t the Institute, where he hs been teching

### 7 - Continuous random variables

7-1 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7 - Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin

### u(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.

Lecture 4 Complex Integrtion MATH-GA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex

### 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

### Best Approximation. Chapter The General Case

Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given

### 10. AREAS BETWEEN CURVES

. AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

### Necessary and Sufficient Conditions for Differentiating Under the Integral Sign

Necessry nd Sufficient Conditions for Differentiting Under the Integrl Sign Erik Tlvil 1. INTRODUCTION. When we hve n integrl tht depends on prmeter, sy F(x f (x, y dy, it is often importnt to know when

### MATH 222 Second Semester Calculus. Fall 2015

MATH Second Semester Clculus Fll 5 Typeset:August, 5 Mth nd Semester Clculus Lecture notes version. (Fll 5) This is self contined set of lecture notes for Mth. The notes were written by Sigurd Angenent,

### a n+2 a n+1 M n a 2 a 1. (2)

Rel Anlysis Fll 004 Tke Home Finl Key 1. Suppose tht f is uniformly continuous on set S R nd {x n } is Cuchy sequence in S. Prove tht {f(x n )} is Cuchy sequence. (f is not ssumed to be continuous outside

### 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

### REVIEW Chapter 1 The Real Number System

Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

### Math 4200: Homework Problems

Mth 4200: Homework Problems Gregor Kovčič 1. Prove the following properties of the binomil coefficients ( n ) ( n ) (i) 1 + + + + 1 2 ( n ) (ii) 1 ( n ) ( n ) + 2 + 3 + + n 2 3 ( ) n ( n + = 2 n 1 n) n,

### spring from 1 cm to 2 cm is given by

Problem [8 pts] Tre or Flse. Give brief explntion or exmple to jstify yor nswer. ) [ pts] Given solid generted by revolving region bot the line x, if we re sing the shell method to compte its volme, then