# From the Numerical. to the Theoretical in. Calculus

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 From the Numericl to the Theoreticl in Clculus Teching Contemporry Mthemtics NCSSM Ferury 6-7, 003 Doug Kuhlmnn Phillips Acdemy Andover, MA

2 How nd Why Numericl Integrtion Should Precede the Fundmentl Theorem The short nswer is: "We mesure re, we don't define it." (Jerry Uhl) HOW TO INTRODUCE NUMERICAL INTEGRATION FIRST. Assumptions: Students know the derivtives of the usul elementry functions, including polynomils, trig functions, exponentil nd logs. In ddition they know the power rule for non-integer exponents, the product, quotient nd chin rules. They hve done few ntiderivtive prolems, minly velocity/distnce types. Introducing re under curve: L n nd R n. On the first dy of integrtion I sk the sic question: Wht is this re? How cn we mesure this re? One pproximtion leds to the left-hnd Riemnn sum in the usul fshion: L n = f (x 0 )h + f (x 1 )h f (x n 1 )h where h =. Note tht I only consider pproximtions using rectngles of equl n width. The first time through clculus is not the time to generlize to things like prtitions, norms of prtions, refinements, etc. We immeditely consider the right-hnd Riemnn sum nd discuss why it is not priori possile to choose which is etter, the Left Hnd Rule or the Right Hnd Rule. At this point the clss writes short TI-83 progrm, clled INTEGRAL or AREA, tht clcultes this sum with, nd n s user inputs. This is good time to tech nd/or review For-End loops. (A listing of the finl version of the INTEGRAL progrm is t the end of these notes.) The first integrl progrm using Left nd Right hnd rules only nd ssumes tht the function is in Y1 is listed here. 1

3 INTEGRAL Progrm :ClrHome :Input "LOWER LIMIT=",A :Input "UPPER LIMIT=",B :Prompt N :(B-A)/N->H :0->L:0->R :A->X :For(J,1,N) :L+Y1*H->L :X+H->X :R+Y1*H->R :End :Disp "L, R",L,R Trpezoid nd Midpoint Rules At this point the clss usully invents the trpezoid rule themselves: T n = 1 (L n + R n ) This leds to modifiction of the INTEGRAL progrm.. I then suggest to the clss nother wy, the Midpoint rule, which uses the y-coordinte t the middle of the suintervl for the height of the rectngles. One cn compre the Trpezoid nd Midpoint rules in the following wy: nd T n = f (x 0 ) + f (x 1 ) h + f (x ) + f (x ) 1 h f (x ) + f (x ) n 1 n h M n = f x 0 + x 1 h + f x 1 + x h f x n 1 + x n We now mke nother, this time more sutle, chnge in our INTEGRAL progrm to llow us to clculte M n. h

4 Simpson's Rule without prols. For L n nd R n there ws no priori reson for choosing one or the other. Ech seems to e eqully ccurte nd so when we verge them for the trpezoid rule, we weight ech of them the sme. Cn we do similr thing with the Trpezoid nd Midpoint rules? We cn compre the ccurcy of the Trpezoid nd Midpoint rules y looking t some exmples tht we lredy know. By using our INTEGRAL progrm we see tht s n increses, our estimtes of the re under y = cos(x) etween x = 0 nd x = π pproch 1. Tking it s fct tht this re is indeed 1, let's compre the errors of the Trpezoid nd Midpoint rules, i.e let's exmine T n 1 nd M n 1. Here re three cses using n = 5, 10 nd 0 with the resulting errors or devitions from 1 for the trpezoid nd midpoint pproximtions. N T n 1 M n Note tht the Mid-point pproximtion error is out hlf tht of the Trp pproximtion nd is of opposite sign. In this cse the Trp rule is too smll nd the Mid rule is too ig ut the Mid rule is out twice s close to the ctul re. We cn lso compre the ccurcies of T n nd M n y the following rgument. Consider one rectngle in the midpoint rule: M M 3

5 Keeping the point mrked M fixed on the curve, rotte the top of the rectngle, extending it t the sme time, until you mke it tngent to the curve t M. Wht is the re under the trpezoid formed with this tngent line s the non-prllel top? With little considertion, one cn show tht this re is the sme s the re of the orginl rectngle. This mens tht the strightforwrd Midpoint rule, s esy to clculte s the Left- nd Right-hnd rules, cn e thought of s the Tngent to the Midpoint Trpezoid Rule. Which is more ccurte, T n or M n? First nswer this question: Which is etter pproximtion to function, tngent line or secnt line. The nswer gives support to the rgument tht M n is etter pproximtion. So does the following heuristic rgument. For functions tht re concve up, the Midpoint rule is too smll nd the Trp rule is too ig. (Other wy round for concve down.) The picture ove lso suggest tht the error for Trp is twice s lrge in mgnitude s the error for Mid nd is of opposite sign, point we discovered numericlly erlier. Since M n is closer to the true vlue thn T n y fctor of two nd M n ndt n re on opposite sides of the true vlue, tht suggests we use weighted verge of the two. Consequently we define Simpson's rule s: S n = M n + T n 3 WARNING: Note the suscript. If we use n sudivisions for the Trp rule nd n sudivisions for the Midpoint rule, we hve effectively used n sudivisions for Simpson's rule this wy. 4

6 WHY DO NUMERICAL INTEGRATION FIRST? By now I hve lso introduced the integrl nottion f( x) dx to represent the re. By doing severl prolems pproximting this integrl, the students come to know tht f( x) dx is rel numer, not clss of functions. Additionlly, they c cn verify the ovious f (x) dx + f (x)dx = f (x) dx nd f (x)dx = 0. They cn even verify nd think out why c f (x)dx = f (x)dx (h is negtive in one of the sums). Finlly, I hve the clss perform this exercise: Let Y1=cos(X) nd evlute the following for vrious vlues of 0 cos(x)dx A little discusion llows them to see tht we hve creted new function: For every '' there is exctly one numer ssocited with tht, nmely the re from 0 to. I then sk them to ech find the vlue of this integrl for = 0,.,.4,.6,.8, 1.0, 1.,... s mny s there re students, mye twice s mny with every one doing two. When I first did this exercise we plotted the points on the ord: (0,0) (.,.199) (.4,.389) (.6,.565) (.8,.717) etc These questions were sked y students s we plotted them; 1. Why re they going down? (Eventully it does). Why re they negtive? (Ditto) 3. Why is it sin? If students cn sk these questions on their own, I ws convinced tht this method of introducing the integrl ws vlule. At this point we cn prove the Fundmentl Theorem in the usul wy, hving motivted it with the ove nd severl other exmples. However we cn lso explore ntiderivtives y solving differentil equtions grphiclly vi slope fields nd numericlly with Euler's method. 5

7 Slope Fields nd Euler's Method Suppose we wnted to find solutions to differentil eqution of the form y = g(x, y). Exmples of such differentil equtions include: y = y (1) y = x () y = x y (3) y = cos y (4) y = exp( x ) = e x (5) y = 1 + x 3 x (6) Equtions (1) through (4) re solvle y nlytic techniques ut equtions (5) nd (6) re not. Nevertheless, we cn get some qulittive informtion out the solutions of ll of the ove y looking t the slope fields for ech. SLOPE FIELDS If first-order differentil eqution cn e put in the form y = g(x, y), then we cn determine the slope of the solution y = f (x) through ny point (x, y). Grphiclly, we cn drw short line of the proper slope through ech of severl points (x, y) in the plne. The resulting picture of short slope lines is clled slope field. I think of slope field s series of conditionl sttements: If solution psses through the point (x, y), this is the slope it would hve. Consider simple cse: f (x) = x. On piece of grph pper, here chosen very smll to mke it esier, select grid of points nd t ech point selected drw short segment whose slope is x, the first coordinte of the point. This is the resulting picture: y x We know the solutions to this differentil eqution re f (x) = x + C, nd if we mentlly overly these on the grid ove, we cn see how the solutions follow the slope field. 6

8 In prctice this cn e very time consuming or t lest tedious, ut computer or progrmmle clcultor cn do the tsk for us. In the following exmples, TI-83 progrm drew the slope fields. The listing for this progrm is given t the end. Before we proceed, let's remind ourselves of the following theorem found in lmost ny text on elementry differentil equtions. Theorem. For first order differentil eqution y = g(x, y) with intil condition y(x 0 ) = y 0, sufficient (though not necessry) condition tht unique solution y = f (x) exist is tht g nd g e rel, finite, single-vlued, nd continuous over y rectngulr region of the plne contining the point (x 0, y 0 ). In simpler terms, if g(x, y) is suitly nice, we cn sfely mke the ssumption tht unique solution exists. In ll of the cses we shll exmine, the ove conditions re stisfied. Wht this mens is tht given specific point in our slope field there is exctly one solution tht psses through this point. Exmple 1. y = y Before we sketch the slope field, notice tht if y = y then y = y = y. Similrly, ny order derivtive of y is equl to y. Below is the slope field for this eqution on the viewing window [-4.7, 4.7] X [-3.1, 3.1] (Unless otherwise stted, ll viewing windows will hve these dimensions.) y = y We esily recognize the exponentil functions f (x) = Ae x s solutions nd cn esily visulize them on top of the slope field. 7

9 Exmple. y = x (Left to reder to expnd ove slope field.) Exmple 3. y = x y Note tht y = 1 y =1 (x y) = 1 x + y. We then conclude tht y > 0 if y < x y > 0 if y > x 1 y = 0 if y = x y = 0 if y = x 1 y < 0 if y > x y < 0 if y < x 1 The slope field for this eqution shows tht ny solution stisfies the ove conditions. y = x y Note tht nywhere long the line y = x, the slope is zero. It is esy to sketch qulittive solutions to the differentil equtions y following the slope fields, ut here gin the computer cn help using numericl methods to pproximte solutions. 8

10 EULER'S METHOD The first numericl method is Euler's method, strightforwrd, simple (nd hence prone to inccurcies t times) method for pproximting solutions to y = g(x, y). It is nlogous to using the left hnd rule for pproximting integrls. If we know tht the point (x 0, y 0 ) is on our curve nd we define our new x 1 to e x 1 = x 0 + h where h is the step-size or increment. Then the new y is given y y 1 = y 0 + g(x 0, y 0 )h. We pproximte y y using the tngent line which hs slope given y g(x 0, y 0 ). The new point is connected to the old point nd the process is repeted, generting n pproximte solution to the differentil eqution. Tht is, we find x = x 1 + h nd y = y 1 + g(x 1,y 1 )h. Note tht we reclculted the slope t the point (x 1,y 1 ). We repet this s mny times s needed. Below is the originl slope field nd next to it is slope field with severl solutions sketched on it using Euler's method for y = x y One question tht immeditely rises, is the existence of possile symptote to ll of the solutions. They ll seem to converge to line. In fct, if we you nlytic techniques (using integrting fctors) to solve y = x y we get the generl solution to e y = x 1+ Ce x It is now cler tht the line y = x 1 is n symptote for ll solutions to our eqution. 9

11 Exmple 4. y = cos y Using nlytic techniques, nmely seprting vriles, we get sec y dy = dx Integrting nd rememering the trick for integrting secnt, we get ln sec y + tn y = x + C But wht do these curves look like? Cn you sketch one? Question: Wht does the solution tht goes through the point (-4, 3) look like? Answer: Huh? Slope fields cn help. Below is the slope field for y = cos y. y = cos y Exmining it crefully, we see tht the lines y = π nd y = π re importnt. Both re fixed points in the sense tht if y = π or y = π, then cos y = 0 nd so there is no chnge in y. However, there is very ig difference etween the two lines. The line y = π is n ttrctive fixed point. If you strt within neighorhood of y = π the solution will converge to the line: It will e "ttrcted" to the line. On the other hnd, the line y = π is repelling fixed point. If you strt even slightly wy from y = π further wy nd eventully converges to nother fixed point. the curve moves 10

12 Below re Euler solutions with different strting points ll close to ech other. x 0 = 4, y 0 = π, x 0 = 4, y 0 = π, +.01 x 0 = 4, y 0 = π,.01 I leve it to the reder to determine wht the curve through (-4, 3) looks like. Euler's Method nd the Fundmentl Theorem If the derivtive is function of x lone, i.e. dy dx = f (x), then Euler's method cn e used to illuminte the Fundmentl Theorem of Clculus. Suppose we know the derivtive nd strting point for function so tht we re given point (, f ()) nd the derivtive f (x) nd we wish to evlute (estimte) f () where for this rgument < (lthough it doesn't hve to e). If we define h = n following: nd then use Euler's with step size h nd n steps, we get the 11

13 x 0 = f (x 0 ) = f ()... x 1 = x 0 + h f (x 1 ) f (x 0 ) + f (x 0 )h x = x 1 + h f (x ) f (x 1 ) + f (x 1 )h x n 1 = x n + h f (x n 1 ) f (x n ) + f (x n )h x n = x n 1 + h f (x n ) f (x n 1 ) + f (x n 1 )h Note tht x n = nd f (x n ) = f (). If we sustitute the next to lst eqution for f (x n 1 ) into to the lst one we get f (x n ) f (x n ) + f (x n )h + f (x n 1 )h Sustitute the previous eqution ove nd get f (x n ) f (x n 3 ) + f (x n 3 )h + f (x n )h + f (x n 1 )h. Repeting this we get f (x n ) f (x 0 ) + f (x 0 )h + f (x 1 )h f (x n 1 )h Rememering tht f (x 0 ) = f () nd f (x n ) = f () we see tht the lst eqution just sys tht f () f () f (x 0 )h + f (x 1 )h f (x n 1 )h nd the right hnd side of this eqution is just the Left-hnd Riemnn sum pproximting the integrl of f (x) nd so we re led to f () f () = f (x)dx which is the Fundmentl Theorem. 1

14 Progrm listings: INTEGRAL Progrm :ClrHome :Input "LOWER LIMIT=",A :Input "UPPER LIMIT=",B :Prompt N :(B-A)/N->H :0->L:0->M:0->R :A->X :For(J,1,N) SLOPEFLD Progrm :Func:FnOff :PlotsOff :ClrDrw :ClrHome :Disp "AXES ON (1) :Disp "OR OFF (0)" :Input A :If A=0 :Then:AxesOff :Else:AxesOn :End :Disp "HOW MANY MARKS?" :Input "ACROSS: ",A :Input "DOWN: ",D :(Xmx-Xmin)/A->H :(Ymx-Ymin)/D->V :For(I,1,D) :For(J,1,A) :Xmin+(J-1)H+H/->X :Ymin+(I-1)V+V/->Y :Y1->M :Y-.30MH->S :Y+.30MH->Z :X-.30H->P :X+.30H->Q :If s (Z-S)>.6V :Then :Y+.30V->Z :Y-.30V->S :(Z-Y)/M+X->Q :(S-Y)/M+X->P :End :Line(P,S,Q,Z) :End:End :L+Y1*H->L :X+H/->X :M+Y1*H->M :X+H/->X :R+Y1*H->R :End :(L+R)/->T :(M+T)/3->S :Disp "L, R, T,M,S",L,R,T,M,S EULER Progrm :Func:FnOff :Input "X START=",A :Input "Y START=",B :Input "STEP SIZE=",H :Input "NO. STEPS=",N :A->X:B->Y :For(I,1,N) :X->U:Y->V :Y+Y H->Y :X+H->X :Line(U,V,X,Y) :End 13

15 Runge-Kutt techniques re nlogous to Simpson's rule. Below is the code for progrm tht uses fourth order Runge-Kutt technique to numericlly pproximte solution to first order differentil eqution y = g(x,y). RK4GRAPH Progrm :Func:FnOff :Input "X START=",A :Input "Y START=",B :Input "STEP SIZE=",H :Input "NO. STEPS=",N :A->X:B->Y :For(I,1,N) :X->U:Y->V :Y1->K :X+H/->X :V+H*K/->Y :Y1->L :V+H*L/->Y :Y1->S :X+H/->X :V+H*S->Y :Y1->T :V+(H/6)(K+L+S+T)->Y :Line(U,V,X,Y):End Doug Kuhlmnn Phillips Acdemy Andover, MA Ferury

### 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

### 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

### 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.

### Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...

Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting

### Section 4: Integration ECO4112F 2011

Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

### 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

### 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

### 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

### 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:

### 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

### Chapter 9 Definite Integrals

Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

### Interpreting Integrals and the Fundamental Theorem

Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

### 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

### 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

### 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)

### 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

### 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

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

### The Trapezoidal Rule

_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

### 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

### 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

### 3.4 Numerical integration

3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,

### 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 =

### 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

### Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

### 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

Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

### 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:

### 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

### 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 3B Final Review

Mth 3B Finl Review Written by Victori Kl vtkl@mth.ucsb.edu SH 6432u Office Hours: R 9:45-10:45m SH 1607 Mth Lb Hours: TR 1-2pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems

### 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

### 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..............................................

### 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

### 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

### 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

### 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:

### 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

### 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

### 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

### 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

### 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

### Linear Systems with Constant Coefficients

Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

### set is not closed under matrix [ multiplication, ] and does not form a group.

Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed

### Math Calculus with Analytic Geometry II

orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

### 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

### dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.

Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd

### LECTURE 19. Numerical Integration. Z b. is generally thought of as representing the area under the graph of fèxè between the points x = a and

LECTURE 9 Numericl Integrtion Recll from Clculus I tht denite integrl is generlly thought of s representing the re under the grph of fèxè between the points x = nd x = b, even though this is ctully only

### 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

### How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

### 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

### (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

### 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

### Convex Sets and Functions

B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line

### COSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III) - Gauss Quadrature and Adaptive Quadrature

COSC 336 Numericl Anlysis I Numericl Integrtion nd Dierentition III - Guss Qudrture nd Adptive Qudrture Edgr Griel Fll 5 COSC 336 Numericl Anlysis I Edgr Griel Summry o the lst lecture I For pproximting

### 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

### 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

### Section 6.1 INTRO to LAPLACE TRANSFORMS

Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform

### x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b

CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick

### 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

### 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

### 2. VECTORS AND MATRICES IN 3 DIMENSIONS

2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

### 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

### Lecture 19: Continuous Least Squares Approximation

Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for

### THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS

THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS CARLOS SUERO, MAURICIO ALMANZAR CONTENTS 1 Introduction 1 2 Proof of Gussin Qudrture 6 3 Iterted 2-Dimensionl Gussin Qudrture 20 4

### Section 7.1 Area of a Region Between Two Curves

Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

### Best Approximation in the 2-norm

Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion

### 13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

### Homework 3 Solutions

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

### 12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS

1 TRANSFORMING BIVARIATE DENSITY FUNCTIONS Hving seen how to trnsform the probbility density functions ssocited with single rndom vrible, the next logicl step is to see how to trnsform bivrite probbility

### 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

### Numerical Integration

Chpter 1 Numericl Integrtion Numericl differentition methods compute pproximtions to the derivtive of function from known vlues of the function. Numericl integrtion uses the sme informtion to compute numericl

### 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

### 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

### METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS

Journl of Young Scientist Volume III 5 ISSN 44-8; ISSN CD-ROM 44-9; ISSN Online 44-5; ISSN-L 44 8 METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS An ALEXANDRU Scientific Coordintor: Assist

### 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

Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics - A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the

### 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

### Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

### 31.2. Numerical Integration. Introduction. Prerequisites. Learning Outcomes

Numericl Integrtion 3. Introduction In this Section we will present some methods tht cn be used to pproximte integrls. Attention will be pid to how we ensure tht such pproximtions cn be gurnteed to be

### 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

### Series: Elementary, then Taylor & MacLaurin

Lecture 3 Series: Elementry, then Tylor & McLurin This lecture on series strts in strnge plce, by revising rithmetic nd geometric series, nd then proceeding to consider other elementry series. The relevnce

### Calculus - Activity 1 Rate of change of a function at a point.

Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus - Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus

### u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + \$

### 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

### Sample Questions PREPARING FOR THE AP (AB) CALCULUS EXAMINATION. tangent line, a+h. a+h

Smple Questions PREPARING FOR THE AP (AB) CALCULUS EXAMINATION B B A B tngent line,, f '() = lim h f( + h) f() h +h f(x) dx = lim [f(x ) x + f(x ) x + f(x ) x +...+ f(x ) x n] n B B A B tngent line,, f

### 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

### The Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5

The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle

### 1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q.

Chpter 6 Integrtion In this chpter we define the integrl. Intuitively, it should be the re under curve. Not surprisingly, fter mny exmples, counter exmples, exceptions, generliztions, the concept of the

### ECO 317 Economics of Uncertainty Fall Term 2007 Notes for lectures 4. Stochastic Dominance

Generl structure ECO 37 Economics of Uncertinty Fll Term 007 Notes for lectures 4. Stochstic Dominnce Here we suppose tht the consequences re welth mounts denoted by W, which cn tke on ny vlue between

### USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.

### 5.2 Volumes: Disks and Washers

4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of cross-section or slice. In this section, we restrict

### 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

### 5.1 Estimating with Finite Sums Calculus

5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during

### 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

### STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

### 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