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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "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"

Transcription

1 . 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 u to get: e d e u du e u + C e + C d) We hve: + d d + d rcsin + d After substituting u we obtin: d u du u + C Hence: + d rcsin() + C. Compute the following integrls: ) b) c) 4 (t + ) 5 dt 4 d d

2 d) π/ sin() cos() d Solution: ) Substitute u t + to get: (t + ) 5 dt 5 ] 5 u 5 du u b) Note tht 4 < for nd 4 > for 4. Hence we cn write: 4 4 d ( ( 4)d + 8 ) + (4 c) We perform the substitution u : u / du 4 [ ( 4)d 4 ] 6 ( 8)) ( u) d du u u / du + d) After substituting u cos() we get [ u u / / du / ] u + u du u / [ u / / ( / / ) 4 (/ / ) + 5 (5/ 5/ ) π/ sin() d cos() u ] du u u du ] [ ] [ u 5/ + 5/ ]. Find n ntiderivtive F () of f() Solution: The ntiderivtives of f() re of the form: such tht F (). F () rcsin() + C for some constnt C. We hve F () rcsin() + C C, hence the nswer is F () rcsin() + 4. Use the definition of derivtive to show tht the derivtive of f() + is f () +. Solution: We hve: f f( + h) f() ( + h) + ( + h) ( + ) () lim lim h h h h + h + h + + h h + h + h lim lim lim + + h + h h h h h

3 5. Clculte the following limit: Solution: We hve: cos() cos() Since lim +cos(), we hve: lim cos() + cos() + cos() cos() + cos() sin() + cos() lim cos() sin() lim, sin() so tht we only hve to compute lim. Let us compute the side limits by L Hopitl s rule. We hve: sin() sin() cos() lim lim lim sin() sin() cos() lim lim lim sin() Since the two side limits re different, it follows tht lim does not eist. 6. Consider the function f() e. does not eist nd hence lim cos() ) Where is f continuous? Where is it differentible? b) Find ll the criticl points of f. Find ll the locl mim nd minim nd the intervls where f is incresing or decresing. c) Find the inflection points of f nd the intervls where f is concve down/up. d) Does f hve ny horizontl symptotes for f? e) Sketch the grph of f. Solution: ) The function g(), being polynomil, is continuous nd differentible everywhere. The function e is composition of two functions tht re differentible for every, hence it is lso differentible everywhere (hence continuous everywhere). b) We compute the first derivtive of f() using the chin rule: f () ( + )e Since f() is differentible everywhere, the criticl points re the points c stisfying f (c). We hve: f (c) (c + )e c c c + c To find out whether c is locl mimum or locl minimum, we need to compute f (): Hence: f () ( + ) e e (( + ) )e f ( ) (( + ) )e ( ) e 4 < so we conclude tht c is locl mimum, f() is incresing for < (since f () is positive there) nd f() is decresing for > (since f () is negtive there). c) To find the inflection points of f(), we need to solve f (). We hve: f () (( + ) )e ( + ) +,

4 Thus there re two inflection points. We hve: < < < + > + f () > f () < f () > hence f() is concve upwrd in ], [ ] +, + [ nd concve downwrd in [, + ]. d) Since lim + ( ) lim ( ), we hve: lim f() lim + + (e ) e lim + ( ) 4

5 7. Find the volume of pyrmid with squre bse of length l nd height h. Solution: Plce the pyrmid on coordinte system so tht the its verte coincides with the origin nd the bse is perpendiculr to the -is. Then its volume cn be obtined by the formul: V h A()d where A() is the re of the intersection of the pyrmid with plne prllel to its bse t distnce from the verte (the re of the red squre in the picture). We hve A() l(), where l() is the side of the red squre in the picture. We need n epression for l() in terms of. To find it, consider the following side view: Note tht there re two similr tringles which give: l()/ l/ h l() l h 5

6 Hence we hve: h ( V l ) ( ) l h d d h h ( ) [ ] l h ( ) l h h h l h 8. Compute the volume of the solid obtined by considering the region bounded by y nd nd y nd rotting it long the line y. Solution: To compute this volume, we cn shift the whole region up by units nd then rotte long the line y. The new region is bounded by y +, nd y. Applying the formul for the volume of the solid obtined by rotting region bounded between two curves, we obtin: V π(( + ) )d π( )d π[ ] 8π 7 9. A rectngulr poster is to be produced with design in the middle nd blnk spce round the edges. The blnk spce will etend 4cm from the top nd bottom edges nd cm from the left nd right edges. The re of the design in the middle will be cm. Find the dimensions of the poster with the smllest totl re. Solution: The centrl design is rectngle of sides (horizontl) nd b (verticl). Then the totl re of the poster is: A ( + 4)(b + 8) Using the reltion b, we get b A() ( + 4)( nd hence: + 8) To minimize A, we compute its derivtive with respect to : A () Hence A () Since A ( 5) 4 >, we obtin tht 5 minimizes 5 A. Hence 5, b 5 5 re the dimensions tht give the poster the smllest totl re.. Find the re enclosed between the two curves y nd y 4 +. Solution: First we hve to find the intersection points of the two curves: 4 + 4, If is in the intervl [, ], then < 4 + nd hence the re cn be found s the following integrl: A (4 + )d (4 )d Since this is the integrl of n even function on the intervl [, ], it equls: (4 )d ] [4 (4 ) 6

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

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

More information

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

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

More information

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

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:

More information

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

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

More information

Math 0230 Calculus 2 Lectures

Math 0230 Calculus 2 Lectures Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two

More information

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

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

More information

APPM 1360 Exam 2 Spring 2016

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

More information

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 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

More information

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions ) - TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the

More information

B Veitch. Calculus I Study Guide

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

More information

EXERCISE I. 1 at the point x = 2 and is bisected by that point. Find 'a'. Q.13 If the tangent at the point (x ax 4 touches the curve y =

EXERCISE I. 1 at the point x = 2 and is bisected by that point. Find 'a'. Q.13 If the tangent at the point (x ax 4 touches the curve y = TANGENT & NORMAL EXERCISE I Q. Find the equtions of the tngents drwn to the curve y 4y + 8 = 0 from the point (, ). Q. Find the point of intersection of the tngents drwn to the curve y = y t the points

More information

CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS

CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS 6. VOLUMES USING CROSS-SECTIONS. A() ;, ; (digonl) ˆ Ȉ È V A() d d c d 6 (dimeter) c d c d c ˆ 6. A() ;, ; V A() d d. A() (edge) È Š È Š È ;, ; V A() d d 8

More information

Week 10: Riemann integral and its properties

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

More information

Calculus 2: Integration. Differentiation. Integration

Calculus 2: Integration. Differentiation. Integration Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is

More information

P 1 (x 1, y 1 ) is given by,.

P 1 (x 1, y 1 ) is given by,. MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce

More information

Convex Sets and Functions

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

More information

AP Calculus BC Review Applications of Integration (Chapter 6) noting that one common instance of a force is weight

AP Calculus BC Review Applications of Integration (Chapter 6) noting that one common instance of a force is weight AP Clculus BC Review Applictions of Integrtion (Chpter Things to Know n Be Able to Do Fin the re between two curves by integrting with respect to x or y Fin volumes by pproximtions with cross sections:

More information

[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves

[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.

More information

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

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.

More information

Prerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,

Prerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) , R rern Tower, Rod No, Contrctors Are, Bistupur, Jmshedpur 800, Tel 065789, www.prernclsses.com IIT JEE 0 Mthemtics per I ART III SECTION I Single Correct Answer Type This section contins 0 multiple choice

More information

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

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

More information

SCHEME OF WORK FOR IB MATHS STANDARD LEVEL

SCHEME OF WORK FOR IB MATHS STANDARD LEVEL Snnrpsgymnsiet Lott Hydén Mthemtics, Stndrd Level Curriculum SCHEME OF WORK FOR IB MATHS STANDARD LEVEL Min resource: Mthemtics for the interntionl student, Mthemtics SL, Hese PART 1 Sequences nd Series

More information

Math 113 Exam 1-Review

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

More information

Math Calculus with Analytic Geometry II

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 information

38 Riemann sums and existence of the definite integral.

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

More information

FINALTERM EXAMINATION 2011 Calculus &. Analytical Geometry-I

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...

More information

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

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

More information

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

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

More information

5.5 The Substitution Rule

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

More information

7.6 The Use of Definite Integrals in Physics and Engineering

7.6 The Use of Definite Integrals in Physics and Engineering Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems

More information

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

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

More information

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

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

More information

Section 4: Integration ECO4112F 2011

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

More information

MATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC

MATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot

More information

I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=

I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r= 11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel

More information

5.2 Volumes: Disks and Washers

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

More information

Big idea in Calculus: approximation

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:

More information

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

More information

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

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:

More information

Math 107H Topics for the first exam. csc 2 x dx = cot x + C csc x cotx dx = csc x + C tan x dx = ln secx + C cot x dx = ln sinx + C e x dx = e x + C

Math 107H Topics for the first exam. csc 2 x dx = cot x + C csc x cotx dx = csc x + C tan x dx = ln secx + C cot x dx = ln sinx + C e x dx = e x + C Integrtion Mth 07H Topics for the first exm Bsic list: x n dx = xn+ + C (provided n ) n + sin(kx) dx = cos(kx) + C k sec x dx = tnx + C sec x tnx dx = sec x + C /x dx = ln x + C cos(kx) dx = sin(kx) +

More information

Some Methods in the Calculus of Variations

Some Methods in the Calculus of Variations CHAPTER 6 Some Methods in the Clculus of Vritions 6-. If we use the vried function ( α, ) α sin( ) + () Then d α cos ( ) () d Thus, the totl length of the pth is d S + d d α cos ( ) + α cos ( ) d Setting

More information

Best Approximation in the 2-norm

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

More information

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK WRITTEN EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/MC-UPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION

More information

Section 7.1 Area of a Region Between Two Curves

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

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

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

More information

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

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

More information

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,

More information

Chapter 9 Definite Integrals

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

More information

Fact: All polynomial functions are continuous and differentiable everywhere.

Fact: All polynomial functions are continuous and differentiable everywhere. Dierentibility AP Clculus Denis Shublek ilernmth.net Dierentibility t Point Deinition: ( ) is dierentible t point We write: = i nd only i lim eists. '( ) lim = or '( ) lim h = ( ) ( ) h 0 h Emple: The

More information

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 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

More information

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

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

More information

Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes

Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl

More information

BRIEF NOTES ADDITIONAL MATHEMATICS FORM

BRIEF NOTES ADDITIONAL MATHEMATICS FORM BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht

More information

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

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

More information

Math 4200: Homework Problems

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,

More information

Sum: lim( f + g) = lim f + limg. Difference: lim( f g) = lim f limg. Product: lim( f g) = (lim f )(limg)

Sum: lim( f + g) = lim f + limg. Difference: lim( f g) = lim f limg. Product: lim( f g) = (lim f )(limg) MATH 5: Clculus, SET8 SUMMARIES [Belmonte, 207] 2 Limits nd Derivtives 2.2 The Limit of Function Limit of sclr function: We write lim x f (x) = L nd sy the limit of f (x) s x pproches equls L if nd only

More information

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

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

More information

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. 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

More information

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

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..............................................

More information

Not for reproduction

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

More information

The Trapezoidal Rule

The Trapezoidal Rule 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 Approimte

More information

Chapter 1 - Functions and Variables

Chapter 1 - Functions and Variables Business Clculus 1 Chpter 1 - Functions nd Vribles This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited. Business Clculus 1 Ch 1:

More information

20 MATHEMATICS POLYNOMIALS

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

More information

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

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

More information

Chapter 6: Transcendental functions: Table of Contents: 6.3 The natural exponential function. 6.2 Inverse functions and their derivatives

Chapter 6: Transcendental functions: Table of Contents: 6.3 The natural exponential function. 6.2 Inverse functions and their derivatives Chpter 6: Trnscendentl functions: In this chpter we will lern differentition nd integrtion formuls for few new functions, which include the nturl nd generl eponentil nd the nturl nd generl logrithmic function

More information

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

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

More information

Phys 7221, Fall 2006: Homework # 6

Phys 7221, Fall 2006: Homework # 6 Phys 7221, Fll 2006: Homework # 6 Gbriel González October 29, 2006 Problem 3-7 In the lbortory system, the scttering ngle of the incident prticle is ϑ, nd tht of the initilly sttionry trget prticle, which

More information

Riemann Integrals and the Fundamental Theorem of Calculus

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

More information

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write

More information

Chapter 6. Riemann Integral

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

More information

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

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

More information

Summary of Elementary Calculus

Summary of Elementary Calculus Summry of Elementry Clculus Notes by Wlter Noll (1971) 1 The rel numbers The set of rel numbers is denoted by R. The set R is often visulized geometriclly s number-line nd its elements re often referred

More information

LINEAR ALGEBRA APPLIED

LINEAR ALGEBRA APPLIED 5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order

More information

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

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.

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

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

More information

5: The Definite Integral

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

More information

Shape and measurement

Shape and measurement C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

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

More information

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

More information

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 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.

More information

Section 14.3 Arc Length and Curvature

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

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. 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

More information

Math 113 Exam 2 Practice

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

More information

( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12.

( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12. Stright line grphs, Mied Eercise Grdient m ( y ),,, The eqution of the line is: y m( ) ( ) + y + Sustitute (k, ) into y + k + k k Multiply ech side y : k k The grdient of AB is: y y So: ( k ) 8 k k 8 k

More information

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

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 =

More information

Partial Differential Equations

Partial Differential Equations Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for One-Dimensionl Eqution The reen s function provides complete solution to boundry

More information

Math 554 Integration

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

More information

Math 100 Review Sheet

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

More information

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81 FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first

More information

The Trapezoidal Rule

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

More information

Trignometric Substitution

Trignometric Substitution Trignometric Substitution Trigonometric substitution refers simply to substitutions of the form x sinu or x tnu or x secu It is generlly used in conjunction with the trignometric identities to sin θ+cos

More information

Practice final exam solutions

Practice final exam solutions University of Pennsylvni Deprtment of Mthemtics Mth 26 Honors Clculus II Spring Semester 29 Prof. Grssi, T.A. Asher Auel Prctice finl exm solutions 1. Let F : 2 2 be defined by F (x, y (x + y, x y. If

More information

Distance And Velocity

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

More information

ragsdale (zdr82) HW2 ditmire (58335) 1

ragsdale (zdr82) HW2 ditmire (58335) 1 rgsdle (zdr82) HW2 ditmire (58335) This print-out should hve 22 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc

More information

STATICS VECTOR MECHANICS FOR ENGINEERS: and Centers of Gravity. Eighth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.

STATICS VECTOR MECHANICS FOR ENGINEERS: and Centers of Gravity. Eighth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr. 007 The McGrw-Hill Compnies, Inc. All rights reserved. Eighth E CHAPTER 5 Distriuted VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinnd P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Wlt Oler Tes Tech

More information

New Expansion and Infinite Series

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

More information

FUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x- x1)(x- x2) a y

FUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x- x1)(x- x2) a y FUNCTIONS: Grde 11 The prbol: ( p) q or = +b + c or = (- 1)(- ) The hperbol: p q The eponentil function: b p q Importnt fetures: -intercept : Let = 0 -intercept : Let = 0 Turning points (Where pplicble)

More information

Math 360: A primitive integral and elementary functions

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:

More information

Physics 2135 Exam 1 February 14, 2017

Physics 2135 Exam 1 February 14, 2017 Exm Totl / 200 Physics 215 Exm 1 Ferury 14, 2017 Printed Nme: Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the est or most nerly correct nswer. 1. Two chrges 1 nd 2 re seprted

More information