Student Handbook for MATH 3300

Transcription

1 Student Hndbook for MATH If people do not believe tht mthemtics is simple, it is only becuse they do not relize how complicted life is. John Louis von Neumnn Contents: Deprtment of Mthemtics Ohio University Jnury 208. Review the Student Hndbook for MATH 230 p Mteril You Should Know Before 3300 p Syllbus for 3300 p Assigned Problems for MATH 3300 p. 6 Course Web Site: O.U. Mtlb Web Site: Clculus Coordintor: Kelly Bubp, bubp@ohio.edu Mth Deprtment Office: Morton 32, mthemtics@ohio.edu,

2 Review the Student Hndbook for MATH 230 Welcome to MATH 3300 Clculus III t Ohio University. We hope tht you excel. If you did not tke Clculus I or Clculus II t Ohio University, you need to get the Student Hndbook for MATH 230 nd review it crefully. If you did tke Clculus I or II here, then you need to review the Hndbook for MATH 230. It contins mong other things: Policy on Students with Disbilities nd Policy on Acdemic Integrity. Some generl dvice bout studying Clculus. Mteril you should know before studying Clculus. Study Aids & instructions bout Mtlb Assignments. 2 Mteril You Should Know Before MATH 3300 Mth is cumultive. To help you succeed in Clculus III these pges summrize the mteril from Clculus I & II. Your Mth 3300 instructor will tke for grnted tht you know nd cn use the following mteril. You should review it before strting the course nd before ech test. If you re new t Ohio U. or did not tke MATH 230 nd 2302 here you should obtin copies of the Student Hndbook for MATH 230 & Student Hndbook for MATH 2302 from the course web site nd red them crefully. If you lredy hve copies of the hndbooks for MATH 230 nd 2302 then you should review them now. Pre-Clculus Mteril As generl rule you should understnd nd be ble to use ll the mteril in reference pges - 4 in the inside front nd bck covers of the textbook. You should memorize most of the formuls on those pges. Pge 4 of the Student Hndbook for MATH 230 contins more detiled informtion. Limits nd Derivtives. See the Student Hndbook for Definitions relted to Integrtion: F (x) is n Antiderivtive of f(x) mens: F (x) = f(x) Riemnn Sum - R n, L n nd M n re exmples. Definite Integrl - The limit s n of ny Riemnn sum. Averge of Function: f vg = b f(x) dx b Integrtion Theorems: If f(x) is continuous on [, b] then b f(x) dx exists, however, it might not be expressible in terms of elementry (usul) functions. Fundmentl Theorem of Clculus: If f is continuous, nd F is n ntiderivtive of f, then d x b Prt : f(s)ds = f(x). Prt 2: f(x) dx = F (b) F (). dx 2

3 Riemnn sums nd numericl integrtion: Let (x 0, x, x 2,..., x n ) be evenly spced, = x 0, b = x n, x = x i x i = (b )/n Let (y 0, y, y 2,..., y n ) be vlues of f(x), i.e. y i = f(x i ) n left sum - L n = x y i = x(y 0 + y y n ) right sum - R n = x i=0 n y i = x(y + y y n ) i= trpezoid rule - T n = x/2 (y 0 + 2y + 2y y n + y n ) Simpson s rule - S n = x/3 (y 0 + 4y + 2y 2 + 4y y n 2 + 4y n + y n ) midpoint sum/rule - M n = x (f( x ) + f( x 2 ) f( x n )) where x i = (x i + x i )/2, i.e. the mid-points of the intervls: [x i, x i ]. L Hopitl s rule: f(x) lim x g(x) = lim x f (x) g (x) if the first limit is 0 0 or nd the 2nd limit exists. Chnge 0 nd to 0 0 or. Use ln for 00, 0, Integrls to memorize: u n du = un+ + C, n. n + du = ln u + C u e u du = e u + C cos u du = sin u + C sin u du = cos u + C du = u 2 sin u + C + u 2 du = tn u + C sec 2 u du = tn u + C sec u tn u du = sec u + C sinh u du = cosh u + C cosh u du = sinh u + C f(x) + g(x) dx = f(x) dx + g(x) dx Three mjor integrtion techniques: Substitution: Recognize f(g(x)) g (x) dx, set u = g(x), nd get: f(u) du. By prts: u dv = uv v du, first identify dv tht cn be integrted. Prtil frctions: First reduce by dividing Rel roots: (x + b) n A x + b + A 2 (x + b) A n (x + b) n Complex roots: x 2 + bx + c Ax + B x 2 + bx + c Arc Length: Prmetric: x = f(t), y = g(t), α t β: L = Polr: if r = r(θ), L = β α r 2 + (dr/dθ) 2 dθ β α f (t) 2 + g (t) 2 dt 3

4 Vectors: Length: ū = u 2 + u2 2 + u2 3 Addition: ū + v = u + v, u 2 + v 2, u 3 + v 3 Multipl. by constnts: cū = cu, cu 2, cu 3 Unit vectors: ū =, ū = v/ v Coordinte unit vectors: î, ĵ, ˆk ū v = u v + u 2 v 2 + u 3 v 3 (sclr) ū v = det î ĵ ˆk u u 2 u 3 v v 2 v 3 ū v is vector. proj v ū = ū v ū 2 ū (vector) comp v ū = ū v ū (sclr) Geometry ū v iff ū = c v ū v = ū v cos θ ū v iff ū v = 0 ū v = ū v sin θ ū v is perpendiculr to both ū, v A = ū v = re of prllelogrm V = ū v w = volume of prllelepiped Algebr of vectors ū + v = v + ū ū + ( v + w) = (ū + v) + w ū + 0 = ū ū + ( ū) = 0 c(ū + v) = cū + c v (c + d)ū = cū + dū (cd)ū = c(dū) v v = v 2 ū v = v ū ū ( v + w) = ū v + ū w (cū) v = c(ū v) 0 ū = 0 ū v = v ū (cū) v = c(ū v) ū ( v + w) = ū v + ū w (ū + v) w = ū w + v w ū ( v w) = (ū v) w ū v w = (ū w) v (ū v) w Physics: Work: W = F D (constnt force & liner motion) Torque: τ = r F 4

5 3 Syllbus for 3300 Text: Essentil Clculus with Erly Trnscendentls, Jmes Stewrt, 2nd edition. Review of Conic Sections* 0.5 Equtions of Lines nd Plnes 0.6 Cylinders nd Qudric Surfces 0.7 Vectors Functions nd Spce Curves 0.8 Arc Length nd Curvture** 0.9 Motion in Spce. Functions of Severl Vribles.2 Limits nd Continuity.3 Prtil Derivtives.4 Tngent Plnes nd Approximtion.5 The Chin Rule.6 Directionl Derivtives nd Grdient.7 Mximum nd Minimum Vlues.8 Lgrnge Multipliers 2. Double Integrls Over Rectngles 2.2 Double Integrls On Generl Regions 2.3 Double Integrls in Polr Coord. 2.4 Applictions of Double Integrls 2.5 Triple Integrls 2.6 Triple Ints. in Cylindricl Coords. 2.7 Triple Ints. in Sphericl Coords. 2.8 Chnges of Vribles in Integrls 3. Vector Fields 3.2 Line Integrls 3.3 Fund. Theorem for Line Integrls 3.4 Green s theorem 3.5 Curl nd Divergence 3.6 Prmetric Surfces 3.7 Surfce Integrls 3.8 Stokes Theorem 3.9 The Divergence Theorem * Note tht Conic Sections re no longer covered in MATH 300 ** Skip Norml nd Binorml Vectors. This is TAGS (Ohio Trnsfer Assurnce Guides) course nd the bove topics follow closely the mteril prescribed by TAGS. 5

6 4 Homework Problems for MATH 3300 Section Problems odd, 6, 7-39 odd odd, odd, 7-22, 23, 33-5 odd, 57-6 odd 0.8-5, -9 odd, ML* Spce Curves odd, 5-25 odd. -2 odd, odd, 4-46, 5; ML2* Functions odd, 23, 25, 27, 29, 30; ML3* Contour Plots.3 3, 4, 7-23 odd, 37-40, 43-5 odd; ML4* Prtil Deriv..4, 3, 5, -4, 7, 25, 27, odd, 7, 22, 23, 25, 27, , 22, , 37, 38, 44; ML5* Grdients.7-3 odd, odd, 3-43 odd.8-5 odd, 6, 7, 38, 39; ML6* Lgrnge Multipliers 2., 3, 4, 7-3 odd odd, 7-25 odd, 3-4 odd 2.3-6, 7-27 odd; ML7* Double Integrls odd, 43, 45; ML8* Approximte Integrls 2.6-3, 5-27 odd 2.7-0, -27 odd, 35, 36, 40; ML9* Volume in 4-spce odd 3., 3, -8, odd, 6, 27, 28, 33, 35, odd, 27, odd, 7, 9, odd 3.6, 3, -4, 5-2 odd, 29, odd odd, 7-27 odd, 37, odd, odd, 7, 23, 27 * - optionl, but your instructor might ssign some problems. 6