DOCUMENT RESUME ED 033 667 JC 690 400 AUTHOR Capper, Michael R., Comp. TITLE Instructional Objectives for a Junior College CCurse in. Calculus and Analytical Gecmetry. INSTITUTION California Univ., Los Angeles. ERIC Clearinghouse for Junior Coll. Information. Pub Date Ncv 69 Note 77p. ELKS Price EDRS Price ME-A0.50 HC Not Available from EDRS. Descriptors *Analytic Gecnietry, *Behavioral Objectives, *Calculus, *Junior Colleges Abstract See JC 690 392 above. [Not availablein hard copy because of marginal reproducibilitycf original.] ikELIA..421,11,..541,11)42f U.S. DEPARTMENT OF HEALTH, EDUCATION & WELFARE OFFICE OF EDUCATION THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROM THE . PERSON OR ORGANIZATION ORIGINATING IT.POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILY REPRESENT OFFICIAL OFFICE OF EDUCATION POSITION OR POLICY. INSTRUCTIONAL OBJECTIVES FOR A JUNIOR COLLEGE COURSE IN CALCULUS AND ANALYTICAL GEOMETRY Compiled by Michael R. Capper ERIC CLEARINGHOUSE FOR JUNIOR COLLEGES University of California Los Angeles, California 90024 November 1969 +4, 4.+3 i4:41 .,++ +++, +++ fr. UNIT I FUNDAMENTAL CONCEPTS OFALGEBRA An introduction to modernmathematics is presented, followed by discussion ofthree basic algebraic concepts: sets, relations, andoperations. All behaviors will be performed under testconditions: in class, without open books or notes, and allwork is to be shownwithout scratch paper. I Goal: The student willunderstand the basic elements of set theory. Objective: 1. The student willdefine what is meant by A = B where A and B aresets. 100 Sample item: Define what is meant by A = B where A and B aresets. 2. Given a list of pairs of sets,the student will relate the sets,A and B, in each pair by showingwhether A = B, ALB, BeA, ALB, orBcA. 66 Sample item: Write 1 if A= B 2 if AFB 3 if BszA 4 if A cr-B 5 if B4=A 6 if none of the above are true a. A =fintegersi B =(integersgreater than zeros b. A =;integralmultiples of 3.1 B {integralmultiples of 2j . Giventwo sets, Aand B, the student Objective: 3. will inacatethe elements of A UB and A /1 B. 80 Sample item: Indicate the unionand intersection of thefollowing sets: a. A = 11,3,61 B =13,5,7} b. A = tmegativeodd numbers) B = integers divisible by-lj 4. Given two orthree sets, thestudent will draw a Venndiagram representing the sets andgiven expressionsinvol- ving the sets. 66 Sample item: If U is the setof inte- gers and A =11,2,3i B = (1,2,3,4,53 C = (3,4,5,6,71 draw a Venndiagram representing A,B, and C and cross-match neatly the area representingA u(B/1 0). concepts II Goal: The studentwill understand the of cross- product,relation, and equivalence relation. Objective: 5. Giventwo sets, S andT, the student SxS, SxT, and TxS. will form 90 Sample item:Let S =40,1,21and T = 4-1,0,1i. ;:zorm SxS, SxT,and rl'L S. 6. The studentwill define termstaken from the followinglist: ordered pair, product set(cross-product), relation, reflexive, symmetric,transitive, equivalence relation. 100 Sample item: Define the termsymmetric in terms of arelation R. will identifywhich pro- Obiective: 7. The student perties of aneklivalence relation for a givenrelation. are true 66 Sample item: Given a relation =f(a,a),(a,b)1(a,c),(b,b),(bla),(c,01 on a setS = fa,b,c1,identify those properties of anequivalence relation which are truefor R. will understandthe terms III Goal: The student distributive. commutative,associative, and student willdefine the terms ra Objective: S.- The distri- commutative,associative, and butive. 100 with Sample item: Define commutative respect toaddition ofintegers. will understandthe concept of IV Goal: The student it to anoperation and beable to apply groupproperties. will define termstal:en Objective: 9. The student from thefollowing list: operation, inverse of anelement. identity element, 100 Sample item: Define the termoperation. student will 10. Given anoperation, the determine whetherthat operationis commutative,associative, ordistri- butive. 66 Sample item: If a*b meanstake a+b+2 where a andb are integersand * is anoperation defined onI, is * commu- tative? associative?distributive? Given a tableof operations,the student 11. whether the setis a will determine the group. If the setis not a group, student willidentify thoseproperties which the setlacks to makeit a group. 75 tables, Sample item: Given the following ) «mom in each case indicate whether ornot the set A =l.avb,c1 is a group. If the set is not a m-oup,.indicatewhich property or properties arelacking. a. O ab c a -a cb bcba bac b. abc a bca b cab a bc 4 UNIT II THE REALNUMBER SYSTEM properties of thereal numbers This unitdiscusses subsequent areasof study. which areprerquisite for behaviors will beperformed Unless otherwisenoted, all without open books or under testconditions: in class, withoutscratch paper. notes, and allwork is tobe shown will be able todemonstrate I Joel: The student that a given setis a field. permitted, the At homewith open books Ob'ective: 1. show that the setof student will ordered rational numbers,Q, defined as pairs, is afield by provingall the for Q. field properties 95 will be able toutilize the I/ Goal: The student absolute rules governinginequalities and values for realnumbers. will indicatewhether the Objective: 2. The student true given mathematicalsentences are orfalse andthen either prove or respectively. give acounterexample, 90 the Sample item: Indicate whether following is true orfalai; if true, proveand if false,give a counterexample; If a4b40,thenb2 ca2. student will Objective : 3. Given anexpression, the indicate theabsolute value ofthat expression. 90 Sample item: Evaluate thefollowing: a. 13 -51 b. 114-x1 if y"0"x will understarydthe representa- III Goal: The students line. tion of thereal numbersby a numbr will define termsfrom the Objective: 4. The student following listusing setnotation: interval, half- openinterval, closed openinterval. Sample item: Define the term open interval usingset notationand indicate thesymbol for the open interval from-3 to 8. will provegiven statements Oblective: 5. The student the concerning theconcept of length on real line. 80 Sample item: Show that lb-al<la-cl if a4b4c. of real numbers,the student, Objective: 6. Given a set will sketchits graph on thereal number line. 90 of the Sample item: Sketch the graph following sets: a.lx/Ix143.1 b. inineli3 "Cr^ , UNIT III ALGEBRAIC EXPRESSIONS' proficiency in manipulativeskills'is necessary for the application ofmathematical principles. This unit discusses types of algebraicexpressions and the manipu- lation of suchexpressions. All behaviors will be performed under testconditions: in class, without open books or notes, and allwork is to be shown without scratch paper. I Goal: The student will gainproficiency ih manipulating expressions involvingintegral and rational exponentsand radicals. Ittketive: 1. The student will simplify alist of expressions involving integral exponents. 70 Sample item: Simplify the following and remove all negative exponents: b.(stur a . 2.-. 1113 y (st) u"4 Objective: 2. The student will simplify alist of expressions involving rational exponents. 70 Sample item: Simplify the following and remove all negative exponents: in a. (a1-4)1)* b. ab--)w 2 b15 t Objective: 3. The student will simplify alist of expressions involving radicalsand rational exponents. 70 Sample item: Simplify the following: a.34x6(16x8)* b.3 x1375- II Goal: The student will be able toperform operations with and factorpolynomials. Objective: 4. The student will perform theindicated operation and find the degreeof the resulting polynomial. 70 Sample item: Perform the indicated operation and find the degree of the resulting polynomial: a.(x3+3x+2) (x2.3x+5) b.6x2 + 4x 2:c Objective: 5. The student will factor eachof the given polynomials by completingthe square. 75 Sample item: Factor by completing the square: a.16x2 + 64x-80 b. a2 + 8a + 1 Objective: 6. The student will factor eachof the given polynomials by any method. 75 Sample item: Factor by any method: a.16x4-y8 b.x3+x2+ x + 1 Objective: 7. The student will perform thegiven operation and simplify the givenrational integral algebraic expressions. 70 Sample item: Simplify thefollowing: b a a. 1 + 1 b. 417 a2 b--2- a x 4.1 UNIT IV EQUATIONS ANDINEQUALITIES and inequalitiesis one of The ,solvingof equations alge- in the studyof elementary the moreimportant tasks mathematics to otherfields bra. Manyapplications of interpreting a are based onthe conceptof finding or other-knownparameters. All single variablein terms of conditions: in behaviors willbeperformed'Under test and all workis to be class, withoutopenbooks or notes, shown withoutscratch paper. will be able tosolve linear I Goal: The student equations andapplicationsinvolving linear equations. list of lin- 1. The studentwill solve a Ob'ective: equations in onevariable. ear 85 solution set Sample item: Give the for eachof thefollowingequations: a. 6x +3 = 4 b. 2/(4-x) =1/(2x-8) of formulas,the student Objective: 2. Given a list will solveeach for theindicated of the othervariables. variable in terms 85 Sample item: Solve fortheindicated variable interms of theremaining vari- ables in thefollowing equations: a. p =2(1 + w) for1 b. F =(9/5)C 32 for C 3. Given au aipplication (word problem) , th.4 student will translate the problem into mathematical symbols involving a linear equation. 66 Sample item: Translate the following problems into mathematical symbols. Do not solve the problems. a. If a car travels 18 miles on a gallon of gas, how far will a $1*00 purchase ahe him if one gallon costs 320? Objective : 4. Given al. application, the student will solve for the unkl.ovn variable. 60 Sample 1..:31m: Solve the following problem: The sum of tU of a 2-digit number ia 13, the difference equals 5. Wat is the numlc,c? II Goal: The student will be able to use the quadra-' tic formula to solve quadratic equations-and applications involving quadratic equations.