MATH CALCULUS FALL Scientia Imperii Decus et Tutamen 1

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1 MATH CALCULUS FALL 2018 Scientia Imperii Decus et Tutamen 1 Robert R. Kallman University of North Texas Department of Mathematics 1155 Union Circle Denton, Texas office telephone: office fax: office robert.kallman@unt.edu July 15, Taken from the coat of arms of Imperial College London.

2 FALL 2018 COURSE: MATH , CALCULUS I. PREREQUISITES: MATH 1650 (Pre-Calculus) or {MATH 1600 (Trigonometry) and MATH 1610 (Functions, Graphs and Applications)}. CLASS MEETS: We have GAB 461, MWF, 10:00 a.m. - 10:50 a.m. and GAB 461, TR, 9:30 a.m. - 10:50 a.m. reserved for our use. In despite of the fact that this is a 4 hour class, initially we will meet MTWRF, 10:00 a.m. - 10:50 a.m. in the assigned classroom. An exception to this is that tests will be given on Tuesday or Thursday, 9:30 a.m. - 10:50 a.m. This will give us some flexibility if we need to cancel some future classes and/or if we become ill. Later on these meeting days and time might well be modified. FINAL EXAM DATE AND TIME: The final is scheduled for Saturday, December 8, 2018 in GAB 461, 8:00 a.m. - 10:00 a.m. TEXT: James Stewart, Calculus, Eighth Edition, Cengage Learning, Boston, MA 02210, copyright 2016, 2012, ISBN INSTRUCTOR: Robert R. Kallman, 315 GAB [office], [office telephone], [fax], kallman@unt.edu [ ] OFFICE HOURS: Monday - Friday, 8:00 a.m. - 9:30 a.m. & Monday, Wednesday, 11:00 a.m. - 1:50 p.m. and before and after any class ATTENDANCE POLICY: Mandatory. Specifically for TAMS students: if you are absent for any reason, you are required to file an absence report with Dr. Fleming of the TAMS Academic Office. ELECTRONIC DEVICES: No electronic devices of any sort are to be on let alone used during the class. Repeated flouting of this will result in a grade penalty. HOMEWORK: Homework will be assigned and some designated subset of it will be graded. The designated homework assigned during a given week are to be handed in at the beginning of class on the Wednesday of the following week. Late homeworks will not be accepted under any circumstances. Each homework problem will receive a grade of 0, 1, or 2 points. Failure to turn in a homework set on time will result in a grade of -1 for that homework set. Take the homework seriously, for your performance on the homework will constitute 40 ACADEMIC INTEGRITY: There is no reason why TAMS students should not demonstrate complete academic integrity at all times, particularly on tests. Any transgression of this will result in a grade of zero on the test and a grade of F for the course. Consistent with this policy the instructor will retain xerox copies of a random sampling of all tests. GRADING POLICY: Grades will be based on the total number of points accrued from the assigned graded homeworks, from three in class 80 minute examinations (5 problems plus 1 bonus question), given circa late September, late October and late November, and from an in class 120 minute final (8 problems plus 1 bonus question). I will retain xerox copies of a sample of these tests. The number of points per test problem and final problem will normally be 10. There is no excuse for missing a test and no makeup tests will be given under any circumstances. A student missing a test will receive a grade of -1 on that test. If a 1

3 student is unavoidably absent from a test and makes arrangements with the instructor well before the test date, then the grade assigned to the missing test will be prorated by the student s performance on the final examination minus 10 points. It is difficult a priori to determine the precise break points for the final grades. However, the golden rule in determining the final assigned grade is that if the number of points earned by person A is to the number of points earned by person B, then person A has a grade which is to the grade of person B. The only possible exception is that you must take the final examination and receive a passing grade on the final in order to get a grade greater than F. TOPICS: The topics to be covered can be found in most of Chapter 1 - Chapter 6 plus a few extra topics. It is a rather ambitious goal to cover these topics in some depth. This will require considerable work on the part of the students and the instructor. Some supplementary notes will be handed out. At 5 class days per week we will have 71 class days (plus a reading day and a final day) to cover 42 sections. This should give us plenty of slack for three tests and lots of time for review if we stick to 5 classes per week. APPROXIMATE ITINERARY: The following is a first attempt, very rough approximation to what our schedule will be. This will perhaps be dynamically reconfigured as the semester progresses since it is of course impossible to make such a schedule with hard-and-fast rules, particularly since we are starting the academic year relatively late and ending so early. Test #1, Thursday, October 4, 2018, covering Chapters 1-2. Test #2, Thursday, November 1, 2018, covering Chapters 3-4. Test #3, Tuesday, November 27, 2018, covering Chapter 5. Final, Saturday, December 8, 2018, cumulative, Chapter 1-5. ASK QUESTIONS in class so that we may all benefit. If you need help, it is your responsibility to seek me out. See me during my office hours. Empirical evidence suggests that there is a strong correlation between the amount of work done by the student and his/her final grade. STUDENTS WITH DISABILITIES: It is the responsibility of students with certified disabilities to provide the instructor with appropriate documentation from the Dean of Students Office. STUDENT BEHAVIOR IN THE CLASSROOM: The Powers That Be have strongly suggested that students be given the following statement: Student behavior that interferes with an instructor s ability to conduct a class or other students opportunity to learn is unacceptable and disruptive and will not be tolerated in any instructional forum at UNT. Students engaging in unacceptable behavior will be directed to leave the classroom and the instructor may refer the student to the Center for Student Rights and Responsibilities to consider whether the student s conduct violated the Code of Student Conduct. The university s expectations for student conduct apply to all instructional forums, including university and electronic classroom, labs, discussion groups, field trips, etc. The Code of Student Conduct can be found at 2

4 In other words, cause trouble in the classroom and you will probably be cast into the Darkness and sent to the KGB. 3

5 How to Study for This Class Attend every class. Pay attention in class, take careful notes, and ask questions if needed. The evening of every class go over your classroom notes, list topics on which you have questions or need clarification, read the relevant section of the textbook, do the assigned homework to be graded, look over the additional homeworks to verify that you understand how to do them and make note of those additional homeworks that you do not understand to ask about them during the next class. It is important that you put a great deal of effort into the homework, both those to be turned in and those that are less formally assigned. One becomes adroit at any human activity - e.g., hitting a fast ball, throwing a slider, making foul shots or jump shots, or driving off a tee - only with a great deal of practice. The same applies to calculus. Do not waste your time memorizing endless lists of derivatives and antiderivatives. This in fact is counterproductive. Instead, know a few basic computational techniques (e.g., product rule for differentiation, chain rule for differentiation, sin = cos, etc.), and try to understand the big picture and concepts involved in problem solving. All of the problems encountered in this class should be first approached by asking oneself what is a reasonable way to proceed. Then given the proper path or direction, you can then solve the problems by small, logical steps that inevitably lead one to the final solution. 4

6 Science is the belief in the ignorance of experts. Richard Phillips Feynman (5/11/1918-2/15/1988) 5

7 Section 1.1. Four Ways to Represent a Function. Symbols. = for all or for every, = there exists, = in or an element of, / = not in or not an element of, = such that, = = implied by, = = implies, = if and only. Naive Set Theory. We should be familiar with the primitive notion of set and x A, A = {x S(x)} or A = {a 1, a 2,... a n } (note {x} = {x, x}), the empty set, A B or B A is equivalent to x A = x B, A = B is equivalent to A B and B A which is equivalent to x A x B. A B = {x x A or x B}, A B = {x x A and x B}, A B = {x x A and x / B}, A B = (A B) (B A), the symmetric difference of A and B, if A U, where U is the universal set under discourse, then A c = U A and A B = A B c. We have (A c ) c = A and therefore A = B if and only if A c = B c. We also have (A B) c = A c B c and (A B) c = A c B c. We have the commutative laws A B = B A and A B = B A, the associative laws (A B) C = A (B C) and (A B) C = A (B C), and the distributive laws A (B C) = (A B) (A C) and A (B C) = (A B) (A C). An elementary but quite hard fact to prove is that the symmetric difference operation is associative, viz., (A B) C = A (B C). I ll be quite impressed if you can prove this. Proposition 1. Prove that A = A, A B = B A, A c B c = A B, (A B) c = A c B = A B c, A (B C) = (A B) (A C), and (A B) C = A (B C). This proves that P(U) is a commutative ring with identity with = and + =. Proof: A B = (A B c ) (B A c ). From this formula A = A, A A =, and A B = B A follow easily. A c B c = (A c B cc ) (B c A cc ) = (A B c ) (B A c ) = A B. (A B) c = ((A B c ) (B A c )) c = (A B c ) c (B A c ) c = (A c B) (B c A) = (A c B c ) (A c A) (B B c ) (B A) = (A B cc ) (B c A c ) = A B c. Therefore (A B) c = (B A) c = B A c = A c B. A (B C) = A ((B C c ) (C B c )) = (A B C c ) (A C B c )) = ((A B) (A c C c )) ((A C) (A c B c )) = ((A B) (A C) c ) ((A C) (A B) c )) = (A B) (A C). (A B) C = ((A B) C c ) (C (A B) c ) = ((A B) C c ) (C (A B c )) = (((A B c ) (B A c )) C c ) (((A B) (B c A c )) C) = (A B c C c ) (A c B C c ) (A B C) (A c B c C). This last expression is obviously invariant under permutations of A, B and C. Hence, (A B) C = (C B) A = (B C) A = A (B C). Note that A + = A = A so is an additive identity. Note that A + A = A A =, so A is its own additive inverse. Finally if U is the universal set lurking in the background with respect to which we are taking complements, then A U = A U = A and thus U is a multiplicative identity. It is now simple to combine these observations together with the set theoretic identities already deduced to conclude that P(U) is a commutative ring with identity with * = and + =. Ordered pairs (a, b) = {{a}, {a, b}}. Try and prove the following lemma on your own. 6

8 Lemma 2. (a, b) = (c, d) if and only if a = c and b = d. Proof: If a = c and b = d, then certainly (a, b) = {{a}, {a, b}} = {{c}, {c, d}} = (c, d). Conversely, suppose that {{a}, {a, b}} = (a, b) = (c, d) = {{c}, {c, d}}. First, suppose that a = b. Then {{a}} = {{a}, {a}} = {{a}, {a, a}} = {{a}, {a, b}} = {{c}, {c, d}}, {c} {{a}} = {c} = {a} = c {a} = c = a, {c, d} {{a}} = {c, d} = {a} = d {a} = d = a, so a = b = c = d and the exercise is proved in this case. A symmetric argument shows the exercise follows if c = d. Next, suppose that a b and c d and {{a}, {a, b}} = {{c}, {c, d}} = {a} {{c}, {c, d}} = {a} = {c} or {a} = {c, d}. If {a} = {c, d}, then c {a} = c = a and d {a} = d = a, a contradiction since c d. Thus {a} = {c} = c {a} = c = a. {c, d} {{a}, {a, b}} = {c, d} = {a} or {c, d} = {a, b}. If {c, d} = {a} = d {a} = d = a = c, a contradiction. Hence, {c, d} = {a, b} = d {a, b} = d = a or d = b. If d = a = d = a = c, a contradiction. Therefore d = b. Note that {a, b} (a, b) - this is strange. Note also that (a, b) {a, b}. a is called the first coordinate of (a, b) and b is called the second coordinate of (a, b). The Cartesian product of the sets X and Y, denoted X Y, is defined to be {(x, y) x X and y Y }. Motivate with pictures. Informally, given sets X and Y, a function from X into Y is a rule that assigns to each element x X a unique y Y. In this case X is called the domain of the function. Functions are usually denoted by letters such as f, g, h, F, G, etc. f : x y = f(x). The set f(x) = {y Y y = f(x) for some x X} Y is called the range of f, denoted range(f). More formally, a function (or map or mapping) f from X to Y, denoted f : X Y, is a subset f X Y such that for each x X there is some y Y such that (x, y) f and with the additional property that (x, y 1 ) f and (x, y 2 ) f = y 1 = y 2. In other words, for every x X there is a unique y Y such that (x, y) f. In this circumstance y is said to be the value of f at x X and the notation f(x) = y is equivalent to (x, y) f. There is nothing special about the symbol x, any symbol would do, but x is easier to write than a picture of a tyrannosaurus rex. Functions are therefore identified with their graphs. Recall that f : X Y is into (injective, injection, one-to-one) if f(x 1 ) = f(x 2 ) = x 1 = x 2 and f : X Y is onto (surjective, surjection) if range(f) = Y, i.e., for every y Y there is some x X such that f(x) = y (x, y) f, f is a bijection if f is one-to-one and onto. If so, then f 1 = {(y, x) Y X (x, y) f} is a function, f 1 : Y X. Check the identities f f 1 : Y Y is the identity and f 1 f : X X is the identity. If f : X Y is a function and = A X, the restriction of f to A, denoted by f A, f A : A Y, is defined to be f (A Y ). Thus if a A, then (f A)(a) = f(a). Notice that if f : X Y and g : X Y are two functions and f g, then f = g. Composition of functions. If g : X Y and f : Y Z, then the composition of f with g, denoted f g, is defined by (f g)(x) = f(g(x)) Z, so f g : X Z. Representing functions, polynomial functions, even functions, odd functions, ratinal functions, algebraic functions. For this section, there will be no formal homework assigned. Instead read Section 1.1, make sure you understand compositions, look over the chapter s homeworks and let us know if there is anything interesting there. Nonformal homework: look over problems # 62 and # 63 and volunteer to present on the board. 7

9 Section 1.2. Mathematical Models: A Catalog of Essential Functions. Polynomials, rational functions with a suitable domain, even functions, odd functions, f + g, fg, f/g with a suitable domain, functions defined by tables (as might arise in physics, engineering, economics, experiments). No exponentials, no logarithms. For this section, there will be no formal homework assigned. Instead read Section 1.2, look over the chapter s homeworks and let us know if there is anything interesting there. 8

10 Section 1.3. New Functions From Old Functions. Sums, linear combinations, products, quotients, compositions. Trig Functions. Recall the definition of angles, the definition of the six basic trig functions, the Pythagorean Theorem, the Law of Cosines in both the acute and obtuse angle cases, basic identities, the addition formulas, double angle formulas, half angle formulas, addition formulas for tangent, even and odd. In what follows Z is the set of integers. cos(0) = 1 and sin(0) = 0 cos(π/2) = 0 and sin(π/2) = 1 cos(π) = 1 and sin(π) = 0 cos(3π/2) = 0 and sin(3π/2) = 1 cos(2π) = 1 and sin(2π) = 0 1 = cos 2 (A) + sin 2 (A) 1 + tan 2 (A) = sec 2 (A) 1 + cot 2 (A) = csc 2 (A) cos(a + 2πn) = cos(a) for all n Z sin(a + 2πn) = sin(a) for all n Z cos( A) = cos(a) sin( A) = sin(a) tan( A) = tan(a) cos(π/2 A) = sin(a) sin(π/2 A) = cos(a) cos(a ± B) = cos(a) cos(b) sin(a) sin(b) sin(a ± B) = sin(a) cos(b) ± cos(a) sin(b) 9

11 cos(a + π) = cos(a) and cos(π A) = cos(a) sin(a + π) = sin(a) and sin(π A) = sin(a) tan(a + nπ) = tan(a) for all n Z sin(2a) = 2 sin(a) cos(a) cos(2a) = cos 2 (A) sin 2 (A) = 2 cos 2 (A) 1 = 1 2 sin 2 (A) cos 2 (A) = 1+cos(2A) 2 sin 2 (A) = 1 cos(2a) 2 tan 2 (A) = 1 cos(2a) 1+cos(2A) cos 2 (A/2) = 1+cos(A) 2 sin 2 (A/2) = 1 cos(a) 2 tan 2 (A/2) = 1 cos(a) 1+cos(A) tan(a + B) = tan(a)+tan(b) 1 tan(a) tan(b) tan(a B) = tan(a) tan(b) 1+tan(A) tan(b) tan(2a) = 2 tan(a) 1 tan 2 (A) sin(a + B) + sin(a B) = 2 sin(a) cos(b) sin(a + B) sin(a B) = 2 cos(a) sin(b) cos(a + B) + cos(a B) = 2 cos(a) cos(b) cos(a B) cos(a + B) = 2 sin(a) sin(b) sin(a) + sin(b) = 2 sin( 1 2 (A + B)) cos( 1 2 (A B)) sin(a) sin(b) = 2 sin( 1 2 (A B)) cos( 1 2 (A + B)) cos(a) + cos(b) = 2 cos( 1 2 (A + B)) cos( 1 2 (A B)) cos(a) cos(b) = 2 sin( 1 2 (A + B)) sin( 1 2 (A B)) sin(2a) = 2 tan(a) 1+tan 2 (A) 10

12 cos(2a) = 1 tan2 (A) 1+tan 2 (A) Use the addition laws for the tan function to prove: π/4 = arctan(1/2) + arctan(1/3) π = arctan(1) + arctan(2) + arctan(3) π/4 = 4 arctan(1/5) arctan(1/239) Exercise 1. Graded: #1: Compute cos(π/12); #2: Compute tan(3π/8) These homeworks are due at the beginning of class on Wednesday, September 5, Remember that homeworks make up circa 40% of your grade. Look over the homeworks from Section 1.3 and let us know if there is anything interesting there. 11

13 Section 1.4. The Tangent and Velocity Problems. Limits arise naturally. Simple examples, f(x) = x 2, f(t) = t 3. Exercise 2. Graded: Problem #5, page 50. These homeworks are due at the beginning of class on Wednesday, September 5, Remember that homeworks make up circa 40% of your grade. Look over the homeworks from Section 1.3 and let us know if there is anything interesting there. 12

14 Section 1.5. The Limit of a Function. Average velocity of a particle on the line, the instantaneous velocity. Difference quotients and the slope of a tangent line. Let f be a function defined on some open interval containing c but not necessarily at c itself. The number L is said to be the limit of f as x approaches c, denoted, lim x c f(x) = L, means that the function values f(x) approach L arbitrarily closely as x approaches c, or equivalently, f(x) is close to L whenever x is close to c but x c. Examples: f(x) = 4x + 5, c = 2 = L = 13. c = 3 = L = 7 f(x) = 2x 2 3x + 5, c = 1 = L = 10 f(x) = x3 2x+4 x 2 +1, c = 3 = L = 25/10 = 5/2 If a continuous graph, just slide along the graph f(x) = x2 9 x 3, c = 3 = L = 6 by factoring f(x) = x3 8 x 2, c = 2 = L = 12 by factoring f(x) = 3x 4 if x 0 and f(0) = 10, c = 0 = L = 4 One-sided limits f(x) = x /x for x 0, c = 0 = L = 1 and L+ = 1 f(x) = tan(x), c = π/2 = L = + and L+ = f(x) = sin(1/x) for x 0, c = 0 = L+, L and L do not exist f(x) = sin(x)/x for x 0, c = 0 = L = 1 f(x) = x2 1 2x+2 2, c = 1 = L = 4 f(x) = x sin(x) x 3, c = 0 and numerically estimate L = 1/6 Iif A R, define the characteristic function of A be defined by χ A (x) = 1 if x A and χ A (x) = 0 otherwise. Then lim x c χ Q (x) does not exist for any c R Exercise 3. Graded: #1 compute lim x 1 x 3 1 x 1 ; #2 compute lim x 1 x x 1 13

15 These homeworks are due at the beginning of class on Wednesday, September 5, homeworks make up circa 40% of your grade. Remember that 14