MSCI 300 FALL 2015 Calculus 1 Charles Rubenstein
MSCI 300 – FALL 2015 Calculus 1 Charles Rubenstein, Ph. D. Professor of Engineering and Information Science Week 1: Preface, Session 1: Monday 08/24/15 Mondays 6: 30 pm-8: 50 pm PMC 403
- SILS Math 300 Charles Rubenstein, Ph. D. Professor of Engineering and Information Science Calculus 1 http: //www. Charles. Rubenstein. com/300 - Preface Copyright © 2015 C. P. Rubenstein 2
Who is Dr. Rubenstein ? • Subject Background in – – Bioengineering Electrical Engineering Systems Analysis Information Science • Certifications – Microsoft Trainer; – Comp. TIA A+ Certified • Professional Society Memberships – ALISE (Member), IET (Fellow, Chartered Engineer) – IEEE (Senior Member) Member of the 2010 -2011 Board of Directors of this 430, 000 member professional engineering organization that produces more than one third of the world’s electrotechnology information Copyright © 2015 C. P. Rubenstein 3
My wife Rose and our three sons… Adam Scott Jaron Copyright © 2015 C. P. Rubenstein 4
Scott & Liz, Jaron & Steph, Rose & Me and Adam Scott Jaron Copyright © 2015 C. P. Rubenstein 5
Our two Grandchildren Alyssa Michelle and Noah Meir And our Grand. DOGter, Kona Copyright © 2015 C. P. Rubenstein 6
My Style of Teaching • • • Lectures Interactive Lectures = Class participation Design & Technique Demonstrations Individual and Group Assignments Review of Assignments Copyright © 2015 C. P. Rubenstein 7
- SILS Math 300 Charles Rubenstein, Ph. D. Professor of Information Science Session 1 Calculus 1 Copyright © 2015 C. P. Rubenstein 8
Syllabus Review and Highlights Copyright © 2015 C. P. Rubenstein 9
Instructor Contact Information Dr. Charles Rubenstein <crubenst@pratt. edu> Professor of Engineering & Information Science Pratt Manhattan Campus Office: PMC 604 -C Fall 2015 Office hours (by appointment *) • Mondays: 5: 00 pm-6: 00 pm = PMC 604 -C • Tuesdays: 12: 00 pm - 2: 00 pm = ARC G-45 (*Please email me at least a day in advance if you plan on coming to office hours…) Send me an email … crubenst@pratt. edu Subject line: 300 Calc Copyright © 2015 C. P. Rubenstein 10
About Your Grades Assignments are designed to see what you have learned - NOT to see what you DIDN’T learn 1. Homework Quiz value = 25% of final grade – HOMEWORK IS DUE NEXT CLASS SESSION – There will be a five (5) minute quiz on one homework problem for each of ten (10) homework assignments. A set of ALL homework assignments will be provided the first class session. There is NO excuse for not doing homework, quiz. 2. Total Exams value = 75% of final grade – Take Home Midterm = 25% – Final Exam = 50% Copyright © 2015 C. P. Rubenstein 11
Attendance Requirements You are responsible for attending and participating in all class sessions. TWO (2) unexcused absences (Dr. note, etc. ) will result in ONE grade lower final grade (e. g. , you earn a B+, get a C+) THREE (3) unexcused absences (Dr. note, etc. ) will result in the final grade of “F” Copyright © 2015 C. P. Rubenstein 12
Recommended: TI-83 plus Graphing Calculator Any graphing calculator may be used in this class The CM/FM Department has TI-83 Plus units for semester loan. Please See Phil Ramus (pramus@pratt. edu) in CM/FM Office. Copyright © 2015 C. P. Rubenstein 13
Required Textbook* Pub Date: 1991 Publisher: Wellesely-Cambridge Press *FREE Online at this Web. Link: http: //ocw. mit. edu/ans 7870/resources/Strang/Edited/Calculus. pdf Calculus Strang, Gilbert Copyright © 2015 C. P. Rubenstein 14
Math 300 – Chapter Topics Strang, Gilbert. Calculus (1991). Wellesely-Cambridge Press. http: //ocw. mit. edu/ans 7870/resources/Strang/Edited/Calculus. pdf First SIX (6) Chapters ONLY *: CHAPTER 1 : Introduction to Calculus CHAPTER 2 : Derivatives CHAPTER 3 : Applications of the Derivative CHAPTER 4 : The Chain Rule CHAPTER 5 : Integrals CHAPTER 6 : Exponentials and Logarithms *You are expected to be reading at least the chapter we will be working on in class, and doing chapter problems as needed. Copyright © 2015 C. P. Rubenstein 15
Fall 2015 Math/Science TUTORING WHO: TBD WHERE: Pratt Brooklyn Writing & Tutorial Center (North Hall across from Bank) WHEN: TBD BY APPOINTMENT ONLY: TBD or email: TBD Call: Copyright © 2015 C. P. Rubenstein 16
MSCI 300 – Fall 2015 - Class Schedule & Due Dates (* Quizzes on Homework due; Reviewed in same session) Copyright © 2015 C. P. Rubenstein 17
In Today’s Class: • • 2 Do: Review of Syllabus 2 Do: Review Algebra and Trigonometry Distribute: Homework Sets #01 -10 (also online) Distribute: Class Notes Set #01 -#07 (also online) For our next class – Session 2: • • DUE: Homework Set #01 Reading: Strang, Calculus Chapter 1: Introduction to Calculus 2 Do: Functions and Slopes Quiz #01: Homework #01 example Copyright © 2015 C. P. Rubenstein 18
Class Session Archives http: //www. Charles. Rubenstein. com/300/ 15 fa 300_HANDOUTS_Part 1. pdf Which includes: MSCI 300 syllabus 01 HWK. pdf – 14 HWK. pdf (Homework Sets #11 -14 optional) Class Notes 01 Notes. pdf – 07 Notes. pdf 15 fa 01. pdf (Class Power. Point slides) * 15 fa 01 h. pdf (slides in handout format) * *Archive materials normally online by Thursday evenings Copyright © 2015 C. P. Rubenstein 19
Questions? Copyright © 2015 C. P. Rubenstein 20
Algebra & Trigonometry Review R. 1 Algebraic Notation and Rules R. 2 Products of Binomials (FOIL) R. 3 Extending FOIL R. 4 Fractions and Factors R. 5 Equations and Identities R. 6 Manipulating Equations R. 7 Solving Linear Algebraic Equations R. 8 Quadratic Equations and Quadratic Formula R. 9 Rules of Exponents Copyright © 2015 C. P. Rubenstein 21
Algebra and Trigonometry Review Copyright © 2015 C. P. Rubenstein 22
Expressions Examples of Algebraic Expressions: 2 + x 3 + x 2 z 3 + 4 x – y + 3 Each of these expressions contain TERMS Copyright © 2015 C. P. Rubenstein 23
Expressions Examples of Algebraic Expressions: (2 + x) (3 + x 2) + This expression contains two terms: (2 + x) (3 + x 2) and Both are products of two FACTORS = factors: z 3 + 4 and 1/(x – y + 3)2 Copyright © 2015 C. P. Rubenstein 24
Algebraic Notation Multiplication is indicated by ‘adjacency’: 3 ax = 3 times a times x Roman Numeral adjacency: CLV = C + L + V = 100 + 5 Decimal number adjacency (multiplication & addition): 625. 4 = (6 × 100) + (2 × 10) + (5 x 1) + (4 × 1/10) Copyright © 2015 C. P. Rubenstein 25
Rules of Arithmetic and Algebra 1. Commutative law of addition: a + b = b + a 2. Commutative law of multiplication: a • b = b • a 3. Associative law of addition: a + b + c = a + (b+c) = (a+b) + c 4. Associative law of multiplication: (a • b) c = a • (b • c) 5. Distributive law of multiplication: a (b+c) = ab + ac Copyright © 2015 C. P. Rubenstein 26
Commutative Law of Addition 1. Commutative law of addition: a + b = b + a Perhaps the simplest of rules, factors add the same regardless of the order in which you add them… 2 + 3 = 3 + 2 = 5 5 + 25 = 25 + 5 = 30 1 + 2 + 3 + 4 + 5 = 5 + 4 + 3 + 2 + 1 = 15 Copyright © 2015 C. P. Rubenstein 27
Commutative Law of Multiplication 2. Commutative law of multiplication: a • b = b • a 2 • 3 = 3 • 2 = 6 5 • 25 = 25 • 5 = 125 1 • 2 • 3 • 4 • 5 = 5 • 4 • 3 • 2 • 1 = 120 Copyright © 2015 C. P. Rubenstein 28
Associative Law of Addition 3. Associative law of addition: a + b + c = a + (b+c) = (a+b) + c 2 + 3 + 4 = 2 + (3 + 4) = (3 + 2) + 4 = 9 5 + 25 + 10 = 5 + (25 + 10) = (5 + 25) + 10 = 40 (1 + 2) + (3 + 4) + 5 = (5 + 4) + (3 + 2) + 1 = 15 Copyright © 2015 C. P. Rubenstein 29
Associative Law of Multiplication 4. Associative law of multiplication: (a • b) c = a • (b • c) (2 • 3) 4 = 2 (3 • 4) = 24 (5 • 25) 10 = 25(10 • 5) = 1250 (1 • 2) • (3 • 4) • 5 = (5 • 4) • (3 • 2) • 1 = 120 Copyright © 2015 C. P. Rubenstein 30
Distributive Law of Multiplication 5. Distributive law of multiplication: a (b+c) = ab + ac 4 • (2 + 3) = 4 • 2 + 4 • 3 = 20 10 • (5 + 25) = 10 • 5 + 10 • 25 = 300 2 ( 3 + 4 ) = 6 + 8 = 2 ( 7 ) = 14 A common error is to write a (b + c) = ab + c Do you see why? Copyright © 2015 C. P. Rubenstein 31
Distributive Law of Multiplication: Areas Proof of Distributed Law: Products of two binomials using Areas (a)(b+c) = ab + ac (a+b)(c+d) = ac + ad + bc + bd The order of these operations is: FOIL (first, outer, inner, last) Copyright © 2015 C. P. Rubenstein 32
Special Cases of FOIL to Remember 1. Sum squared: (a+b)2 = a 2 + 2 ab + b 2 (a+b)2 = (a+b) = a 2 +ab +ba +b 2 = a 2 +2 ab +b 2 2. Difference squared: (a-b)2 = a 2 - 2 ab + b 2 (a-b)2 = (a-b) = a 2 - ab - ba +b 2 = a 2 - 2 ab +b 2 3. Sum times difference: a 2 - b 2 = (a +b)(a-b) a 2 - ab + ba - b 2 = a 2 - b 2 Copyright © 2015 C. P. Rubenstein 33
Examples of FOIL 1. FOIL used to multiply (10 + 1) • (10 + 5) = 10 • 10 + 10 • 5 +1 • 10 + 1 • 5 = 165 same calculation as 11 • 15 = 165 2. FOIL to multiply Roman numerals for 11 • 15: XI • XV = (X + I) • (X +V) = (X • X) + (X • V) + (I • X) + (I • V) Which yields = C + L + X + V = CLXV Copyright © 2015 C. P. Rubenstein 34
Extending FOIL 1. Binomial × Trinomial: (a+b) (c + d + e) = ac + ad + ae+ bc+ bd+ be 2. Binomial Cubed: (a+b)3 = a 3 +3 a 2 b +3 ab 2 +b 3 OR: (a+b)3 = (a+b) • (a+b)2 = (a+b) • (a+b) = (a+b) • (a 2 +ba +ab +b 2) = (a+b) • (a 2 +2 ab +b 2) = a 3 +2 a 2 b +ab 2 +a 2 b +2 ab 2 +b 3 = a 3 +3 a 2 b +3 ab 2 +b 3 Copyright © 2015 C. P. Rubenstein 35
Pascal’s Triangle and FOIL 3. Higher Power of a Binomial (using the Pascal Triangle) Formulas for the square, cube, and higher powers of a binomial are found by inspection of the Pascal Triangle: (0) 1 (0) The numbers in the Pascal Triangle give the n coefficients of the expansion of (a + b) 1 1 1 2 1 1 3 3 1 Example: (a + b)4 = 1 4 6 4 1 a 4 + 4 a 3 b + 6 a 2 b 2 + 4 ab 3 + b 4 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 Each number in the triangle is the sum of the two numbers above it. Copyright © 2015 C. P. Rubenstein 36
Extending FOIL 4. Trinomial Squared (a + b + c)2 = a 2 +b 2 +c 2 +2 ab +2 bc +2 ac 5. Any Polynomial Squared The square of any polynomial is the sum of the squares of each term plus twice every possible unordered cross term Example: (a+b+c+d)2 = a 2 + b 2 +c 2 +d 2 + 2 ab +2 ac +2 ad + 2 bc +2 bd + 2 cd Copyright © 2015 C. P. Rubenstein 37
FOIL Computational Tricks Squaring Integers that end in ‘ 5’ Pick a number that ends in a 5 and I will give you the Square of it WITHOUT using a calculator… Note the patterns: 152 = 225, 252 = 625, 352 = 1225, 452 = 2025, 552 = 3025, 652 = 4225, 752 = 5625, and 852 = 7225. . . The ‘Trick’ Each square ends in 25 and the number to the left of the 25 is the product of n and n+1 for (10 n+5)2 so if we wanted 952 we would multiply 9 times 100 = 9, 000 and add the last two digits: 9, 025 Copyright © 2015 C. P. Rubenstein 38
FOIL and Computational Tricks Squaring Integers that end in ‘ 5’ e. g. , (10 n+5)2 Why does this work? Just carry out the algebra using the general equation: (n 5)2 = (10 n+5) (10 n+5) = 100 n 2 + 100 n + 25 (n 5)2 = 100 n(n+1) + 25 Last time: (125)2 = (12 x 13 x 100) + 25 156 x 100 + 25 = 15, 625 For homework #01 extra credit: Find the trick to find the squares of the numbers 51, 52, … 59. Copyright © 2015 C. P. Rubenstein 39
Fractions Rule of “ 1” e. g. , (a/a = 1) You can multiply a fraction by a/a without changing its value since a/a =1. Example: where a = 7/5 a This is a drawn out way to illustrate the rule you already know: to divide by a fraction, multiply by the reciprocal of the fraction : a / (b/c) = a • c/b = ac/b Copyright © 2015 C. P. Rubenstein 40
Simplifying Products of Factors Any expression that is a product of factors can be written as a fraction where the numerator and denominator are both products of factors: Example: = Note that the two factors in the left hand expression are: 1 / y and ax 2 /b Copyright © 2015 C. P. Rubenstein 41
Equations are mathematical sentences: the numerical value of the right side is the same as the numerical value of the left side Conditional Equations, like 2 x - 6 = x + 4 hold true for only one value or condition of x: x=10 Unconditional Equations (usually called an identity) Example: x 2 + x - 6 = (x-2) • (x+3) holds true for any value of x! While the left and right hand sides are different expressions, they produce identical numerical values for any value of x. Copyright © 2015 C. P. Rubenstein 42
Identities Some of the most commonly used identities include (x + 1)2 = x 2 + 2 x +1 (x - 1)2 = x 2 - 2 x+1 x 2 - y 2 = (x+y) • (x-y) These were noted earlier, and sin 2(x) + cos 2(x) =1 Copyright © 2015 C. P. Rubenstein 43
Manipulating Equations Think of the left and right sides as weights balancing a seesaw. If you add equal weights to each side, balance is maintained, i. e. , the equation is still true. In fact, any operation, if applied to both sides of an equation maintains the equality. Examples: a. x + 3 = 2 x - 5 x = 2 x - 8 [Add -3 to each side ] b. (x + 3)-1 = (2 x – 5)-1 [Take the reciprocal of each side] c. x +3 = 2 x - 5 (x+3)n = (2 x – 5)n [Raise each side to n] d. If a + b = c + d and e + f = 3 g then d 1. (a + b) + (e + f) = (c + d) + 3 g [Adding equations] d 2. (a + b) • ( e + f) = (c + d) • (3 g) [Multiplying equations] d 3. (a + b) / (e + f) = (c + d) / (3 g) [Dividing equations] Copyright © 2015 C. P. Rubenstein 44
Solving Linear Equations Linear equations with one unknown, x, can appear in the form: ax +b = cx +d The unknown appears only to the first power ( x 1 = x) Rearranging we get (a-c)x = d-b. Dividing both sides by (a-c) we find x = (d-b) / (a-c) Sometimes a linear equation is disguised, as in Cross multiply and this becomes 2(2 x+5) = x - 3 or 4 x+10 = x - 3, a linear equation. Note that the cross multiplication is equivalent to multiplying both sides of the equation by (x-3)(2 x+5). Instead of cross multiplication, you can invert each side. Copyright © 2015 C. P. Rubenstein 45
Quadratic Equations Quadratic equations contain x 2 (x to the second power) and generally also contain x (to the first power). Example: x 2 - 5 x + 6 = 0 How do we solve this? We might rearrange it as x 2 = 5 x - 6 and then take the square root of each side, producing Have we solved the equation? No, this expression gives x in terms of x. We would have to know x to calculate x. Solving by factoring: The equation, x 2 - 5 x + 6 = 0, can be factored into the form (x-2) • (x -3) = 0 (Verify by using FOIL) The product of two factors can be zero if and only if one or both of the factors is zero. In this case x =2 makes the first factor zero and x=3 makes the second factor zero and are the two solutions of this equation. (Quadratic equations have two solutions). Copyright © 2015 C. P. Rubenstein 46
Perfect Squares, Completing the Square Perfect squares: Now consider another quadratic equation, x 2 + 6 x + 9 = 2 In this example, the left hand side is a perfect square, i. e. , we can rewrite the equation as (x +3)2 = 2 If we take the square root of each side, we get x +3 = ± 2 and x = -3 ± 2 Completing the square: When the equation does not include a perfect square, we can fix it up so that it does. Example: x 2 + 6 x + 8 = 2 The left hand side is not a perfect square but it would be if the 8 were a 9. So let us just add 1 to each side of the equation: x 2 + 6 x + 9 = 3 The left hand side is now (x +3)2 and the solutions of the equation are x = -3 ± 3 Copyright © 2015 C. P. Rubenstein 47
Quadratic Formula The trick of completing the square was known to the Babylonians by 400 BC. This method, when applied to the general quadratic equation: ax 2 + bx +c =0 yields the Quadratic Formula: x = There are similar formulas for cubic and quartic equations but, interestingly enough, not for general equations of fifth and higher degrees. Copyright © 2015 C. P. Rubenstein 48
Factoring Quadratics When asked to solve an equation like x 2 - 4 x - 1 = 0 we soon resort to the quadratic formula or completion of the square to find the solutions 2 + 5 and 2 - 5. Knowing these solutions (the roots of the expression x 2 - 4 x - 1 we could go back and rewrite the equation (check it) as ( x - [2 + 5] ) • ( x - [2 - 5]) = 0 When we say the original equation cannot be solved by factoring, we mean simply that the factors are not simple enough that we can guess them by inspection. Transcendental equations such as 2 x = 5 -x generally do not have closed form solutions and are solved by numerical methods. This particular equation has one solution, x = 1. 715620733. . . (Try this example on your calculator. You may have to use trial and error). Copyright © 2015 C. P. Rubenstein 49
Rules of Exponents Since a 3 • a 2 = (a • a)(a • a) = a 5, we recall Rule 1: Exponents Add: an • am = an+m Likewise, since (a 2) 3 = (a • a)( a • a) = a 6 , we recall Rule 2: Multiplication of Exponents: (an) m = anm Finally, (abc)3 = (abc) • ( abc) = a 3 b 3 c 3 , we recall Rule 3: Apply the exponent to every factor (abcd. . )n = anbncndn. . . Copyright © 2015 C. P. Rubenstein 50
Rules of Exponents Using Rule 1: Exponents Add: an • am = an+m We can write a½ • a ½ = a 1 = a from which: a ½ = a and more generally, the 1/n power of a is the nth root of a. Again using Rule 1, we can write a +1 • a -1 = a 0 = 1, from which a -1 = 1/a Copyright © 2015 C. P. Rubenstein 51
Polynomial Cubed (1 of 3) Polynomials Squared Example: (a+b+c+d)2 = a 2 + b 2 +c 2 +d 2 + 2 ab +2 ac +2 ad + 2 bc +2 bd + 2 cd Polynomials Cubed Example: (a+b+c+d)3 = a (a 2 + b 2 +c 2 +d 2 + 2 ab +2 ac +2 ad + 2 bc +2 bd + 2 cd) + b (a 2 + b 2 +c 2 +d 2 + 2 ab +2 ac +2 ad + 2 bc +2 bd + 2 cd) + c (a 2 + b 2 +c 2 +d 2 + 2 ab +2 ac +2 ad + 2 bc +2 bd + 2 cd) + d (a 2 + b 2 +c 2 +d 2 + 2 ab +2 ac +2 ad + 2 bc +2 bd + 2 cd) = Two more slides for Cubed work … Copyright © 2015 C. P. Rubenstein 52
Polynomial Cubed (2 of 3) Polynomials Cubed Example: (a+b+c+d)3 = a 3 + ab 2 +ac 2 +ad 2 + 2 a 2 b +2 a 2 c +2 a 2 d + 2 abc +2 abd + 2 acd + a 2 b + b 3 + bc 2 +bd 2 + 2 ab 2 +2 abc +2 abd + 2 b 2 c +2 b 2 d + 2 bcd + a 2 c + b 2 c +c 3 + cd 2 + 2 abc +2 ac 2 +2 acd + 2 bc 2 +2 bcd + 2 c 2 d + a 2 d + b 2 d +c 2 d +d 3 + 2 abd +2 acd +2 ad 2 + 2 bcd +2 bd 2+ 2 cd 2 One more slide for Cubed polynomial … Copyright © 2015 C. P. Rubenstein 53
Polynomial Cubed (3 of 3) Polynomials Cubed – putting it all together Example: ( a + b + c + d )3 = a 3 + b 3 + c 3 + d 3 + 3 a 2 b + 3 a 2 c + 3 a 2 d + 3 ab 2 + 3 b 2 c + 3 b 2 d + 3 ac 2 + 3 bc 2 + 3 c 2 d + 3 ad 2 + 3 bd 2 + 3 cd 2 + 6 abc +6 abd + 6 acd + 6 bcd ! Copyright © 2015 C. P. Rubenstein 54
Questions? Copyright © 2015 C. P. Rubenstein 55
For Class Session #02: • • • DUE: Homework Set #01 Reading: Strang - Chapter 1 (see next slide) Quiz #01 on Homework Set #01 2 Do: Review Homework Set #01 2 Do: Functions and Slopes For Session 3: • • • DUE: Homework Set #02 Reading: Strang, Calculus Chapter 2: Derivatives Quiz #02 on Homework Set #02 2 Do: Review Homework Set #02 2 Do: Approximating Slopes, Limits, The Derivative Copyright © 2015 C. P. Rubenstein 56
Chapter 1 CHAPTER 1: Introduction to Calculus 1. 1 Velocity and Distance 1. 2 Calculus Without Limits 1. 3 The Velocity at an Instant 1. 4 Circular Motion 1. 5 A Review of Trigonometry 1. 6 A Thousand Points of Light 1. 7 Computing in Calculus Copyright © 2015 C. P. Rubenstein 57
Any Questions? Send me an email … crubenst@pratt. edu or c. rubenstein@ieee. org Copyright © 2015 C. P. Rubenstein 58
End Copyright © 2015 C. P. Rubenstein 59
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