GSB 420 Math Chapter 1-2 Calculus (Week1 In-Class Homework) Name : ______________________ I. Find X from the following equations (0.5 point each) 1. 3x = 90 2. Ln(ex) = 25 3. Monthly payment (X) of $500,000 mortgage when the mortgage interest rate is 4.5% for 30 years 4. Future value (X) of monthly saving of $1200 for 20 years at 5% interest rate. II. Find the first derivatives of the following functions. Show your work for full credit. (1 point each) 1. 2. 3. 4. 5. Y 34 X 5 3 X 4 10 X 3 17 X 2 3 X 15 Y ( X 2 3)5 Y 1 3 X 4 5X 3 (7 X 2 1) 2 Y 3X 4 Y (25 X 2 11X 100) (2 X 3 10) 6. Y 8 X 2 Z 2 19 X 5 Z 3 7. 8. Y eX 2 3Z 125 2X 2 2 X 5 Y ln(3 X 3 7 X 2 15 X 1000) GSB420 CLASS 2 STATISTICS (MATH CH1-2) Business Mathematics Page 1 of 20 GSB420 CLASS 2 STATISTICS (MATH CH1-2) Page 2 of 20 Chapter 1. Algebra Review I. Properties of Exponents Note that 00 = undefined 1. X 0 =1 2. 1 = X b = X ^ ( b ) Xb 3. X a * X b = X a X b = X a X b = X a +b \u0001 \u0001 1 = X 10 = X ^ ( 10) X 10 \u0001 X ^ (a + b) \u0001 X 2 * X 3 = X 2 X 3 = X 2 X 3 = X 2+3 = X 5 \u0001 2 3 2 4 = 2 3+ 4 = 2 7 4. = 128 ( X a ) b = X a*b = X ab = X ab \u0001 2 3 2*3 23 6 \u0001 (X ) = X = X = X X ^ ab \u0001 (2 3 ) 4 = 2 34 = 212 = 4096 5. 6. Xa = X a X b = X a b b X X2 1 \u0001 3 = X 2 X 3 = X 23 = X 1 = X X ( XY ) a = X a * Y a = X a Y a = X aY a \u0001 ( XY ) 2 = X 2 * Y 2 = X 2 Y 2 = X 2Y 2 7. n X =X 1 n = X 1/ n 1 \u0001 8. 4 = 4 2 = 41 / 2 = 4 0.5 = (2 2 ) 0.5 = 2 20.5 = 21 = 2 q X p / q = ( X 1 / q ) p = ( X p )1 / q = X p \u0001 210/5 = 22 = (21/5 )10 = (210 )1/5 = 5 210 = 22 = 4 \u0001 8 2 / 3 = (2 3 ) 2 / 3 = 2 2 = 3 8 2 = 3 64 = 4 GSB420 CLASS 2 STATISTICS (MATH CH1-2) Page 3 of 20 II. Linear and Nonlinear Functions 1. Linear Functions Linear Functions have the general form of: Y=a+bX 2. Nonlinear Functions There are many different types of nonlinear functions such as polynomial, exponential, logarithmic, trigonometric functions, etc. If we are to find the roots to a quadratic function of: aX 2 + bX + c = 0 we can find their two roots by using the following quadratic formula: X1, X 2 = b b 2 4ac 2a GSB420 CLASS 2 STATISTICS (MATH CH1-2) III. Find the roots, X1 and X2, of the following quadratic equations: X 2 5 X 14 = 0 1. Factoring Method: 2. Quadratic Formula X1 , X 2 = b b 2 4ac 2a Page 4 of 20 GSB420 CLASS 2 STATISTICS (MATH CH1-2) IV. A system of equations in two unknowns 3 x + 2 y = 13 4 y 2x = 2 1. Substitution Method 2. Elimination Method Page 5 of 20 GSB420 CLASS 2 STATISTICS (MATH CH1-2) Page 6 of 20 V. Useful Mathematical Operators n 1. n Summation Operator = Sigma = \u0001 X i = i =1 X i = X i i =1 n X i = X 1 + X 2 + ..... + X n 1 + X n = Sum Xi's where i goes from 1 to n. i =1 n 2. Multiplication Operator = pi = \u0001 X i = X i i =1 n X i =1 i = X 1 X 2 .... X n 1 X n = Multiply Xi's where i goes from 1 to n. GSB420 CLASS 2 STATISTICS (MATH CH1-2) Page 7 of 20 VI. Time in Value 1. The Concept of Periodic Interest Rates Assume that the annual percentage rate (APR) is ( r 100 )%. That is, if an APR is 10%, then r = 0.1. Also, define FV = future value, PV = present value, and t = number of years to a maturity. i) Annual compounding for t years \u0001 ii) Semiannual compounding for t years \u0001 iii) Quarterly compounding for t years \u0001 iv) Monthly compounding for t years \u0001 v) Weekly compounding for t years \u0001 vi) Daily compounding for t years \u0001 vii) Continuous compounding for t years \u0001 FV = PV (1 + r ) t r FV = PV (1 + ) 2t 2 r FV = PV (1 + ) 4t 4 r FV = PV (1 + )12t 12 r FV = PV (1 + ) 52t 52 r 365t FV = PV (1 + ) 365 FV = PV e rt GSB420 CLASS 2 STATISTICS (MATH CH1-2) Page 8 of 20 2. Annuity Calculation Annuity Formulas: FV = A [(1 + i ) n 1] i A [(1 + i ) n 1] PV = i (1 + i ) n where A= the fixed annuity amount; n = the number of periods; and i = a periodic interest rate. Of course, FV = the future (or final or terminal) value and PV = the present (or current) value. GSB420 CLASS 2 STATISTICS (MATH CH1-2) Page 9 of 20 a. If you obtain a 30 year mortgage loan of $100,000 at an annual percentage rate (APR) of 6%, what would be your monthly payment? Answer: 0.06 12 x 30 1] ) 12 100,000 = 0.06 0.06 12 x 30 (1 + ) 12 12 A [(1 + Therefore, A=$599.55 Excel : =PMT(0.06/12,360,100000) b. If you invest $1,000 a month in an account that is guaranteed to yield a 10% rate of return per year for 30 years (with a monthly compounding), what will be the balance at the end of the 30-year period? Answer: 0.1 360 ) 1] 12 FV = = $2,260,487.92 0.1 12 Excel: =FV(0.1/12,360,1000) $1,000 [(1 + GSB420 CLASS 2 STATISTICS (MATH CH1-2) VII. Exponential and Logarithmic Functions 1. Exponential Functions e1 = 0.367879 e0 = 1 e1 = 2.71828 e2 = 7.389056 e5 = 148.4132 ln(e 1 ) = ln(0.367879) = 1 ln(e0 ) = ln(1) = 0 ln(e1 ) = ln(2.71828) = 1 ln(e 2 ) = ln(7.389056) = 2 ln(e5 ) = ln(148.4132) = 5 ln(e a ) = a ln(e x ) = x Find X a) 3 X = 15 b) 3 X = 59049 Page 10 of 20 GSB420 CLASS 2 STATISTICS (MATH CH1-2) 2. Logarithmic Functions a. When Y = b x , log b (Y ) = log b (b x ) = x log b ( XY ) = log b ( X ) + log b (Y ) X ) = log b ( X ) log b (Y ) Y log b ( X a ) = a log b ( X ) log b ( For Example: Y = ALa K b log b (Y ) = log b ( A) + a log b ( L) + b log b ( K ) b. Properties of Logarithms i) Product Property: logb mn = log b m + log b n ii) Quotient Property: log b iii) Power Property: log b m n = n log b m m = log b m log b n n Page 11 of 20 GSB420 CLASS 2 STATISTICS (MATH CH1-2) Page 12 of 20 Chapter 2. Calculus Review dY Y f ( X + X ) f ( X ) = Y ' = lim = lim X 0 X X 0 dX X That is, Slope = rise Y = =b run X Y=a+bX Y Y = rise X = run a 0 X GSB420 CLASS 2 STATISTICS (MATH CH1-2) Page 13 of 20 1. The Constant Function Rule: Given a primary function of Y = a where a is any constant, then, its derivative (or slope function) is: dY = Y'= 0 dX 2. The Power Function Rule: Given a primary function of Y = aX b then, its derivative (with respect to X) is: dY = Y ' = baX b 1 dX GSB420 CLASS 2 STATISTICS (MATH CH1-2) 3. Page 14 of 20 The Sums and Differences Rule: Given a primary function of Y = U W where U = U ( X ) and W = W ( X ) then, its derivative (with respect to X) is: dY dU dW = Y'= dX dX dX GSB420 CLASS 2 STATISTICS (MATH CH1-2) 4. The Product Rule: Given a primary function of Y = U W where U = U ( X ) and W = W ( X ) then, its derivative (with respect to X) is: dY dU dW = Y'= W + U dX dX dX Page 15 of 20 GSB420 CLASS 2 STATISTICS (MATH CH1-2) 5. The Quotient Rule: U W where U = U ( X ) and W = W ( X ) Given a primary function of Y = then, its derivative (with respect to X) is: dU dW W U dY dX dX = Y '= dX W2 Page 16 of 20 GSB420 CLASS 2 STATISTICS (MATH CH1-2) 6. Page 17 of 20 The Chain Rule (=The Function-of-Functions Rule) Given a primary function of Y = f (U ) where U = g ( X ) then, its derivative (with respect to X) is: dY dY dU = dX dU dX GSB420 CLASS 2 STATISTICS (MATH CH1-2) 7. Derivatives of Natural Logarithmic Function f ( x) = ln( x) f '( x) = d ln x 1 = dx x proof ) h( x) = eln( x ) = x d ln x ln( x ) d ln x e = x =1 dx dx d ln x 1 = dx x h '( x) = Examples f ( x) = ln(h( x)) f '( x) = d ln(h( x)) h '( x) = dx h( x) f ( x) = ln( x 2 ) f '( x) = d ln( x 2 ) 2 x 2 = 2 = dx x x f ( x) = x 2 ln( x) d ln( x) dx 2 + ln( x) dx dx 1 = x 2 + 2 x ln( x) = x + 2 x ln( x) x f '( x) = x 2 Page 18 of 20 GSB420 CLASS 2 STATISTICS (MATH CH1-2) E. The First Derivative and the Second Derivative d 2Y \u0001 Leibniz Notation dX 2 or Y " \u0001 Lagrangean Notation Page 19 of 20 GSB420 CLASS 2 STATISTICS (MATH CH1-2) F. Total Derivatives and Partial Derivatives Given Y = f ( X , Z ,W ) = 3 X 2 2 XZ 3 + 5 XW XZ 2 + ZW 2 10 A partial derivative with respect to X: dY dX = Z ,W = cons tan t Y = 6 X 2 Z 3 + 5W Z 2 X A partial derivative with respect to Z: dY dZ = X ,W = cons tan t Y = 6 XZ 2 2 XZ + W 2 Z A partial derivative with respect to W: dY dW = X , Z = cons tan t Y = 5 X + 2 ZW W Page 20 of 20 GSB420: Applied Quantitative Analysis Autumn, 2016 updated: 8/29/2016 INSTRUCTOR: Jin Man Lee Office: Loop Campus (1 E Jackson DPC Room #6230) Phone: (312)362-5970 Email: jlee141@depaul.edu The best way to reach me is to send email. Please use GSB420 as a prefix on the subject line, and that will get my attention immediately. If you don't receive my reply within 24 hours, please remind me again. Due to many email filters and mass email, your email might be lost. OFFICE HOURS: Tuesday & Thursday, 4:00- 5:00 or by appointment (at DPC 6230) COURSE OBJECTIVES The class will provide quantitative, statistical, and analytical methods that would be beneficial to MBAs in the modern business world. The course is demanding because there are so many topics to squeeze in within 10 sessions or less. I expect students to be proactive and ready for each class by challenging me with questions. We will cover essential topics on math, calculus, and statistics in focusing on practical applications in business. Thus, most of examples will be related with business world, and computer software will be extensively used for class and homework. REQUIRED TEXTBOOK Jin W. Choi, Step-by-Step Business Math and Statistics, 3rd, 2010, ISBN: 0536560706 SOFTWARE USED in CLASS (required for homework and lab classes) Minitab (any version): DePaul computer labs, and a 30 day trial version is available from minitab.com. We will start to use Minitab from the third week. RECOMMENDED SUPPLEMENTARY TEXTBOOKS Any Algebra and Calculus books used from previous Math classes. From DePaul library, you can also access two useful books via online; Business Statistics Demystified by Steven M. Kemp and Sid Kemp. Statistics for Dummies by Deborah Rumsey. EXAMS Exam I (Class3 Test for 60 Minutes) Exam II (Class4 Test for 60 Minutes) Exam III (Class6 Test for 60 Minutes) Exam IV (Class8 Test for 60 Minutes) Final Exam (TAKE HOME SUBMISSION TO D2L DROPBOX ) 1 ASSIGNMENTS Pop Quizzes: We will have pop quizzes in class. All require to submit the answers or sign-up sheet before leaving class. Lab Assignments: Depending upon lab classes with availability of license, the lab assignment can be done as a group with two or less. All assigned work needs to be uploaded to D2L. If the work cannot be done in the lab class, one more revised version can be uploaded after class. Homework: Weekly homework will be posted in D2L, and due the following week before class. All homework needs to be uploaded to Dropbox in D2L before the class start (5:30 PM). - All assignments are to be prepared individually unless otherwise stated by me. You risk an academic integrity violation if submit the same work and answers with others. Group study is encouraged but not the submission of homework. - Assignments are graded based on completion. Failure to answer any questions or nonsensical attempts at answering questions will result in an incomplete assignment. - Late submission will get steep penalty without prior approval by instructor. Extra Credit: End of chapter Quizzes on D2L - All work is due in a week with unlimited tries, and the best score will be recorded automatically. GRADE Exam I-IV (10% each, total 40%), Final Exam (25%), In-Class Quizzes(5%), Computer LAB Assignment (10%), Homework (20%), Weekly End of Chapter Quizzes on D2L (10% extra credit) Scale of grade: A: 93 or above, A-: 88-92.9, B+: 85-87.9, B: 80-84.9, B-: 77-79.9, C+: 75-76.9, C: 70-74.9, C-: 68-69.9, D+: 65-67.9, D: 60-64.9, F: Below 60 COMPUTER LABS, room will be announced in class There will be 3 computer lab classes. All computer lab classes starts from the second half of class (7:30-9:00PM) after regular lectures, that means we will move from regular classroom to a computer lab during break. The assigned lab for the class will be announced in class. Computer Lab 1 (Class 6) Minitab for Statistics I Computer Lab 2 (Class 8) Minitab for Statistics II Computer Lab 3 (Class 10) Minitab for Regression Analysis ACADEMIC HONESTY Work done for this course must adhere to the University Academic Integrity Policy. Violations include but are not limited to the following categories: cheating; plagiarism; fabrication and academic misconduct. 2 Cheating: any action that violates University norms or an instructor's guidelines for the preparation and submission of assignments. Such actions may include using or providing unauthorized assistance or materials on course assignments, or possessing unauthorized materials during an examination. Plagiarism: the representation of another's work as your own. You are to prepare your own homework assignments. Violations may result in the failure of the assignment, failure of the course, and/or additional disciplinary actions. Misconduct: This includes but is not limited to attempts to bribe an instructor for academic advantage; persistent hostile treatment of, or any act or threat of violence against, an instructor, advisor or other students. Violations may result in additional disciplinary actions by other university officials and possible civil or criminal prosecution. You may review the Academic Integrity Policy in the Student Handbook or by visiting Academic Integrity at DePaul University (http://academicintegrity.depaul.edu) ATTENDANCE POLICY I do not take attendance. The attendance will be automatically checked by in-class quizzes. Excuses on exam days may be considered under extraordinary circumstances provided by official documentation. CLASSROOM RULES & PROFESSIONAL POINTS Prohibitions: Cell phones must be turned OFF. Use of the internet is not permitted unless specifically directed by me. This includes checking of email and use of instant messengers. You must sit at the front of the classroom if you are using a computer. Tape recorders, unrelated reading materials, and food are also prohibited in the classroom. Behavior: You may not leave the classroom for any reason during an exam (go to the bathroom beforehand!). Further, unprofessional behavior such as inappropriate chatting, leaving in the middle of class, or showing up excessively late, etc. are disruptive and unacceptable. If you need to leave class early, let me know in advance. For first time violations you will receive a warning. In the event that violations continue, I will ask you to leave the classroom. (I reserve the right to add to this list as situations arise.) Student with Disability: Students with Disability may register the The Productive Learning Strategies (PLuS) Program. You may request your exam schedule arrangement by requesting through the PLuS program. For more information on the PLuS program, you may visit http://studentaffairs.depaul.edu/plus/ or call: 312-362-8000. 3 TENTATIVE SCHEDULE OF TOPICS (The instructor may change the order or contents by needs, any special material needs for class will be available on D2L) I. Business Mathematics Class 1, MathCh 1-2, Algebra and Calculus Review Class 2, MathCh 3 Optimization Methods Class 3, EXAM I (MathCh 1-3, 60 Mins) MathCh 4 Applications to Economics II. Business Statistics Class 4, Ch 1-4. EXAM II (MathCh 3-4, 60 Mins) Introduction to Statistics and Descriptive Measures Class 5, Ch 5-6 , Probability, and Discrete probability Distributions Class 6, EXAM III (Ch 1-6, 60 Mins) Ch 7-9 , Normal, and Sample Distribution (t Distribution) COMPUTER LAB 1 (MINITAB1) Class 7, Ch 10-12 Confidence Interval Estimation and Hypothesis Tests Class 8, EXAM IV (Ch 7-12, 60 Mins) COMPUTER LAB 2 (MINITAB2) Ch 13. Introduction of Regression Analysis Class 9, Ch 13. Simple Regression Analysis Class 10, Ch 14. Multiple Regression Analysis COMPUTER LAB 3 (MINITAB3) FINAL EXAM (TAKE HOME EXAM SUBMISSION TO D2L DROPBOX) 4