Math 101: Final Exam Review Sheet
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Math 101: Final Exam Review Sheet (Answers are at the end.) Exam Coverage: Everything we learned in the course. Exam Date: Friday, December 11, 2015 Exam Time: 10:30 am – 12:30 pm (Arrive at least 10 minutes early.) Exam Location: Exam Room Instructors Sections Main Auditorium Falco, Grinshpan, Odintsova, Onderdonk, Wadke, Yang 1, 2, 3, 4, 10, 12, 15, 18, 19, 22, 23, 24, 26 Nesbitt 111 Coppola, Gilman, Patel, Wong 7, 11, 16, 20, 25 PISB 120 Rickert 13, 14, 21 REMEMBER: (1) A formula sheet containing financial math formulas will be provided on the exam. It is on the last page of this review sheet as a reference. (2) BRING YOUR NON-GRAPHING CALCULATOR. No sharing of calculators allowed. Using an unapproved calculator on the exam will result in a grade of 0 on the exam. (3) The exam is closed-book and closed-notes. (4) Bring your student ID card. Page 1 of 16 1. Find the equation of the line passing through (2,7) and (-­β1,1). 2. Find the equation of the line that is perpendicular to 2π₯ − 3π¦ = 5 and passes through the point (-­β2,6). 3. Find the slope of the line: π₯ = −3 . 4. Determine if this is a function: 5. Find the domain: a) π π₯ = −π₯ b) π π₯ = 8π₯ − 24 c) π π₯ = log ! π₯ − 1 d) π π₯ = ! ! ! !!!!!" e) π π₯ = π₯ ! − 1 Page 2 of 16 f) π π₯ = ln(π₯ + 3) g) π π₯ = π ! 6. Let the supply and demand functions for butter pecan ice cream be given by ! ! ! ! π = π π = π and π = π· π = 100 − π where π is the price in dollars and π is the quantity in units of 10-­βgallon tubs. a) Find the equilibrium quantity and equilibrium price. b) Find the quantity supplied when the price is $20. c) Find the price when the quantity demanded is 225 10-­βgallon tubs. 7. Find the vertex, x-­β and y-­βintercepts, and graph: π π₯ = π₯ ! − 2π₯ − 3 . Find the domain and range. Page 3 of 16 8. Solve for x: a) π π₯ = log ! π₯ + log ! π₯ − 12 = 2 b) 2!! = 9 c) 6!!!!! = ! !! ! !"# d) log ! 3 − π₯ = 5 9. A company rents out cars for $45 per day and $0.12 per mile. Find a function for the cost, C(x), of a rented car driven for x miles in a day. 10. A store that installs satellite TV receivers finds that if it installs x receivers per week, then its costs will be πΆ (π₯) = 80π₯ + 1950, and its revenue will be π π₯ = −2π₯ ! + 240π₯ (both in dollars). a) What is the minimum break-­βeven quantity? Page 4 of 16 b) Find the number of receivers the store should install to maximize profit, and find the maximum profit. 11. Find the domain, vertical and horizontal asymptotes, x-­β and y-­βintercepts, x-­βvalues of any holes, and graph: a) π π₯ = !!!! !!! b) π π₯ = ! ! !!!!! !!! 12. Use the properties of natural logarithms to simplify: ln 5π !! − 5π₯ − ln π + ln 1 Page 5 of 16 13. Suppose you deposit $1000 in an account. What rate compounded monthly will allow your money to double in 5 years? How much interest would you have earned? 14. How much must you deposit at the end of each month into an account earning 11.2% interest compounded monthly to accumulate $150,000 in 20 years? How much interest would you have earned? 15. What is the simple interest rate of a loan charging $18 in interest after 2 years on a principal of $150? 16. During the mid-­β1990's the Brandywine long-­βterm growth fund returned 19.6% compounded quarterly. How much would a $5,000 investment in the fund be worth after 4 years? 17. Solve for x: a) π ! ! !!!!! = 1 b) 81!! = 27 !!! Page 6 of 16 18. Solve each system using any method. β§ο£± x + 3 y = 63 β©ο£³4 x + 5 y = 140 a) β¨ο£² β§ο£± 2 x + 3 y = 1, β©ο£³4 x + 6 y = 2. b) β¨ο£² 19. The pharmacist at the Charter Drug Shop filled 92 prescriptions today for antibiotics and cough suppressants. If there were 34 more prescriptions for antibiotics than cough suppressants, how many prescriptions for each were filled? (Hint: Define variables and solve using linear equations. Do not solve by guessing!) Page 7 of 16 20. Solve using the Gauss-­βJordan Method (Matrices): 3π₯ − 5π¦ − 2π§ = −9 −4π₯ + 3π¦ + π§ = 11 8π₯ − 5π¦ + 4π§ = 6 Page 8 of 16 For Problems 21 – 23: Use the given matrices to find each matrix expression: β 2 1 2 βο£Ά β βο£· β − 1 2 − 1βο£Ά β 3 6 8 βο£Ά βο£·βο£· , B = ββ βο£·βο£· , C = β 1 − 2 1βο£· A = ββ βο£ 2 − 1 2 β ο£Έ βο£ 7 5 4 β ο£Έ β 2 1 2 βο£· βο£ β ο£Έ 21. AC 22. AB 23. A+2C Page 9 of 16 24. An electronics manufacturer produces 2 kinds of televisions: 30in and 32in. The 30in requires 2 hours of assembly and 2 hours of finishing and the 32in require 3 hours of assembly and 4 hours of finishing. There are 210 hours of assembly time and 260 hours of finishing time available each week. How many of each type of TV can be produced each week? (Hint: Define variables and solve using linear equations. Do not solve by guessing!) 25. The Art Museum of Philadelphia sells children's tickets for $6 and adult’s tickets for $12. On Saturday, 180 tickets were sold with gross receipts totaling $1,560. How many of each type of ticket was sold? (Hint: Define variables and solve using linear equations. Do not solve by guessing!) Page 10 of 16 26. Laura purchased a house for $360,000 with a 10% down payment and took out a 30-­βyear loan for the balance at 4.25% compounded monthly. a) What will her loan’s monthly payment be? b) After her loan is fully amortized, how much interest will she have paid? 27. At the end of each quarter you deposit $3,500 into a savings account earning 4.25% compounded quarterly. a) How much will you have in your savings account at the end of 10 years? b) How much interest will you have earned? 28. If you purchase a car which requires you to pay $150 a month for 15 years with an interest rate of 8% compounded monthly, and no money down, what is the purchase price of the car? 29. For the function π π₯ = 2π₯ ! + 3π₯ − 1, find the following and simplify: π π₯ + β − π(π₯) β Page 11 of 16 30. Graph the following piecewise function: −π₯ ! , π₯ < 2 π π₯ = 2π₯ + 4, π₯ ≥ 2 31. The graph of a polynomial with 4 turning points has degree ________________. 32. Does the function below have an even/odd degree? Does it have a positive/negative leading coefficient? 33. How long will it take for a $2,000 deposit to grow to $2,500 if it earns 2.6% compounded continuously? 34. If $1,000 earns 4% compounded continuously, what will it be worth in 95 days? Page 12 of 16 35. Find the inverse, if it exists, for each matrix. 5 10 a) −3 −6 −3 −8 b) 1 3 1 −1 c) 2 0 Page 13 of 16 ANSWERS 1. π¦ = 2π₯ + 3 ! 2. π¦ = − π₯ + 3 ! 3. The slope is undefined for any vertical line. 4. No, it is not a function because it fails the vertical line test. 5a. (−∞, 0] 5b. 3, ∞ 5c. (1, ∞) 5d. −∞, −3 ∪ −3, 10 ∪ 10, ∞ 5e. (−∞, ∞) 5f. −3, ∞ 5g. (−∞, ∞) 6a. The equilibrium quantity is 125 10-­βgallon tubs and the equilibrium price is $50. 6b. 50 10-­βgallon tubs 6c. $10 7. vertex: (1, -­β4) x-­βint: (-­β1, 0), (3, 0) y-­βint: (0, -­β3) Domain = −∞, ∞ and Range = [−4, ∞) 8a. π₯ = 16 8b. π₯ ≈ 1.06 ! ! ! ! 8c. π₯ = − , 8d. π₯ = −29 9. πΆ π₯ = 0.12π₯ + 45 10a. 15 receivers 10b. 40 receivers, maximum profit is $1,250 11a. Domain: −∞, −2 ∪ −2, ∞ , VA: π₯ = −2, HA: π¦ = 3, x-­βint: (1/3, 0), y-­βint: (0, -­β1/2), no holes 11b. Domain = −∞, 2 ∪ 2, ∞ , VA: none, HA: none, x-­βint: (-­β4, 0), y-­βint: (0, 4), hole @ x = 2 Page 14 of 16 12. ln 5 + 2π₯ − 1 13. π ≈ 13.94%, πΌ = $1,000 14. deposit $168.89, πΌ = $109,466.40 15. 6% 16. $10,749.24 17a. π₯ = −2, −3 17b. π₯ = ! !" 18a. π₯, π¦ = 15, 16 18b. !!!! ! , π¦ for any # π¦ 19. 63 prescriptions for antibiotics and 29 prescriptions for cough suppressants 20. π₯, π¦, π§ = −3, −2, 5 −2 7 21. −6 6 −2 7 22. No solution 23. No solution 24. It takes 2x hours to assemble the 30in TV’s and 3y hours to assemble the 32in TV’s. Since the total assembly time is 210 hours: 2π₯ + 3π¦ = 210 It takes 2x hours to finish the 30in TV’s and 4y hours to finish the 32in TV’s. Since the total finishing time is 260 hours: 2π₯ + 4π¦ = 260 Solving this system of 2 equations, we get x = 30 and y = 50. 25. 100 children’s tickets and 80 adult’s tickets were sold. 26a. $1,593.89 26b. $249,800.40 27a. $173,324.94 Page 15 of 16 27b. $33,324.94 28. $15,696.09 29. 4π₯ + 2β + 3 30. 31. ≥ 5 32. even degree, positive leading coefficient 33. π‘ = !" !.!" ≈ 8.58 π¦ππππ !.!"# 34. $1,010.61 35a. The inverse does not exist. −3 −8 35b. 1 3 35c. 0 −1 ! ! ! ! Page 16 of 16 Potentially Useful Formulas: 1) πΌ = πππ‘ 2) π΄ = π 1 + ππ‘ 3) π΄=π 1+ 4) π΄ = ππ !" 5a) 6) 7) 8) 9) π =π π=π π =π ! ! ! π! = 1 + π=π ! !" ! !! π΄ = π + πππ‘ or π΄ = π(1 + π)! −1 ! !" !! ! ! ! 5b) π=π or π =π ! !!" ! ! ! or π=π ! ! ! !!" !! ! or π =π !! !! π= ! ! or π! = π ! − 1 or ! ! ! !" !! !! ! !! or !!! ! !! ! ! !!! ! !! !! !!! !! ! ! !! !!! !! π = ππ‘ π΄=π+πΌ
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