Review problems for Exam 3 CE Math 1050 Spring 2018

Transcription

1 CE Math 1050 Exam 3 review with answers - Page 1 Review problems for Exam 3 CE Math 1050 Spring 2018 This is a comprehensive set of review problems for Exam 3, CE Math 1050, Spring term The layout of the document is: Learning objectives are numbered and bolded. Sample problems for each learning objective are beneath the objective. Complete answers begin on page 6. For classes taking their exams as written by the UVU Math Department, the question types (not exact questions) for Exam 3 will be taken from a much smaller subset of this set. This exam will cover sections 4.4 to 5.5 and Linear Programing from the Stewart text. Exam 3 will start being given on March 27 th, Please, direct questions or comments about this document to jennifer.hooper@uvu.edu. THANK YOU to all who help edit, perfect, and polish these documents. Exponential and Logarithmic Functions A student is able to: 4. Use the properties of logarithms to combine or expand logarithmic expressions. Problem I: Write each expression as a sum or difference of logarithms (expand). Please show at least one intermediate step or some of your reasoning for full credit. a) log b ( x y 2 z ) = b) log b ( x3 (x 2) 2 x 2 +5 ) = Problem II: Write each expression as a single logarithm (contract). Please show at least one intermediate step or some of your reasoning for full credit. a) 3 log b x + 5 log b y = b) 1 ln(x + 3) 1 ln(x + 2) lnx = Use the change of base formula for logarithms. Use the change of base formula to change log 4 19 to an expression involving only natural logs.

2 CE Math 1050 Exam 3 review with answers - Page 2 6. Solve logarithmic and exponential equations. Problem I: Solve the following equations for x. Please, give exact answers. In other words, do not use a calculator, but instead leave your answer in a form ready to put into a calculator. a) log 6 (x + 2) + log 6 (x 3) = 1 b) log log 3 (x + 2) = 2 c) log(x 3) + log(x + 2) = log(4x) Problem II: Solve the following equations for x. Please, give exact answers. In other words, do not use a calculator, but instead leave your answer in a form ready to put into a calculator. a) e 2x + 5e x - 6 = 0 b) 2 x2 +12 = 2 7x c) 10 2x 7 = 2

3 CE Math 1050 Exam 3 review with answers - Page 3 7. Solve applied exponential and logarithmic problems using base 10 and base e such as; ph, exponential growth and decay. (Remember for these problems you will use one of three equations: A = Pe rt, or when given the doubling time use A = P(2) t a where a is the doubling time, or when given half-life use A = P(2) t h where h is the half-life.) Problem I: Carbon-14 Dating. The burial cloth of an Egyptian mummy is estimated to contain 59% of the carbon-14 it contained originally. How long ago was the mummy buried? (The half-life of carbon-14 is 5730 years.) Please, solve the problem exactly. In other words, do not use a calculator, but instead leave your answer in a form ready to put into a calculator. Problem II: World Population. The population of the world was 7.1 billion in 2013, and the observed relative growth rate was 1.1% per year. At this rate, how long will it take the population to double? Please, solve the problem exactly. In other words, do not use a calculator, but instead leave your answer in a form ready to put into a calculator. Problem III: Bat population The bat population in a certain Midwestern county was 350,000 in 2012, and the observed doubling time for the population is 25 years. t a a) Find an exponential model n(t) = n 0 2 for the population t years after b) Sketch a graph of the population at time t. c) Using the formula from part a), How long will it take the population to reach 2 million? Please, solve the problem exactly. In other words, do not use a calculator, but instead leave your answer in a form ready to put into a calculator.

4 CE Math 1050 Exam 3 review with answers - Page 4 Systems of Equations and Inequalities A student is able to: 1. Solve a system of equations by substitution/elimination. (Also done below.) Problem I: Health Club Management. A fitness club has a budget of $915 to purchase two types of dumbbell sets. One set costs $30 each and the other deluxe set costs $45 each. The club wants to purchase 24 news sets of dumbbells. How many of each set should the club purchase? [To earn full credit, some work must be shown.] 2. Solve a system of nonlinear equations in two variables using substitution or elimination. a) { x2 + y 2 = 10 3x + y = 0 b) { x2 + y 2 = 16 x 2 y 2 = 2 c) { x2 + y 2 = 9 x 2 y = 3

5 CE Math 1050 Exam 3 review with answers - Page 5 Solve a system of linear equations in three variables using substitution or elimination. If there is no solution, so state. If there are infinitely many solutions, please give the formulas for the solutions, (for a system with 2 variables, state your formulas in terms of x, and for a system with three variables state your formulas in terms of z.) 3. x + 2z = 3 a) { 4x + y = 4 5y + 6z = 12 x + y + z = 6 b) { x y + 2z = 5 3x + z = 6 2x y = 10 c) { 4x + 2y = 20 x + 2y + 4z = 3 d) { y + 3z = 5 x 2z = 7

6 CE Math 1050 Exam 3 review with answers - Page 6 4. Graph the solution set of a system of inequalities. Problem I: Graph the solution set. Label all vertices. If there is no solution, indicate that the solution set is the empty set. a) { y 1 3 x 2 y < x 4 y x b) { y x + 5 y > 1 c) { x 2 + y 2 25 y > 2 x < 3

7 CE Math 1050 Exam 3 review with answers - Page 7 5. Use Linear Programming to solve a problem. Problem I: Maximize z = 4x + 3y for the following feasible region. [To earn full credit, some work must be shown.] Problem II: Packaging Nuts. A confectioner sells two types of nut mixtures. The standardmixture package contains 100g of cashews and 200g of peanuts and sells for $1.95. The deluxe-mixture package contains 150g of cashews and 50g of peanuts and sells for $2.25. The confectioner has 15kg of cashews and 20 kg of peanuts available. On the basis of past sales, the confectioner needs to have at least as many standard as deluxe packages available. How many bags of each mixture should he package to maximize his revenue?

8 CE Math 1050 Exam 3 review with answers - Page 8 6. Compute the partial fraction decomposition of a rational function when the denominator is a product of linear factors or distinct quadratic factors. Problem I: Set up the form for the partial fraction decomposition. Do not solve for A, B, C, and so on. 2x 5 + 3x 3 + 4x x(x + 2) 3 (x 2 + 2x + 7) 2 Problem II: Find the partial fraction decomposition of the following rational functions. a) f(x) = x 37 (x+4)(2x 3) b) g(w) = 6w 7 w 2 +w 6 c) h(x) = 17x2 7x+18 7x 3 +42x

9 CE Math 1050 Exam 3 review with answers - Page 9 ANSWERS Chapter 4: Exponential and Logarithmic Functions 4. Problem I: a) log b x 2log b y log b z b) 3log b x + 2log b (x 2) 1 2 log b(x 2 + 5) Problem II: a) log b (x 3 y 5 ) b) ln ( x+3 ) 3 ( x+2)x 5. ln19 ln4 6. Problem I: a) x = 4 b) x = -1 c) x = 6 Problem II: a) x = 0 b) x = 3 or 4 c) x = ln9 = log9 2ln Problem I: 5,730 ln0.59 ln2 Problem II: ln

10 CE Math 1050 Exam 3 review with answers - Page 10 Problem III: t 25 a) n(t) = 350,000(2 ) b) Graph: horizontal axis is time since 2012 in years, vertical axis is Bat population c) t = 25ln( ) ln2 Chapter 5: Systems of Equations and Inequalities 1. Problem I: 11 of the sets that cost $30 each and 13 of the deluxe sets. 2. a) (1, -3) (-1, 3) b) (3, 7) (3, - 7) (-3, 7) (-3, - 7) c) (0, -3) ( 5, 2) (- 5, 2) 3. a) (-1, 0, 2) b) (1, 2, 3) c) (x, 2x + 10) d) (2z 7, 3z + 5, z )

11 CE Math 1050 Exam 3 review with answers - Page Problem I: a) Graph with intersection point at (3, -1) b) Vertices: ( 1, 6), (2, 3), ( 6, 1), ( 6, 1) c) Vertices: (3, 4), (3, 2), ( 21, 2)

12 CE Math 1050 Exam 3 review with answers - Page Problem I: The point (1, 3) maximizes z to the value 13. Problem II: 90 standard-mixture packages and 40 deluxe-mixture packages. Maximum revenue is $ Problem I: A x + B + C + D + Ex+F + Gx+H x+2 (x+2) 2 (x+2) 3 x 2 +2x+7 (x 2 +2x+7) 2 Problem II: a) f(x) = 3 x+4 7 2x 3 b) g(w) = 5 w w 2 c) h(x) = 2x 1 x x