College Algebra - Homework and Practice Exam
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Lab Module 2 Name _____________________ Show all work for each of the following problems. Simplify and clearly indicate all answers. 1. For f(x) = 3x2 + 5 and , a. Verify: g(x + 2) ≠ g(x) + g(2). b. Find (f – g)(x). c. Using the resulting function in (b), show that (f – g)(2) = f(2) – g(2). (The work should be different for each side of the equation.) d. Is (fg)(0) = (0)? Explain. e. Find , h ≠ 0. g(x) 7x 2= - f g æ ö ç ÷ è ø f(x h) f(x) h + - Lab Module 2 page 2 2. A size 36 shoe in France is size 3.5 in England. A function that converts shoe sizes in France to those in England is . A size 6 shoe in the United States is size 3.5 in England. A function that converts shoe sizes in England to those in the United States is f(x) = x + . a. Use composition of functions to find a function that converts shoe sizes in France to those in the United States. Simplify the function. b. Determine the shoe size in the United States for a size 38 shoe in France. Write the answer in a complete sentence using proper grammar and correct spelling. 3. A dance studio has fixed monthly costs of $1500 that include rent, utilities, insurance, and advertising. The studio charges $60 for each private lesson, but has a variable cost for each lesson of $35 to pay the instructor. a. Write a linear cost function representing the cost to the studio C(x) to hold x private lessons for a given month. b. Write a linear revenue function representing the revenue R(x) for holding x private lessons for the month. c. Write a linear profit function representing the profit P(x) for holding x private lessons for the month. d. Determine the number of private lessons that must be held for the studio to break-even. 3x 94g(x) 4 - = 5 2 Lab Module 2 page 3 4. In a study using 50 foreign-language vocabulary words, the learning rate L (in words per minute) was found to depend on the number of words already learned, x, according to the equation L(x) = 20 – 0.4x. a. State the coordinates of the x-intercept. b. State the coordinates of the y-intercept. c. State the slope. d. Graph the linear function for x ≥ 0. (Label the axes completely!!) e. Is the learning rate increasing or decreasing? Explain why your answer makes sense in the context of the problem. Use a complete sentence with proper grammar and correct spelling. Lab Module 2 page 4 5. Suppose that 650 lb of coffee are sold when the price is $4 per pound, and 400 lb are sold at $8 per pound. a. List the data points (use price as the independent variable). b. Find the slope of the line joining the points. c. Interpret the meaning of the slope in the context of this problem. Write the interpretation in a complete sentence with proper grammar and correct spelling. d. Use the point-slope form to write a linear equation for this data. Write the answer in function notation. e. Use this function to predict how much consumers would be willing to buy at a price of $6 per pound. 6. For each final matrix, state the solution. 1 0 0 3 0 1 0 1 0 0 1 8 é ù ê ú-ê ú ê úë û 1 0 6 11 0 1 4 7 0 0 0 2 é ù- ê ú ê ú ê ú-ë û 1 0 2 9 0 1 8 5 0 0 0 0 é ù ê ú-ê ú ê úë û Lab Module 2 page 5 7. The University of Texas at Austin has three times as many students enrolled as the University of Miami. The University of California, Berkley has 3,000 more than twice the number of students as the University of Miami. If the three schools have a total enrollment of 96,000 students, what is the enrollment at each school? a. Describe what the variables represent: b. Write the system of linear equations: c. Write the Augmented Matrix for the system of linear equations then solve using the Gauss-Jordan Elimination Method. Show all row operations and at least 5 of the resulting matrices. Use proper notation. Write the solution as an ordered triple, if appropriate. d. Now, answer the question in a complete sentence with proper grammar and correct spelling: What is the enrollment at each school? = = = x y z Lab Module 2 page 6 8. The ABC Ink Company is a small family owned company that sells packages of ink cartridge refills for smartpens. The Xavier set contains one blue ink refill and one black ink refill. The Yvonne set includes two blue ink refills, three black ink refills, and one red ink refill. The Zena set includes four blue ink refills, five black ink refills, and one red ink refill. The company has sold most of its stock and has found that it has only 11 blue ink cartridge refills, 14 black ink cartridge refills, and 3 red ink cartridge refills. How many of each set should the company package to sell in order to use all of the remaining ink cartridges so that there will be none left in inventory. a. Describe what the variables represent: b. Write the system of linear equations: c. Solve the system of equations using Gauss-Jordan Elimination. Show all proper row operations and the resulting matrices. Write the solution as an ordered triple, if appropriate. d. Fill out the table to show the possible combinations of sets the company can package: x y z e. In at least one complete sentence with proper grammar and correct spelling, write the solutions in terms of what the variables represent. = = = x y z Lab Module 3 Name _____________________ For each of the following problems, show all work. Simplify and clearly indicate all answers. 1. A ball is thrown vertically upward from the top of a building 160 feet tall with an initial velocity of 48 feet per second. The distance d (in feet) of the ball from the ground after t seconds is d(t) = 160 + 48t – 16t2. a. After how many seconds does the ball strike the ground? Write your answer in a complete sentence with proper grammar and correct spelling. b. When will the ball reach its maximum height? What is the maximum height of the ball? Write your answers in a complete sentence with proper grammar and correct spelling. 2. Laura owns and operates Aunt Linda’s Pecan Pies. She has learned that her profits, P(x), from the sale of x cases of pies, are given by P(x) = 150x – x2. a. The company will “break-even” when the profit is zero. How many cases of pies should Laura sell in order to break-even? (Solve for x when P(x) = 0.) Write your answer in a complete sentence with proper grammar and correct spelling. b. How many cases of pies should she sell in order to maximize profit? What is the maximum profit? Write your answer in a complete sentence with proper grammar and correct spelling. Lab Module 3 page 3 3. For f(x) = x3 – 7x2 + 8x + 16, a. Find f(10) using synthetic division. b. Is 10 a zero of the function? Explain in a complete sentence with proper grammar and correct spelling. c. Use synthetic division to determine if x + 1 is a factor of f(x). d. Is –1 a zero of the function? Explain in a complete sentence with proper grammar and correct spelling. e. List all of the zeros and their multiplicities of the polynomial. f. Write the polynomial function as a product of linear factors. Lab Module 3 page 4 4. For the function a. State the degree of the polynomial. b. State the number of zeros the polynomial function will have. c. Use the Rational Zero Theorem to list all of the possible rational zeros d. Use your calculator to determine which numbers in the list of rational zeros are probable rational zeros. e. Use synthetic division to verify one rational zero. f. Use synthetic division or other algebraic methods to find all remaining zeros. List all of the zeros of the polynomial function. - + + -4 3 2f(x) = 3x 4x x 6x 2 Lab Module 3 page 2 5. Analyze and sketch the polynomial functions and complete the charts below. State the degree and sign of the leading coefficient of the polynomial functions. Determine the end behavior of the graph of the functions. For 5b, write the polynomial function as a product of linear factors (in factored form). a. f(x) = –7(x + 2)3(x – 1)2(x – 3) Zeros Multiplicity Crosses/Touches Degree Sign End behavior b. m(x) = x3 – 4x2 Factored form: m(x) = Degree Zeros Multiplicity Crosses/Touches Sign End behavior 6. Based on data from the U.S. Department of Agriculture, the average number of acres per farm x years after 2000 can be approximated by the model below. (Round answers to 2 decimal places.) a. Use the model to estimate the average number of acres per farm in 2005. b. Use the model to predict the average number of acres per farm in 2012. c. Find and interpret the zero of the rational function. Does this result make sense within the context of the problem? Answer in complete sentences using proper grammar and correct spelling. 2164x13 94458x2078)x(A - - = Lab Module 3 page 5 7. A rare species of insect was discovered in the rain forest. In order to protect the species, environmentalists declare the insect endangered and transplant the insects into a protected area. The population of the insect t months after being transplanted is given by P(t). P(t) = a. How many insects were discovered? In other words, what was the population when t = 0? b. What will the population be after 5 years? Round to the nearest whole insect. c. Determine the horizontal asymptote of P(t). Describe what the horizontal asymptote means in the context of the problem. Use the value of the horizontal asymptote in the explanation. Answer in a complete sentence using proper grammar and correct spelling. d. Sketch the graph of P(t). 8. State the domain, vertical asymptote and slant asymptote of the function C(x) = . 3t01.0 )t4.01(60 + + 6x 11x8x7 2 - +- Lab Module 4 Name _____________________ Show all work for each of the following problems. Simplify and clearly indicate all answers. 1. Complete the table below (use fractions–not decimals!) and use it to graph the function f(x) = 3x. x y Domain _______________ Range ________________ coordinates of the y-intercept ____________ equation of the asymptote _____________ –2 –1 0 1 2 2. Using your knowledge of inverses, prepare a table of values for the inverse of the function in problem #1. Graph the inverse on the same coordinate system as the function. Label each graph appropriately as f(x) or f-1(x). Answer the following for the inverse function: The equation of the inverse is f-1(x) = ______________ x y Domain _______________ Range ________________ coordinates of the x-intercept ____________ equation of the asymptote ____________ 1 2 3 4 5 6 7 8 9 -1 -2 -3 -4 -5 -6 -7 -8 -9 -9 -8 -7 -6 -5 -4 -3 -2 -1 9 8 7 6 5 4 3 2 1 y x Lab Module 4 page 2 3. Linda invests $3000 in a bond trust that pays 8% interest compounded monthly. Her friend Lyla invests $3000 in a certificate of deposit that pays 7.75% compounded continuously. For Linda: a. State which formula should be used to solve this problem. _____________________ b. Write the function for Linda. _____________________ c. Determine how much Linda would have in her account after 20 years. Work: Linda’s amount: _____________________ For Lyla: a. State which formula should be used to solve this problem. _____________________ b. Write the function for Lyla. _____________________ c. Determine how much Lyla would have in her account after 20 years. Work: Lyla’s amount: _____________________ Now, determine who has more money after 20 years, Linda or Lyla? _____________________ Lab Module 4 page 3 4. The population of Collin County, which follows the exponential growth model, increased from 491,675 in 2000 to 782,341 in 2010. a. Find the exponential growth rate, k. Don’t round, instead use the exact value of the growth rate. b. Write the exponential growth function. Use the exact growth rate found in a. c. What should the population be in 2016? Use the function from b. Write your answer in a complete sentence with proper grammar and correct spelling. d. When should the population be 999,999? Use the function from b. Write your answer in a complete sentence with proper grammar and correct spelling. e. How long will it take the population to double? Use the exact growth rate found in a. Write your answer in a complete sentence with proper grammar and correct spelling. Lab Module 4 page 4 For the following problems, use the exact rates. Write your final answers in a complete sentence with proper grammar and correct spelling. 5. The function D(h) = 8e -0.3h can be used to find the number of milligrams D of a certain drug that is in a patient’s bloodstream h hours after the drug has been administered. a. How many milligrams will be present after 4 hours? b. When the number of milligrams reaches 1, the drug is to be administered again. After how many hours will the drug need to be administered? 6. A radioactive isotope, selenium, used in the creation of medical images of the pancreas, has a half-life of 119.77 days. If 100 milligrams are given to a patient, how many milligrams are left after 20 days? 7. The Lualailua Hills Quadrangle of the East Maui (Haleakala) volcano on the island of Maui in Hawaii is no longer active. To find out the date of the last eruption, scientists conducted a chemical analysis of samples from the volcano area. The samples contained approximately 62.31% of its original carbon-14. How long ago was the last eruption of the volcano? (Use 5730 years for the half-life of carbon-14.) Lab Module 4 page 5 8. Cryptology is the science of making and breaking codes. This lab explores how the idea of functions and their inverses can be used to encode and decode messages. To encode and decode a message, first replace each letter of the alphabet with a positive integer using the following scheme, thus rewriting the original message as numbers instead of words: A – 1 F – 6 K – 11 P – 16 U – 21 Z – 26 B – 2 G – 7 L – 12 Q – 17 V – 22 Blank - 27 C – 3 H – 8 M – 13 R – 18 W – 23 D – 4 I – 9 N – 14 S – 19 X – 24 E – 5 J – 10 O – 15 T – 20 Y – 25 ENCODING: A one-to-one function can be used to encode a numerical message. For example, suppose you want to send the message MATH to a friend, and you have decided that the function f(x) = 3x + 4 will be the encoding function. This function simply describes the procedure used to create the encoded message – in this case multiply by 3 and add 4. First change the letters to corresponding numbers as shown above: 13 1 20 8. Then use these as the input values in f(x). f(13) = 3(13) + 4 = 43 f(1) = 7 f(20) = 64 f(8) = 28 So the encoded message that you send to your friend is: 43 7 64 28 NOTE: A graphing calculator can be used to evaluate a function as above: a. Press [Y=] on the calculator. Type 3x + 4. b. Choose [2nd][WINDOW] (TBLSET). Set the independent variable to Indpnt: Ask. c. Then choose [2nd][GRAPH] (TABLE). Input each x value (13 1 20 8). The calculator will return the function values (43 7 64 28) which is the encoded message. DECODING an encoded message: Now it is up to your friend to decode the message. Decoding is the process that “undoes” the encoding process. If f(x) encodes the message, what will decode it? The inverse of f(x) or f-1(x)! So f-1(x) will be the decoding function. In the example, the inverse of f(x) = 3x + 4 can be shown to be f-1(x) = . Take the encoded message (43 7 64 28) and use these values as input values in f-1(x). Again, the calculator can be used to decode – simply enter y = and use the TABLE feature as described above. f-1(43) = 13 f-1(7) = 1 f-1(64) = 20 f-1(28) = 8 That’s the original numerical message! The last step is to convert back to letters using the table given previously. Now your friend knows the message that you sent: MATH. 3 4x - 3 4x - Lab Module 4 page 6 Cryptology Applications: 8.1. Suppose f(x) = 6x – 2 is the encoding function. a. Encode “Collin Cougars”. [The answer will be numbers. Remember to include the space.] Code: _____________________________________________________________________ b. Find the inverse of f(x). c. Using the inverse function you just found, decode the following encoded message. Write the letters above the numbers in the code. [The answer should be words. Put spaces where appropriate!] 16 88 70 70 28 40 28 160 4 70 40 28 10 106 4 160 52 112 160 76 148 160 34 4 130 88 106 52 118 28 160 112 124 10 58 28 16 118 d. Why is it necessary for the encoding function to be one-to-one? Explanations should be specific to this assignment, as well as clearly and thoroughly written using complete sentences, correct spelling and proper grammar. Lab Module 4 page 7 8.2. You intercepted the following message from Boris and Natasha. 25 19 19 30 41 17 15 26 27 41 15 28 18 41 18 29 41 34 22 19 41 27 15 34 22 You do not know the encoding or decoding function but from previous work with Boris and Natasha you know their encoding and decoding functions are always linear and have 1 as the coefficient of x. You also know this message consists of 6 words. Using the encoding system described above, a. How would you decode the message? Explain your strategy in complete sentences using proper grammar and correct spelling. b. Identify the decoding function. c. What is the message? x y -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 x y -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 College Algebra Final Exam Review 1. For each of the following, use the given graph f(x): f(x) a) Find f(–2). b) State the zeros, the domain, and the range of the function f. c) Determine the intervals where the function is increasing and decreasing. d) Find the values of x at which the given function f has a relative minimum and maximum. What are the relative minimum/minima and maximum/maxima? 2.) Determine the intervals where the following function is increasing, decreasing and constant. Then, state the domain and range of the function. 3.) Find the difference quotient , h ≠ 0 for each of the following functions. a) f(x) = x2 – x – 2 b) f(x) = 3x2 – 2x + 6 4.) If the square root function is reflected over the y-axis and shifted 2 units down, what is the resulting function? Sketch the graph and state the domain. 5.) If the function f(x) = |x| is shifted 3 units to the left and vertically stretched by a factor of 4, what is the resulting function? Sketch the graph and state the domain. 6.) For the function of f(x) shown below, use transformations to sketch the graph of f(x – 3) + 2. 7.) Determine algebraically whether the following functions are even, odd, or neither. a) f(x) = 3x4 – 2x2 b) g(x) = 2x5 + x c) h(x) = 7x2 + 5x – 1 8.) Let f(x) = x2 – 9x and g(x) = 2x + 3. Find each of the following. a) (f + g)(x) b) (f – g)(x) c) (fg)(x) d) (f/g)(x) e) (g o f)(x) f) (f o g)(x) h )x(f)hx(f -+ x y -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 Final Exam Review page 2 9.) The formula S = can be used to approximate the speed S, in miles per hour, of a car that has left skid marks of length D, in feet. How far will a car skid at 70 mph? Round to one decimal. 10.) The function f(x) = 27 models the number of plant species, f(x), on an island in terms of the area, x, in square miles. What is the area of an island that has 54 species of plants? 11.) Use the piecewise-defined function a) Find f(–2), f(1), and f(3) b) Graph this function. 12.) Use the piecewise-defined function . a) Find g(–2), g(0), and g(2) b) Graph this function. 13.) A company charting its profits notices that the relationship between the number of units sold, x, and the profit, P, is linear. a) If 200 units sold results in $3100 profit and 250 units sold results in $6000 profit, write the profit function for this company. b) State the slope and interpret it in the context of the problem. 14.) A certain commodity has fixed costs of $1500 with a variable cost per unit of $22. The commodity is sold for $52 per unit. a) Write the cost function. b) Write the revenue function. c) Write the profit function. d) Find the break-even point. 15.) Write the resulting matrix after the row operations have been applied: 3R1 + R3 à R3 R2 à R2 16.) Solve using Gauss-Jordan Elimination: D2 2 7 3 1 x ï î ï í ì >- = <+ = 2xfor1x2 0xfor2 0xfor1x xf 2 )( î í ì ->+ -£ = 2xfor2x 2xfor1 xg )( ú ú ú û ù ê ê ê ë é -- - 7 1 4 123 550 321 5 1 ï î ï í ì =+- -=+- =+- 14z2yx3 14y4x2 22z3y3x page 3 Final Exam Review 17.) State the solution of each of the following: A = B = C = 18.) Find the augmented matrix of the corresponding linear system. a) Volunteers X, Y, and Z offered to stuff 864 envelopes with newsletters for a charity. Volunteer X could assemble 170 per hour, volunteer Y, 121 per hour, and volunteer Z, 146 per hour. They worked a total of 6 hours. The sum of the number of hours that volunteers Y and Z spent was twice what volunteer X spent. How long did each of them work? b) Ron attends a cocktail party. He wants to limit his food intake to 133 g protein, 120 g fat, and 165 g carbohydrate. According to the health conscious hostess, the marinated mushroom caps have 3 g protein, 5 g fat, and 9 g carbohydrate; the spicy meatballs have 14 g protein, 7 g fat, and 15 g carbohydrate; and the deviled eggs have 13 g protein, 15 g fat, and 6 g carbohydrate. How many of each snack can he eat to obtain his goal? 19.) Solve 2x2 + 16x + 26 = 0. 20.) A ball is thrown vertically upward from the top of a building 800 feet tall with an initial velocity of 80 feet per second. The distance d (in feet) of the ball from the ground after t seconds is d(t) = 800 + 80t – 16t2. What is the maximum height of the ball? After how many seconds will the ball hit the ground? 21.) P(x) = –x2 + 90x – 300 is a profit function where x is the number of items sold, and P(x) is the profit from that sale. Find the maximum profit and the number of items that must be sold to reach that profit. 22.) Analyze the polynomial f(x) = 7(x – 4)3(x + 1)2(x + 3) and complete the chart below. State the degree of the polynomial function. Determine the end behavior of the graph of the function. zeros multiplicity (how many times it occurs) Does the graph touch or cross at this intercept? 23.) Find all of the complex zeros for f(x) = 2x5 + 12x4 + 18x3. Write the function as a product of linear factors. 24.) Find all of the complex zeros for f(x) = x4 – 7x3 + 14x2 + 2x – 20. Also, use the Rational Zeros Theorem to list the possible rational zeros. 25.) Use the Rational Zeros Theorem to list the possible rational zeros of each of the following: a) f(x) = 2x3 + 12x2 + 18 b) g(x) = 5x4 + 7x3 + 3x2 – 4x – 19 é ù ê ú-ê ú ê úë û 1 0 0 5 0 1 0 1 0 0 1 8 é ù- ê ú ê ú ê úë û 1 0 4 7 0 1 6 13 0 0 0 0 é ù- ê ú ê ú ê úë û 1 0 3 8 0 1 5 9 0 0 0 2 Final Exam Review page 4 26.) Use synthetic division and the Remainder Theorem to find the indicated function value. Complete the following synthetic division to show a function, f(x), divided by x – 3. 3| 1 –2 –12 18 47 By the Remainder Theorem, state the value of f(3). ___________________ 27.) Find f(–3) using Synthetic Division and the Remainder Theorem if f(x) = 2x3 – 5x2 – 23x – 10. 28.) Suppose a polynomial function of degree 4 with rational coefficients has zeros of –4, 1, and 2i. Find the remaining zero. Write the polynomial function as a product of linear factors. 29.) Suppose that a polynomial function of degree 5 with rational coefficients has 4, –2i, and 4 + i as zeros. Find the remaining zeros. 30.) For each polynomial function graphed, determine the minimum possible degree, the zeros and if the multiplicity of the zeros is even or odd. a) b) 31.) When a person gets a single flu shot, the concentration of the drug in milligrams per liter after t hours in the bloodstream is modeled by the following equation. Find the horizontal asymptote of the function, F(t), and interpret what the horizontal asymptote represents with respect to the concentration of flu medication in the bloodstream as time passes. . 32.) For each rational function below, find the equation(s) of the vertical asymptote (VA), and equation of the horizontal asymptote (HA), if it exists. If non-existent, write NONE. a) b) c) 33.) A large group of students is asked to memorize a list of 60 Italian words, a language that is both unfamiliar and phonetically foreign to them. The group is then tested regularly to see how many of the words are retained over a period of time. The average number of words retained is modeled by the function W(t) = where W(t) represents the number of words retained after t days. Find the horizontal asymptote for the graph of the function and describe what it means in this context. 2.0t t6.7)t(F 2 + = 1x2 x5)x(R - = 1x x7xQ 2 - =)( 2x3 4x3xZ 3 + - =)( t t 256 + page 5 Final Exam Review 34.) State the domain of the following functions. a) f(x) = b) f(x) = c) f(x) = d) f(x) = e) f(x) = ln(3x + 4) 35.) Graph by applying transformations to . Give the equations of the asymptotes. 36.) State the domain and range of each graph. From the following graphs, determine whether each graph represents a one-to-one function. a) b) c) d) e) f) g) h) 37.) Find : a) f(x) = 3x3 + 2 b) c) h(x) = . 38.) State the domain (D), range (R), & equation of the horizontal asymptote (HA) for each function. a) b) g(x) = e-x – 5 c) h(x) = –ex – 5. 39.) State the domain (D), range (R), & equation of the vertical asymptote (VA) for each function. a) f(x) = log(2x – 8) b) g(x) = ln(3x + 12) c) h(x) = –log½(x) 40.) Convert to a logarithmic equation: 4x = 2.8. 41.) Simplify: a) log25(5) b) c) d) 5x3 - 22x + x+10 2 3x+2 3e 2x -+ 3 2x 1)x(f + - = x 1)x(f = -1f (x) 7 8x)x(g -= 3 1x + x+1f(x) = e +5 )(log 5aa 5 a alog ÷ ø ö ç è æ 5a a 1log Final Exam Review page 6 42.) Write the following log expression as the sum and/or difference of logs with no exponents or radicals remaining: . 43.) Write as a single logarithm: log(3) + 2log(x) – log(5) 44.) Solve: = 42x + 1. 45.) Solve 6 + e –4x = 9. 46.) Solve 3log2(8x) = 30. 47.) Solve log3(x + 1) – log3(x – 3) = log3(2). 48.) Solve: log2(x) + log2(x + 7) = 3. 49.) If a couple needs $14,500 for a down payment on a house and they invest the $7,250 they have at 5.8%, compounded continuously, how long will it take for their money to grow to the $14,500 needed? 50.) A radioactive substance decays according to the model A(t) = A0e–0.00472t, where A0 is the initial amount present and t is the time in years. a) If there are 20 grams present initially, when will there be 12 grams remaining? b) In how many years will 80% of the original amount remain? 51.) Write the first four terms of the sequence defined by an = n2 + 1. 52.) Write the first four terms of the sequence defined by an = 2n – 5. 53.) Write the first four terms of the sequence defined by an = (–2)n-1. 54.) Find and evaluate: . 55.) Find and evaluate: . 56.) Find and evaluate: . ÷ ÷ ø ö ç ç è æ - + 34 1zy4 2x3 )( log 64 1 å + = 4 1k 1k3 )( ( ) 1k4 1k 32 - = å -× )(å = 5 3k 2k Final Exam Review Answers page 1 1.) a) –4 b) –5, 0, 2; Domain: (–∞, 4); Range: [–4,∞) c) Intervals decreasing: (–∞, –2); (1, 3) Intervals increasing: (–2, 1); (3, 4) d) f has a relative maximum at x = 1, the relative maximum is 2. f has a relative minimum at x = –2, the relative minimum is –4; and f has a relative minimum at x = 3, the relative minimum is –2. 2.) Interval increasing: (–4, –3); Interval Decreasing: (2,4); Interval Constant: (–3, 2) Domain: [–4, 4]; Range: [–1, 3] 3.) a) 2x + h – 1 b) 6x + 3h – 2 4.) ; Domain: (–∞, 0]; 5.) f(x) = 4(x + 3); Domain: (–∞, ∞); 6.) 7.) Recall the algebraic definitions of even & odd: g(–x) = g(x) => g is even => symmetry about the y-axis g(–x) = –g(x) => g is odd => symmetry about the origin a) Even since f(–x) = f(x) b) Odd since –g(x) = g(–x) f(x) = 3x4 – 2x2 g(x) = 2x5 + x f(–x) = 3(–x)4 – 2(–x)2 = 3x4 – 2x2 g(–x) = 2(–x)5 + (–x) = –2x5 – x –g(x) = –(2x5 + x) = –2x5 – x c) Neither: Show that h(–x) does not equal h(x) and h(–x) does not equal –h(x) h(x) = 7x2 + 5x – 1 h(x) = 7(–x)2 + 5(–x) – 1 = 7x2 – 5x – 1 h(x) = –(7x2 + 5x – 1) = –7x2 – 5x + 1 2xxf --=)( x y -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 Final Exam Review Answers page 2 8.) a) (f + g)(x) = x2 – 7x + 3 b) (f – g)(x) = x2 –11x – 3 c) (fg)(x) = 2x3 –15x2 – 27x d) (f/g)(x) = e) (g o f)(x) = 2x2 – 18x + 3 f) (f o g)(x) = 4x2 – 6x – 18 9.) 200 feet 10.) 8 square miles 11.) a) f(–2) = 5, f(1) is undefined, f(3) = 5 b) 12.) a) g(–2) = 1, g(0) = , g(2) = 2 b) 13.) a) P(x) = 58x – 8500 b) The slope is 58. The profit increases at a rate of $58 for each unit sold. 14.) a) Cost function: C(x) = 22x + 1500 b) Revenue function: R(x) = 52x c) Profit function: P(x) = 30x – 1500 d) Break-even point: 50 units 15.) 16.) Solution: (1, –3, 4); Set of Row Operations: 2R1 + R2 à R2 –3R1 + R3 à R3 R2 à R2 3R2 + R1 à R1 –8R2 + R3 à R3 R3 à R3 6R3 + R1 à R1 3R3 + R2 à R2 3x2 x9x2 + - 2 ú ú ú û ù ê ê ê ë é - 19 5/1 4 1040 110 321 2 1 - 17 1 Final Exam Review Answers page 3 17.) a) (5, –1, 8) b) (4z + 7, –6z + 13, z) c) No Solution 18.) a) b.) 19.) 20.) maximum height: 900 feet; It will take 10 seconds for the ball to hit the ground. 21.) maximum profit: $1725; number of items: 45 22.) zero: 4 has a multiplicity of 3, crosses degree: 6 zero: –1 has a multiplicity of 2, touches end behavior: up to the left and up to the right zero: –3 has a multiplicity of 1, crosses 23.) zeros: –3 has a multiplicity of 2, 0 has a multiplicity of 3; f(x) = 2x3(x + 3)2 24.) zeros: –1, 2, 3 + i, 3 – i possible rational zeros: 25.) a) b) 26.) 3| 1 –2 –12 18 47 The remainder is 20, so f(3) = 20. 3 3 –27 –27 1 1 –9 –9 |20 27.) –40; synthetic division: –3| 2 –5 –23 –10 The remainder is –40, so f(–3) = –40. –6 33 –30 2 –11 10 |–40 28.) remaining zero: –2i; f(x) = (x + 4)(x – 1)(x – 2i)(x + 2i) 29.) 2i, 4 – i ú ú ú û ù ê ê ê ë é - 864 0 6 146121170 112 111 ú ú ú û ù ê ê ê ë é 165 120 133 6159 1575 13143 34x ±-= 20,10,5,4,2,1 ±±±±±± 2 9 2 3 2 11896321 ±±±±±±±±± ,,,,,,,, 5 19 5 1191 ±±±± ,,, Final Exam Review Answers page 4 30.) a) Degree 3; zeros: –2 (odd multiplicity), 0 (odd multiplicity), 1 (odd multiplicity) b) Degree: 4; zeros: –5 (odd multiplicity), 0 (even multiplicity), 5 (odd multiplicity) 31.) HA: y = 0; As time passes (t increases), the concentration of flu medication in the bloodstream approaches 0. 32.) a) VA: x = ; HA: y = b) VA: x = –1 and x = 1; HA: y = 0 c) VA: x = ; HA: None 33.) HA: y = 6. This means that most students will remember about 6 of the words for life. 34.) a) b) (–∞, ∞) c) d) (–∞, ∞) e) 35.) VA: x = 2 HA: y = 3 36.) a) (–∞, ∞) b) (–∞, ∞) c) (2, ∞) d) [–4,∞) e) (–∞, ∞) f) [0, ∞) g) (–∞, 2)U(2,∞) h) (–∞, ∞) (–∞, 3] [–3,∞) (–∞, ∞) (–∞, ∞) (2, ∞) [2, ∞) (–∞, 3)U(3,∞) (–∞, ∞) No No Yes No Yes Yes Yes Yes 37.) a) b) g-1(x) = 7x + 8 c) h-1(x) = x3 – 1 38.) a) D: (–∞, ∞); R: (5, ∞); HA: y = 5 b) D: (–∞, ∞); R: (–5, ∞); HA: y = –5 c) D: (–∞, ∞); R: (–∞, –5); HA: y = –5 39.) a) D: (4, ∞); R: (–∞, ∞); VA: x = 4 b) D: (–4, ∞); R: (–∞, ∞); VA: x = –4 c) D: (0, ∞); R: (–∞, ∞); VA: x = 0 40.) x = log4(2.8) 41.) a) b) 5 c) d) –5 2 1 2 5 3 2 - ÷ ø ö êë é ¥, 3 5 ÷ ø ö ç è æ ¥-È÷ ø ö ç è æ -¥- , 3 2 3 2, ÷ ø ö ç è æ ¥- , 3 4 31 3 2x)x(f -=- 2 1 5 1 Final Exam Review Answers page 5 42.) log4(3) + ½log4(x + 2) – 1 – log4(y) – 3log4(z – 1) 43.) 44.) x = –2 45.) x = – ≈ –0.2747 46.) x = 128 47.) x = 7 48.) x = 1 49.) 11.95 years 50.) a) in 108 years b) in 47.3 years 51.) 2, 5 10, 17 52.) –3, –1, 1, 3 53.) 1, –2, 4, –8 54.) 34 55.) –40 56.) 50 æ ö ç ÷ è ø 23xlog 5 4 )3ln(
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