(MATH 1203, 1204, 1204R)

Transcription

1 College Algebra (MATH 1203, 1204, 1204R) Departmental Review Problems For all questions that ask for an approximate answer, round to two decimal places (unless otherwise specified). The most closely related question from the text (and MML) is indicated next to the problem number. GC indicates technology should be applied in solving the problem. Lastly, the associated student learning outcome is in parentheses, e.g. (SLO2) means the question assesses understanding of the second student learning outcome. Chapter 1 1] (SLO1) Determine the domain and range of the following relation, and decide whether the relation represents a function or not. 2] (SLO1) {(3,6), (18, 4), (35,6), (3,2), (57,2)} Determine the domain D and range R of the following relation from its graph, and decide whether the relation represents a function or not. 3] (SLO1) - see problem 21 For f(x) = x 2 + x + 9, find f(2). 4] (SLO1) Use the graph to find f(1).

2 5] (SLO2) Given the linear function: f(x) = 2x + 6 graph the function and find the following: a) x-intercept b) y-intercept c) domain d) range e) slope 6] (SLO8) By noon, 5 inches of rain had fallen during a storm. Rain continued to fall at a rate of 1/5 inch per hour. a) Find a formula for a linear function f that models the amount of rainfall x hours past noon. b) Find the total amount of rainfall by 1:30PM (using the formula from part a.) 7] (SLO2) Find the slope-intercept form of the line passing through (5,-8) and (7,-5). NOTE: Could be given one point and the slope instead. 8] (SLO2) Write an equation of the line containing the point (6, 9) and perpendicular to the line: 9x + y = 7 Note: May instead ask for the parallel line through the point. 9] (SLO8) A person is driving a car on a straight road. The graph shows the distance y in miles that the individual is from home after x hours. Answer the following: a) Find the slope-intercept form of the line. b) How fast is the car traveling?

3 c) How far from home was the person initially? d) How far was the individual from home after 3 hours and 15 minutes? 10] (SLO2) see problem 40 Find the zero of the following function, algebraically: f(x) = 4x 8 11] (SLO2) Solve the equation, algebraically: 2[x (4x + 16) + 6] = 2(x + 10) 12] (SLO2) Solve the following inequality and give the solution in interval notation. 1 5 x 1 7 x 4 13] (SLO8) The length of a rectangular mailing label is 5 cm less than twice the width. The perimeter is 50 cm. Find the dimensions of the label by setting up an appropriate equation and solving. Note: Could involve a triangle instead. 14] (SLO2) see problem 33 Solve for m: q = 3m + 3w 15] (SLO8) In planning retirement, Liza deposits some money at 3.5% interest and twice as much at 4.5%. Find the amount deposited at each rate if the total annual interest income is $1250.

4 Chapter 2 16] (SLO2) Given the graph of the function below answer the questions that follow: a) On what open interval(s) is the function increasing? b) On what open interval(s) is the function decreasing? c) On what open interval(s) is the function constant? d) What is the domain? e) What is the range? 17] (SLO1) Describe the series of transformations applied to the graph of y = x that would result in the graph of y = 1 x Be 4 specific about direction and magnitude. Note: May involve any of the basic functions and any series of transformations. 18] (SLO1) Give the equation of the function whose graph is the same shape as y = x 2 but shrunk vertically by a factor 1/4 and shifted 7 units downward. Note: May involve any of the basic functions and any series of transformations. 19] (SLO1) The given figure shows a transformation of the graph of y = x. Write the equation for the transformed graph.

5 20] (SLO8) The graph of y = f(x) represents the amount of water in thousands of gallons remaining in a swimming pool after x days. Answer the following questions: a) Estimate the initial and final amounts of water in the pool. b) When did the amount of water remain constant? c) Approximate f(2) and f(4). d) At what (average) rate was water being drained from the pool when 1 x 5? 21] (SLO2) For the piecewise linear function: f(x) = { 2x, x 2 find the following: x 2, x > 2 a)f( 3) c)f(0) b)f( 2) d)f(2) 22] (SLO2) 2 x, x 1 Graph the piecewise-defined function: f(x) = { 3 + 2x, x > 1 23] (SLO1) Let f(x) = 6x + 5 and g(x) = 4x 7. Find the following and give the domain of each: a) (f + g)(x) b) (f g)(x) c) (fg)(x) d) ( f ) (x) g e) (f g)(x) f) (g f)(x) Note: May involve non-linear functions.

6 24] (SLO1) Find and simplify the difference quotient f(x+h) f(x) for the function: f(x) = 5x x + 4 h 25] (SLO8) The fixed cost of a certain business is $700 and the cost to produce each item is $15. The selling price of each item is $35. Find each of the following: a) The cost function C(x) b) The revenue function R(x) c) The profit function P(x) d) Determine the break-even point (the number of items that must be produced before a profit is realized). e) Graph P(x). 26] (SLO8) An oil well off the Gulf Coast is leaking, with the leak spreading oil over the water s surface as a circle. At any time t, in minutes, after the beginning of the leak, the radius of the circular oil slick on the surface is r(t) = 3t feet. Let A(r) = πr 2 represent the area of a circle of radius r. a) Find (A r)(t). b) The answer to part a) defines the in terms of. c) What is the area of the oil slick after 5 minutes? Chapter 3 27] (SLO2) Given the quadratic function: P(x) = 2x 2 2x 8: a) Write the function in the form P(x) = a(x h) 2 + k b) Give the vertex of the parabola. c) Sketch a graph of the function. Note: Does not have to be done by completing the square. 28] (SLO2) Use the graph of the function P(x) = 2(x + 5) to answer the following: a) Give the coordinates of the vertex. b) Give the domain and the range. c) Give the equation of the axis of symmetry. d) Give the largest open interval over which the function is increasing. e) Give the largest open interval over which the function is decreasing.

7 f) State whether the vertex is a maximum or minimum point, and give the corresponding maximum or minimum value of the function. 29] GC (SLO2) Graph the function f(x) =.22x 2 + 2x +.53 in an appropriate viewing window and approximate the following: a) the coordinates of the vertex b) the x-intercepts 30] (SLO2) Find all solutions (real or imaginary) of the quadratic equation: (5k 1) 2 = 8 31] (SLO2) Find all solutions (real or imaginary) of the following equation: x(5x 4) = 28 32] (SLO2) Solve the inequality analytically. Verify your answer graphically. 5x + 24 < x 2 33] (SLO2) Solve the following equation for v. F = qmv2 r 34] (SLO8) The height, h, of a frog after jumping off of a stump is a function of its distance, x, from the base of the stump and is given by h(x) =.5x x where h is in feet. a) How high is the frog when its horizontal distance from the base of the stump is 2 feet? b) At what two distances from the base of the stump after it jumped was the frog 4.3 feet above the ground? c) At what distance from the base did the frog reach its highest point? d) What was the maximum height reached by the frog?

8 35] GC (SLO3) For the function: P(x) = 2x 3 14x 2 + 2x + 88, use technology to answer the following: a) Sketch a graph on the window: [-10, 10, -100, 100] b) Determine all local minimum points and tell whether any are absolute minimum points. c) Determine all local maximum points and tell whether any are absolute maximum points. d) Determine the range. e) Determine the x and y intercepts. f) Give the open interval(s) over which the function is increasing. g) Give the open interval(s) over which the function is decreasing. 36] (SLO3) Completely factor P(x) = 3x 3 8x 2 31x + 60 into linear factors, given that -3 is a zero. 37] (SLO3) Find a cubic polynomial in standard form with real coefficients, having the zeros 2 and 4i. Let the leading coefficient be 1. Note: Could be given one real zero and one irrational zero instead. 38] (SLO3) For the polynomial P(x) = x x 2 324, -3 and 3 are zeros. Find the remaining zeros. 39] (SLO3) Find a polynomial of least degree with the following graph:

9 40] (SLO3) Given the polynomial P(x) = 6x x x + 18 a) List all possible rational zeros b) Find all rational zeros and all remaining zeros, if any. c) Factor P(x). Note: May involve two irrational or complex zeros. 41] (SLO3) Find all solutions to the following equation: 6x 3 = 36x 2 54x 42] (SLO3) Find all solutions to the following equation: 2x 3 + x 2 = 8x ] GC (SLO3) Use graphical methods to find all real solutions of the following equation:. 95x x x + 20 = 0 44] GC (SLO8) a) A piece of rectangular sheet metal is 12 inches wide. It is to be made into a drain gutter by turning up the edges to form parallel sides. Let x represent the length of each of the parallel sides. a) Determine a function A that gives the area of a cross-section of a gutter. b) What is the domain of A (i.e. what are the restrictions on x)? c) For what value of x will the cross-sectional area be a maximum? d) What is the maximum cross-sectional area? e) The cross sectional area will be less than 10 square inches for what values of x? Chapter 4 45] (SLO4) Sketch the graph of f(x) = 8 2x by hand. Include the asymptotes as dashed lines, clearly indicate the intercepts and the coordinates of any holes that 10 x exist. Note: May involve y=0 as a horizontal asymptote.

10 46] (SLO4) Sketch the graph of f(x) = 3x2 12 by hand. Include the asymptotes as dashed lines, clearly indicate the intercepts and the coordinates of any holes that x 2 6x+8 exist. 47] (SLO4) Solve the following equation, algebraically: 3 x 2 4x 1 x 2 16 = 0 48] (SLO8) Suppose that an insect population in millions is modeled by f(x) = 11x+1, where x 0 is in months. x+1 a) Sketch a graph of f in the window [0, 14] by [0, 14]. b) Determine the initial insect population. c) Find AND INTERPRET the equation of the horizontal asymptote. 49] (SLO4) see problem 62 Determine the domain of the function f(x) = 2x + 3 Note: Could also involve a cube root function. 50] (SLO4) 3 Consider the following function: f(x) = 6x 12 and answer the following: a) Sketch a comprehensive graph using a suitable graphing window. b) Determine the range. c) Determine the interval(s) over which the function is decreasing, if any. d) Solve the equation: f(x) = 0 either analytically or by using the graph. Note: Could also involve a square root function. 51] (SLO4) Solve the following equation, algebraically: 52] (SLO4) Solve the following equation, algebraically: x 5 = 5x x 1 = 2

11 Chapter 5 53] (SLO1) For the function: f(x) = 3x 5, determine whether f(x) is one-to-one. If so: a) Write an equation for the inverse function y = f 1 (x) b) Graph f and f 1 on the same axes c) Give the domain and range of both functions 54] (SLO1) Find a formula for the inverse, f 1 (x), of: f(x) = 3x ] (SLO5) Suppose f(x) = 2 x 1 a) Graph f(x) by hand (by making a table of values). b) Find the domain and range of f(x). c) What is the equation of the asymptote? d) Is the function increasing or decreasing on its domain? Note: This question could contain any base (including e ) and involve a vertical translation. 56] (SLO5) Sketch the graph of f(x) = 3 x by hand by making a table of values. Then use your knowledge of transformations to sketch the graph of g(x) = 3 x 1 on the same axes. Clearly label your graphs. Note: This question could contain any base (including e ) and involve a horizontal translation. 57] (SLO5) Solve the equation analytically. 2 2x 3 = 16 58] (SLO8) Use the appropriate compound interest formula to find the amount that will be in each account, given the stated conditions. $39,000 invested at 2% annual interest for 3 years compounded (a) annually; (b) daily 59] (SLO5) Use the properties of logarithms to rewrite the expression as a sum or difference of logarithms (with no exponents). Assume all variables are positive. log b ( xy8 z 7 )

12 60] (SLO5) Write the expression as a single logarithm with coefficient 1. Assume all variables are positive. 4 log c x 6 log c y 5 61] (SLO5) The graph of the exponential function f(x) = ( 1 6 )x is given, with three points. a) Graph f 1 (x) = log 6 x. b) Find the domain and range of the inverse. c) Does the inverse increase or decrease over its domain? d) What is the equation of the vertical asymptote for f 1 (x)? 62] (SLO5) Find the domain of the function y = log (4x + 1) analytically. 63] (SLO5) Sketch the graph of f(x) = log 3 x by hand by making a table of values. Then use your knowledge of transformations to sketch the graph of g(x) = 5 + log 3 x on the same axes. Clearly label your graphs. Note: This question could contain any base (including e ) and involve a horizontal translation. 64] (SLO5) For the exponential function f, find f 1 analytically and graph both on the same set of axes. f(x) = 2 x 1 65] GC (SLO5) Use technology to approximate the solution(s) to the nearest hundredth: e x = 6 ln(x)

13 66] (SLO5) Solve the following equation analytically, and give the exact answer in terms of ln: 5e 3x + 5 = 7 67] (SLO5) Solve the following equation analytically, and give the exact answer in terms of log: 2(10 x ) = 8 68] (SLO5) Solve the exponential equation 2 x = 10 and express the solution in exact form in terms of either log or ln. Then give an approximation of the solution to the nearest thousandth. 69] (SLO5) Solve the logarithmic equation 5 ln x = 25 and express the solution in exact form. 70] (SLO5) Solve the logarithmic equation log 6 (2x 1) = 1 71] (SLO5) Solve the logarithmic equation log 4 x + log 4 (x 6) = 2 72] (SLO5) Solve the equation: ln(6x + 5) ln(x) = ln 7 73] (SLO8) The half-life of a radioactive substance is 21.8 years. Answer the following: a) Find the exponential decay model for this substance. b) How long will it take a sample of 1000 grams to decay to 700 grams? c) How much of the same of 1000 grams will remain after 20 years? 74] (SLO8) The table lists future concentrations of a certain greenhouse gas in parts per billion (ppb), if current trends continue. a) Let x=0 correspond to 2010 and x=20 correspond to Find C and a so that f(x) = Ca x models the data. b) Using the model, estimate the gas concentration in the year Year Gas (ppb)

14 75] (SLO8) How much time will be needed for $33,000 to grow to $37, if deposited at 4% compounded quarterly? Use the formula A = P (1 + r n )nt Note: This question could also involve continuous compounding. 76] (SLO8) In 2000 the population of country A reached 3 million, and in 2025 it is projected to be 7.5 million. a) Find values for P 0 and a so that the following models the population of country A in year x. f(x) = P 0 a x 2000 b) Estimate the country s population in 2010 to the nearest hundredth of a million. c) Use f to determine the year during which the country s population might reach 10 million. Chapter 6 77] (SLO6) Use Gaussian or Gauss-Jordan elimination with matrices to solve the system. x y = 6 y z = 0 x + z = 2 78] (SLO6) Solve the system of equations using Gaussian or Gauss-Jordan elimination. 35x 2y + z = 96 20x + y = 57 40x 2y + z = 11 Note: The system of equations could have no solution or infinitely many solutions instead. 79] (SLO8) The money manager for a small company puts some money in a bank account paying 3% per year. He uses some additional funds, amounting to 1/3 the amount placed in the bank, to buy bonds paying 4% per year. With the remainder of the funds he buys a certificate of deposit paying 9% per year. The first year the investments return a total of $578 in interest. If the total of the investments is $10,000 how much is invested at each rate? 80] (SLO6) Graph the solution set of the system of inequalities. x 6y < 2 6x + y > 2

15 81] (SLO6) Graph the solution set of the system of inequalities. 2x + y 5 x 8 x 0 y 0 82] (SLO6) Theo, who is dieting, requires two food supplements, I and II. He can get these supplements from two different products, A and B, as shown in the table. Theo s physician has recommended that he include at least 19g of each supplements in his daily diet. If product A costs 27 cents per serving and product B costs 25 cents per serving, how can he satisfy his requirements most economically (for the lowest cost)? Supplement I II (g/ serving) Product A 3 2 Product B ] (SLO6) Cabinet X costs $40, requires 12 square foot of floor space, and holds 36 cu ft. of files. Cabinet Y costs $50, requires 4 sq. ft. of floor space, and holds 20 cu ft. To get maximum storage, how many of each should be purchased with a budget limit of $500 and floor space of 84 sq. ft.? Chapter 8 84] (SLO7) For the arithmetic sequence, find a 19 and a n when a 1 = 6 and d = 4. 85] (SLO7) Find the sum of the first 62 terms of the arithmetic sequence: 7, 9, 11, 13, 86] (SLO7) Find the sum: 37 ( 4j + 3) j=1 87] (SLO8) A growing antelope population increases by 3 animals per year. If the current population is 33 animals, what will it be in 16 years? Use an appropriate formula.

16 88] (SLO8) How much material would be needed for the rungs of a ladder with 29 rungs, if the rungs taper uniformly from 18 inches to 30 inches? Use an appropriate formula. 89] (SLO7) For the geometric sequence, find a 7 and a n when a 1 = 6, r = 2. 90] (SLO7) Use an appropriate formula to find the sum of the first 12 terms of the following geometric sequence: 18, 9, 9 2, 9 4, 91] (SLO7) Find the sum: ( 1 j 3 ) j=1 92] (SLO7) Find the sum of the infinite geometric series: ] (SLO8) Find the future value of an annuity with payments of $6000 at the end of each year for 10 years at 8% interest compounded annually. 94] (SLO8) Each person has two parents, four grandparents, eight great-grandparents, and so on. What is the total number of ancestors a person has, going back four generations? Fifteen generations? Use an appropriate formula to find the answers.