College Algebra. Some of the vocabulary and skills to review are included in the following table:

Transcription

1 MATH 11 College Algebra FINAL EXAM Review The Final Exam is cumulative. Problems will be similar to those you have seen on previous exams. The exam is worth 1 points. Formulas provided on Exam will also be provided on the Final. Part I. Multiple Choice. The exam will have 15 multiple choice questions which will cover concepts, definitions, and symbolism. The questions will be similar to the MC questions on previous exams. Part II. The remaining questions will be graded on partial credit. Graphs should be clearly labeled. Some of the vocabulary and skills to review are included in the following table: Vocabulary (concepts) Skills (some) function logarithm long division zero of a function exponential synthetic division domain difference quotient interpreting a graph range slope factoring matrix transformations factoring a polynomial row echelon form composition quadratic formula discriminant piecewise function u-substitution asymptote inverse function squaring a binomial rational function polynomial function composing functions systems of equations: end behavior interval notation - dependent increasing/decreasing/constant sign test for inequalities - independent maximum/minimum - consistent rate of change - inconsistent basic function 1. f(x) = x x g(x) = 1 x 1 (a) Find and simplify f( ) (b) Find and simplify (g h)(x) (c) Find and simplify (i) (f h)(x) (d) Simplify the difference quotient: h(x) = 3x + 5 (ii) (f h)( 1) f(x + h) f(x) h. Write the equation of the line through the points (3, 7) and (-3, ). 3. Write the equation of the line through (-,5) that is perpendicular to the line x + 3y + 6 =.

2 . Describe each transformation. Use the graph of y = f(x) given to sketch each transformation. (a) y = f(x 1) + (b) y = 1 f(x) (c) y = f(x) Math 11 Exam Final Fall 11 Grading Guide 61. MC::Function::Substitution (V: ) 5. Graph the piecewise function: ( points) If { f(x) = 3x + (x + ), then f(x + h) = 3 x < f(x) = a) 3x +3h + x b) x 3x < +h ( a ) ( points) c) 3x + + 3h + d) 3x + + h 6. MC::Function::Substitution (V: 1) ( d ) ( points) ( points) If f(x) =x, then f(x + h) = a) x + h b) x + h c) x 3 + x h d) x +xh + h 63. Graphing::IntersectingFunctions::Inequality (V: ) (randomization) {x x < 1} or (, 1) (3 points) (3 points) Use the following figure to find the value(s) of x for which f(x) <g(x). 6. Use the graph to solve f(x) < g(x) f(x) g(x)

3 7. Graph the function f(x) = x + x 6. [Label the scale clearly.] (a) What is the vertex? (b) Over what interval(s) is f(x) increasing? (c) Over what interval(s) is f(x) decreasing? (d) Find the real zero(s) for f(x). 8. A cell phone company develops a pay-as-you-go cell phone plan in which the monthly cost varies directly as the number of minutes used. If the company charges $17.7 in a month when 36 minutes are used, what is the constant of variation? What should it charge for a month in which 5 minutes are used? 9. Demand for a product varies inversely with the price per unit of the product. Demand for the product is 1, units when the price is $5.75 per unit. What is the constant of variation? What is the demand (to the nearest hundred units) when the price is $6.5? 1. The profit that a vendor makes per day by selling x pretzels is given by the function P (x) =.x +.8x. Find the number of pretzels that must be sold to maximize profit. 11. The height of a model rocket launched from a 11 foot cliff is given by h(t) = 16t +96t+11, where t is the time, in seconds, from when the rocket is launched. Find the maximum height the rocket attains above the cliff. 1. Decide whether (x + ) is a factor of P (x) = x 3 + 1x + 3x.

4 13. Describe the end behavior of the graph of f(x) = x + x For the polynomial function P (x) = (x 3)(x )(x + 1), list all the zeros and their multiplicities. Determine whether the graph of P (x) touches or crosses at each zero. 15. Factor f(x) = 3x 3 + x 3x + into linear factors given that is a zero of f(x). 16. Use the Rational Zeros Theorem to list the possible rational zeros for f(x) = 1x 3 13x 13x. 17. Give the equations for any vertical, horizontal, or oblique asymptotes of (a) f(x) = x x x (b) g(x) = 3x x + 5 (c) h(x) = x + 3 x 1 x x x Sketch a graph of f(x) = x 3. Label all intercepts and asymptotes. x Solve each inequality analytically (sign test). Verify graphically. (x )(x + 1) (a) (b) x 5x + < (x + ). Find a formula for the inverse of f(x) = x 3 (a) What is the domain for f(x)? (b) What is the domain for f 1 (x)? 1. (a) Write in logarithmic form: 3 =.15 (b) Evaluate: log 7 8 (c) Evaluate: ln e 3.6. Solve for x: log (6x ) =.1 3. Solve for x: x 5 = 1 6

5 . Solve for x: x 3 = 7 x 5. Solve for x: e 5x 1 = 6. Solve for x: 3x x + 11 = 7. Solve for x: x 3 = 9 8. Use the properties of logarithms to express the following as a sum, difference, or product of simpler logarithms: log m3 n y 9. Suppose $15, is invested at 3.5%. Find the total amount present at the end of 5 years if interest is compounded (a) quarterly (b) continuously 3. If Tim has $1 to invest at 7% compounded continuously, how long will it be before he has $15? How long before his investment is doubled? 31. If Seattle had a population of.3 million in 199 and 1.75 million in 195, use the continuous growth model A(t) = A e kt to find the rate of growth and predict the population for The radioactive isotope of potassium K which is used in the diagnosis of brain tumors, has a half-life of 1. hours. If 5 milligrams are taken, how many milligrams will remain after hours? 33. A sheet of heavy-duty cardboard measuring inches by 3 inches is to be made into an open box by cutting out equal-sized squares of side length x from each corner and folding up the sides. a) What are the restrictions on the values for x? b) Write a function V, which will give the volume of the open box. c) Find the volume if a square that is.5 inches on a side is cut out. d) For what value of x will the volume be a maximum? What is the maximum volume?

6 3. Solve the following linear systems by substitution, elimination, or matrix row operations. { { 6x + 3y = 1 x + 3y = 7 (a) (b) x y = y x = 1 (c) { 1 3 x y = 1 5x 3y = 18 x + y + z = 3 (d) x + 7y + 15z = 1 x + 7y + 13z = 1 (e) { x + 1 y = 7 8x + y = The given matrix represents a linear system. Write the solution to the system Write the augmented matrix for the linear system below. Then, perform the indicated row operations (in order) to transform the matrix. x y + 3z = x + y z = 5 y + z = 1 a) R 1 R b) R + R 3 R Set up a system of equations to solve. Then solve by substitution, elimination, or matrix row operations. An inheritance of $5, is divided among three investments yielding $ in simple interest per year. The interest rates for the investments are 3%, %, and 6%. The difference in the amount of money invested in the fund paying 6% and the amount invested for the least return is the same as the amount invested at %. Find the amount of each investment.

7 Answers: 1. (a) 8 (b) 1 3x + (c) (i) x 5x 5 (i) 1 (d) x + h. y = 1 x y = 3 x (a) The new graph is shifted 1 unit right and units up. Key points: (-1, ), (1, ), (3, ) (b) The new graph is a vertical shrink by a factor of 1. Key points: (-, -1), (, 1), (, -1) - - (c) The new graph is reflected over the x-axis. Key points: (-, ), (, -), (, ) open circle on (,) and (,-) (, 1) (a) (-1, -7) (b) [ 1, ) (c) (, 1] (d) x = 1 ± 7

8 8. k = $.75 charge per minute; $ k = 57, 5; 88 units 1. 7 pretzels 11. maximum height above the cliff: 1 feet 1. yes; P ( ) = 13. falls on the left (as x, f(x) ); falls on the right (as x, f(x) ) 1. crosses at x = ± 3, (each multiplicity 1) and touches at x = 1 (multiplicity ) 15. f(x) = 3(x )(x + )(x 5 3 ) or f(x) = (x )(x + )(3x 5) 16. ±{1,,, 5, 1,, 1, 1 5, 1 1, 5, 5, 5 } (a) (b) (c) VA: x = 7, x = 3 VA: x = VA: x = 3, x = 1 HA: y = 1 HA: none HA: y = OA: none OA: y = 3x + OA: none VA: x = 1 HA: y = y-int: (, -3) x-int: ( 3, ) 19. critical values: (a), -1, - (b) 1, test intervals: (, ), (, 1), (, 1), (1, ), (, ) ( 1, ), (, ) work: show tests answer: (, ) [ 1, ] (1, ) f 1 (x) = (x + 3), x 3 (a) Domain f : [, ) (b) Domain f 1: [ 3, ) 1. (a) log.15 = 3 (b) 1.71 (c)

9 . x x = 1. x = 5. x = ln log log log x = 1 ± i x = 3 and x = log m + log n ( 1 ) log y 9. (a) $17,635.1 (b) $17, years; 9.9 years 31. k = ln ( ) ;.73 million people in k = ln(.5) ; 13.7 milligrams remain after hours 33. a) < x < 15 b) V (x) = ( x)(3 x)x = 1x 1x + x 3 c) 99.5 cu.in. d) x 5.7in., V (5.7) 33. cu.in. 3. (a) ( 1, 3) ((b) (6, 5) (c) (d) (1, -, ) (e) (7 1 y, y) 35. (z +, 1 z 3, z) augmented matrix (a) (b) x = $ invested at 3% y = $ invested at % z = $ invested at 6% x + y + z = 5.3x +.y +.6z = z x = y x = $1, y = $15, z = $5