COLLEGE ALGEBRA PRACTICE FINAL (Revised 3/04)

Transcription

1 Sketch the following graphs:. y x 0 COLLEGE ALGEBRA PRACTICE FINAL (Revised /0) + =. ( ) ( ) f x = x+. ( ) g x = x + 8x 7. y = x. y = x + 6. f ( x) = x + 7. h( x) x + = x + 8. g( x) = x x 9. y = x( x+ )( x )

2 0. f ( x ) = x. g( x) = x x. g( x ) = +. g( x) x + =. log x. ( ) ( f x = log x ) Use the graph of f ( x ) to sketch the following graphs: g( x) = f ( x ) 7. h( x) = f ( x) + 8. j( x) f ( x ) =

3 ( ) 9. Find the solution set for x + x x= 6. 6 b) { } a) 6 { } 8 0. Find the solution set for x, + x= a) { } b) { 7, 7} { + 7, 7 } + {, }. Find the solution set for x+ 7+ x+ 6 = a) { 9, } b) { 9, } { }. Find the solution set for x + 7 > a) { x x> 6} x 6< x< b) x x< 6 or x> x x>. Find the solution set for x + x a) { x x } b) { x x } x x { }. Find the solution set for x, a) ( ) b) ( ) ( ) (, 6) { x x } x > 6x. Express your answer in interval notation., 0,6 (, 0) ( 6, ) x + x+. Find the solution set for 0. Express your answer in interval notation. x a) (, ] [,] b) (, ] [,) (, ) (,] (, ) (,) 6. A chemist has an 80% alcohol solution and a 0% alcohol solution. How many liters of each should he use to obtain liters of a 60% alcohol solution? a) L 80%, L 0% b).l 80%,.L 0% L 80%, L 0%.8L 80%,.L 0%

4 7. If f ( x) x ( ) = + ( f g)( ) = and g x x, find +. a) b) 8. Given the two points P (, ) and Q (, ), find the equation of the line through P and Q. a) x+ y = 0 b) x+ y = x+ y = x y = 9. Find the domain of y = ( x )( x + x 0) x + x a) All real numbers b) { x x } { x x,} { x < x< } 0. Find the coordinates of the vertex for f ( x) = x x. a) (,) b) (, 7) (, ) (, 7). Determine the asymptotes of the graph of f ( x) = x + ( x ) a) VA: x = ; HA : y = b) VA: x = ; HA: y = 0 VA: x = ; HA : y = VA: x = ; HA : y = 0. What is the set of possible rational zeros for ( ) p x = x 9x + x + x 8? 9 a) ±, ±, ±, ± 6, ± 9, ± 8, ±, ±, ± b) ±, ±, ±, ±, ±, ±, ±, ± ±, ±, ±, ± 6, ± 9, ± 8 { } { ±, ± } ( ). Find all the zeros of p x = x 9x + x + x 8. a),,, b),,,,,, {,, i, i}

5 x+ y. Match the solution region of with a x y 0 region of the accompanying graph. B A C D a) A b) B C D. Find the center and radius of the circle x + y 6x+ y 6= 0 a) C: (,) r = 7 b) C: (,) r = 6 C: (, ) r = 7 C:, r = 6 ( ) 6. If f ( x) x ( ) = then f x =. a) x b) y x + x A = and B = : 0 i) Find AB. a) [ 0 ] b) [ ] Impossible 7. For the matrices [ ] ii) Find A. a) [ 00 ] b) [ ] Impossible iii) Find the determinant of B. 0 a) Impossible b) [ ]

6 8. For the function f ( x) = x, find ( + ) ( ) f x h f x h h x+ b) 0 a) h h h Find a polynomial of least degree, with lead coefficient, that has, + i, i for zeros. a) x x + 9x b) x + x + 9x+ x 6x + x 6 x + ( i ) x + ( 6i 7) x+ ( i ).x 0. Solve to decimal places: 8 = 00e a).9 b) Write using summation notation with index k, starting with k =. n 6 k + k k a) ( ) k b) ( ) ( k ) ( ) k = k k = k =. Find the fourth term in the expansion of ( p q) 7 a) 0 p q b) 760 p q. Find the solution set for log( x ) log( x ) a) { } { } 0 p q + = +. b) 7 0 k = 0 p q ( k ) +. Find the inverse of. 6 a) Does not exist 6 b) 6 6

7 x + y =. Solve the system of equations. Give the x coordinates only. x+ y = a) x = i, x= i b) x=, x= x=, x= x=, x= 6. Let Q represent a mass of radioactive radium ( 6 Ra ) (in grams), whose half-life is 60 years. The quantity of radium present after t years is given by Q = a) Determine the initial quantity (when t = 0 ). b) Determine the quantity present after 000 years. 7. An investment of $000 is compounded continuously for 0 years, when it is worth $ What is the interest rate, to the nearest percent? 8. On her 8 th birthday a woman planning for retirement invests $000 in a trust fund that pays 8% interest compounded continuously. How old is she when the account is worth a million dollars? t /60 9. What is the tripling time of an $8000 investment at % interest compounded monthly? 0. The population of a town increases according to the model ( ) 0.09 with t = 0 corresponding to 990. a) Estimate the population in 000. b) Estimate the population in 00. P t. t = 00e, where t is the time in years,. The work (in foot- pounds) done in compressing a volume of 9 cubic feet at a pressure of pounds per W = 90 ln 9 ln. Find W. square inch to a volume of cubic feet is ( ). Identify the quotient and remainder when x x is divided by x.. Identify the vertex, focus, and directrix of the parabola ( ) (. What is? 7 y = x + ) th. Write a formula for the n term of the arithmetic sequence,,, 8, and identify the 0 th term.

8 th 6. Write a formula for the n term of the geometric sequence, 9, 7, 8, and identify the 0 th term. 7. Which is the correct partial fraction decomposition setup for A B C x x x + A B Cx+ D + + x x x + a) + + A Bx + C Dx + E x x x + b) + + x x + ( x + )? Ax+ B Cx+ D x x + + P P k + n+ 8. Indicate the correct and in the mathematical induction proof of the proposition n = n + n a) P : = b) P : = P := P : 9. Find the center and foci of the ellipse ( x ) ( y+ ) P k + P k + P k + P k + + =. 9 y 60. What are the asymptotes of the hyperbola x =? 9 : k = k + k + k k+ k+ : = + k + k + : = k + k k+ : =

9 COLLEGE ALGEBRA PRACTICE FINAL SOLUTIONS (Revised /0) x=- y= x=- x= y=x y=

10 0. y= y= y= y= x=0 x=

11 9. D 0. B. C. B. D. C. B 6. A 7. C 8. E x y = 0 9. E { x x, } 0. A. B. A. B. A. C 6. D 7. i) A ii) D iii) B 8. D 9. A 0. D. C. B. B. D. B 6. a) grams b) 6. grams 7. 6% years 0. a) b) 9. 7 foot-pounds. x + x+ 6+ x. Vertex: (,) Focus: (,) Directrix: x =. 0. an = 7n+ 0 ; a0 = 60 n 6. an = ( ) ; a0 = D 8. C 9. Center: (, ) Foci: (, ) ( 7, ) 60. y =± x