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MATH 22 FINAL EXAM 1. Find an equation of the line that passes through the point (2, −3) which is parallel to the line 6x + 2y = 5 . a) 3x + y = 3 SAMPLE A 6. A manufacturer cuts squares from the corners of a rectangular piece of sheet metal that measures 5 in by 9 in. The metal is then folded up to make an open-top box. Letting x represent the cut side-lengths (in inches) of the squares, which of the following functions represents Volume in terms of x? b) x + 3y = −7 a) V = x3 c) x − 3y = 11 b) V = x(5 − x)(9 − x) d) 3x − y = 9 c) V = (5 − 2x)(9 − 2x) 2. The point (5, 7) lies on a circle whose center is at (2, 3). Find an equation of this circle. 7. Solve the inequality. a) (x + 3)2 + (y + 2)2 = 5 2 d) V = x(5 − 2x)(9 − 2x) 3x3 + 10x2 − 8x ≤ 0 2 b) (x − 2) + (y − 3) = 25 c) (x + 2)2 + (y + 3)2 = 5 d) (x − 2)2 + (y − 3)2 = 25 3. Let f (x) = 2x2 + 4x + c where c > 2. The function f has how many real zeros? a) The function has no real zeros a) (−∞, −4] 2 b) − 4, 0 ∪ ,∞ 3 2 c) − ∞, 0 ∪ ,∞ 3 2 d) − ∞, −4 ∪ 0, 3 b) The function has ONE real zero y c) The function has TWO real zeros d) More information is required 8. Which function is represented by the graph of y = f (x)? -5 -4 -3 -2 -1 2 4. Find the solution to the quadratic inequality x − 10x < −24. a) (0, 10) b) (4, 6) a) f (x) = (x − 3)(x2 + 1)(x + 4) c) (−24, 0] b) f (x) = −(x + 3)(x − 1)2 (x − 4) d) (−∞, 4) ∪ (6, ∞) c) f (x) = (x − 3)(x − 1)(x − 4) d) f (x) = −(x + 3)(x2 + 1)(x − 4) 5. Find the range of the function. √ f (x) = − x − 4 + 2 9. Solve the inequality. 3 ≤2 x−6 a) (−∞, 4] 15 − ∞, 6 ∪ ,∞ 2 b) [4, ∞) a) c) (−∞, 2] b) (−∞, 6) 15 c) ,∞ 2 d) [2, ∞) d) (−∞, 6) ∪ (6, ∞) 1 1 2 3 4 MATH 22 FINAL EXAM SAMPLE A 10. Use polynomial long division to find the quotient Q(x) and remain- 13. The function f has vertical asymptotes at x = −3 and x = 2, and der R(x) when a horizontal asymptote at y = 0. Which is the appropriate form for 4 3 2 (3x − 2x + 2) is divided by (x − 1). f (x)? a) Q(x) = 3x2 − 1, 2 b) Q(x) = −3x + x, R(x) = −3 y 4 R(x) = x + 2 3 c) Q(x) = 3x2 − 2x + 3, R(x) = −2x + 5 d) Q(x) = 3x2 − 2x + 1, R(x) = −2x + 3 2 1 -5 -4 -3 -2 -1 -1 −7x . 11. The one-to-one function f is defined by f (x) = 1 − 9x −1 Find f , the inverse of f . 1 2 3 4 5 x -2 -3 x a) f −1 (x) = 9x − 7 b) f −1 -4 1 − 9x (x) = −7x c) f −1 (x) = − d) f −1 (x) = x 7 + 9x 1 + 9x 7x a) f (x) = a (x − b)(x − c) b) f (x) = a(x − b)(x − c) (x − d)(x − e) c) f (x) = a(x − b) (x − c)(x − d) a(x − b) d) f (x) = (x − c) 12. Consider the polynomial P (x) = 2x4 − 4x2 + kx − 4. Find the value of k such that the remainder is 6 when P (x) is divided by x − 2. 14. Find all asymptotes. f (x) = a) k = 5 (3x − 1)(2 − x) (x + 2)(x − 7) b) k = −3 a) y = 0, c) k = 14 d) k = −21 b) y = 1 , 3 x = −2, x = 7, y = 2, x = −2, c) y = −3, x = −2, d) y = −3, x = 2, 1 , 3 x=7 x=7 1 15. The expression (x2 + 2) 2 − x2 (x2 + 2)− 2 one of the following expressions? a) √ 2 x2 + 2 p b) x2 + 2 3 c) (x2 + 2) 2 (1 − x2 ) 1 − x2 d) √ x2 + 2 2 x=2 x=7 x = −2, 1 x= is equivalent to which MATH 22 FINAL EXAM SAMPLE A 20. Which of the following is the graph of the function g(x) = 3 − ex ? 16. Solve for x. log2 3 − log2 8 = log2 (x) 8 3 4 b) x = 6 3 3 c) x = 8 2 a) x = y 1 a) 1 d) x = 6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 1 2 3 4 5 x -2 17. Given f (x) = 5x−3 , find f −1 (x). -3 -4 a) f −1 (x) = log5 (x) + 3 b) f −1 (x) = ln(x) + 3 c) f −1 y 4 ln(x) + 3 (x) = ln 5 3 2 d) f −1 (x) = log5 (x − 3) 1 b) 18. Solve for x. -5 -4 -3 -2 -1 -1 1 log36 x = 2 x -2 1 a) 6 -3 -4 b) 18 c) y 1 18 2 1 d) 6 c) 19. Solve for x. 5x 2 +5x+8 = 25−3x−8 -1 -1 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 x -2 -3 -4 a) 4 and −4 -5 b) 8 and 3 -6 c) −8 and −3 y d) −4 6 5 4 d) 3 2 1 -1 -1 -2 3 x MATH 22 FINAL EXAM 21. Identify the false statement. SAMPLE A 25. Find the domain of the function. f (x) = − ln(3x + 2) + 4 √ 1 a) ln( e) = 2 x log x b) log = y log y a) (−∞, 4] 2 b) (− , ∞) 3 c) log3 (20) is between 2 and 3 c) [4, ∞) d) The range of f (x) = ln(x) is the set of all real numbers 2 d) (e 3 , ∞) 22. Solve for x. log2 (2x2 − 4) = 5 r a) x = ± 29 2 b) x = ±6 r c) x = ± 2+ 1 log2 (5) 2 √ d) x = ±3 2 FINAL EXAM - SAMPLE A - KEY 1. A 2. B 3. A 4. B 5. C 6. D 7. D 8. B 9. A 10. C 11. A 12. B 13. C 14. C 15. A 16. C 17. A 18. D 19. C 20. A 21. B 22. D 23. A 24. A 25. B 23. Solve for x. ln(x) + ln(x + 1) = ln(2x + 6) a) x = 3 b) x = 3, −2 c) x = 5 d) no solution 24. Solve for x. Write the answer in terms of base-10 logarithms. 48x = 7x+9 a) x = 9 log(7) 8 log(4) − log(7) b) x = 9 log(7) 8 log(4) − 1 c) x = 9 log(7) 7 log(4) d) x = log(7) log(4) 4