Final+exam+review+Spring+2024-1.pdf

MAC 1147 FINAL EXAM REVIEW SPRING 2024 1. Use the graphs of f and g to evaluate the composite function. (f ◦ g)(1) = 1. Use the graphs of 𝑓 and 𝑔 to evaluate (𝑓𝑜𝑔)(1) (f g)(1) ◦ y 10...

MAC 1147 FINAL EXAM REVIEW SPRING 2024 1. Use the graphs of f and g to evaluate the composite function. (f ◦ g)(1) = 1. Use the graphs of 𝑓 and 𝑔 to evaluate (𝑓𝑜𝑔)(1) (f g)(1) ◦ y 10 y = f(x) 8 6 4 2 x -10 -8 -6 -4 -2 2 4 6 8 10 -2 y = g(x) -4 -6 -8 -10 2. Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. log 4 (𝑥 − 4) + log 4(𝑥 − 10) = 2 √𝑥−1 3. Find the domain of the function 𝑔(𝑥) = 𝑥−9 √2 4. Find the exact value of sin−1 ( 2 ) 5. Solve the triangle below 7 30° 115° 6. Evaluate 𝑓(−1) for the given piecewise function 4𝑥 − 3 𝑖𝑓 𝑥 < −1 𝑓(𝑥) = { −2𝑥 + 4 𝑖𝑓 𝑥 ≥ −1 7. Form a polynomial whose zeros and degree are given below Zero: − 8 multiplicity 1 Zero: 1, multiplicity 2 Degree: 3  −3𝜋 8. Use reference angles to find the exact value of sec ( 4 ). Do not use a calculator. 9. Find the domain of 𝑓(𝑥) = √14 − 𝑥 𝑦2 𝑥2 10. Find the vertices and locate the foci for the hyperbola 25 − 100 = 1 11. Solve 3𝑥+6 = 8 3𝜋 12. Evaluate cot ( ) 2 13. Graph the ellipse and locate the foci given 4𝑥 2 = 64 − 16𝑦 2 14. Find the domain of 𝑓(𝑥) = log 9 (𝑥 − 6) 15. Complete the identity sin2 (𝑥) + sin2(𝑥) cot 2(𝑥) = ? 16. Find and simplify the difference quotient for 𝑓(𝑥) = 𝑥 2 + 9𝑥 − 2 17. Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator 4 𝑥5𝑦 log 7 √ 49 𝑥−5 18. Solve 4 2 =4 19. For the given piecewise function, evaluate𝑓(−2), 𝑓(0), 𝑓(2) 4𝑥 + 4 𝑖𝑓 𝑥 < 0 𝑓(𝑥) = { 4𝑥 + 7 𝑖𝑓 𝑥 ≥ 0 20. Find 5 ∑(𝑘 − 10) 𝑘=1 𝜋 1 21. Use substitution to determine whether 𝑥 = 4 is a solution of the equation cos(𝑥) = √2 7𝑥+6 22. Find the inverse of the (one-to-one) function 𝑓(𝑥) = ,𝑥 ≠ 5 𝑥−5 10𝑥 3 23. Find the horizontal asymptote, if any, of ℎ(𝑥) = 2𝑥 2 +1 2 24. Fid the exact value of cos(𝜃), given tan(𝜃) = − 3 and 𝜃 in quadrant 2 25. Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. log(𝑥 + 30) − log(3) = log(2𝑥 + 1) 26. Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. 1 1 5log 7 (4) + log 7(𝑟 − 8) − log 7 𝑟 5 2 27. A building 260 feet tall casts a 100-foot-long shadow. If a person looks down from the top of the building, what is the measure of the angle between the end of the shadow and the vertical side of the building? (Assume the person's eyes are level with the top of the building.) 2𝑥+2 28. Find the inverse of the one-to-one function, 𝑓(𝑥) = 3 𝑥−2 29. Find the domain of the function 𝑓(𝑥) = 𝑥 3 −36𝑥 30. Solve the triangle 31. The functions 𝑓 and 𝑔 are defined by the following tables. Use the tables to evaluate (𝑔𝑜𝑓)(−3). x f(x) −3 3 0 4 1 5 5 −1 x g(x) −4 0 3 −4 6 2 9 −1  𝑥 32. Find the domain of 𝑔(𝑥) = 𝑥 2 −9 𝑥+2 33. Find the vertical asymptotes, if any, of ℎ(𝑥) = 𝑥 2 −4 34. A radio transmission tower is 240 feet tall. How long should a guy wire be if it is to be attached 10 feet from the top and is to make an angle of 20° with the ground? 35. Find the exact value of tan−1 (√3 ) 36. Solve 𝑥(𝑥 + 3)(5 − 𝑥) ≥ 0 37. Use transformations to graph 𝑓(𝑥) = −2𝑥+4 − 3. Determine the domain, range, and find the equation of the horizontal asymptote. Plot at least 3 points on the graph of the basic function and use them to perform transformations. Do one transformation at a time and write the equation for each function. Base function: 1st transformation: 2nd transformation: 3rd transformation: 38. Find the exact value of the expression. Do not use a calculator a. 1 + sin2(20°) + sin2 (70°) b. 1 − tan2 (4°) + csc 2(86°) c.cos(55°) sin(35°) + sin(55°) cos(35°) 4 5 39. Suppose sin 𝛼 = 5, and 𝛼 is in quadrant I, and cos 𝛽 = 13, and 𝛽 is in quadrant IV. Find the exact value of sin(𝛼 + 𝛽). 40. Solve the following equation: 2 cos 𝑥 − √3 = 0 in the interval [0, 2𝜋) 𝜋 41. Find the rectangular coordinates of the point that has polar coordinates(2, 6 ). 42. Analyze the graph of the function𝑓(𝑥) = −𝑥 2 (𝑥 − 1)(𝑥 + 3) as follows a. Determine the end behavior: Find the power function that the graph of 𝑓 resembles for large values of x b. Find the x-intercepts c. Determine whether the graph crosses or touches the x –axis at each x-intercept d. Use the information in a)-c) to sketch the graph of 𝑓. 43. Find the exact value of a. tan(𝜋) 𝜋 b. tan ( 2 ) 𝜋 c. sin ( 2 ) 3𝜋 d. cos ( 2 ) 44. Answer each question for the following functions.  What is the amplitude?  What will be the period?  What is the phase shift?  What will be the final key points?  Graph the function 𝜋 a. 𝑦 = 2cos (2𝑥 + 4 ) b. 𝑦 = −2sin(4𝑥 − 𝜋) 45. For an arithmetic sequence with 𝑎1 = −19, 𝑑 = 7. a. Write a formula for the general term (nth term) of the sequence b. Find 𝑎20 Answers 1. 6 2. 12 3. [1,9) ∪ (9, ∞) 𝜋 4. 4 5. 𝐶 = 35° , 𝑎 = 14 sin(115°), 𝑐 = 14 sin(35°) Calculator answer: 𝐶 = 35° , 𝑎 = 12.68, 𝑐 = 8.04 6. 6 7. 𝑥 3 + 6𝑥 2 − 15𝑥 + 8 8. −√2 9. (−∞, 14] 10. Vertices: (0, −5), (0,5) foci: (0, −5√5), (0,5√5) ln 8 11. ln 3 − 6 12. 0 13. foci: (2√3, 0), (−2√3, 0) 14. (6, ∞) 15. 1 16. 2𝑥 + ℎ + 9 5 1 1 17. log 7(𝑥) + log 7 (𝑦) − 4 4 2 18. 7 19. -4, 7, 15 20. -35 𝜋 21. 𝑥 = is a solution of the equation 4 5𝑥+6 22. 𝑥−7 23. There is no horizontal asymptote 3√13 24. − 13 27 25. 5 5 1024 √𝑟−8 26. log 7 √𝑟 5 27. 𝜃 = tan−1 (13) 3𝑥−2 28. 𝑓 −1 (𝑥) = 2 29. (−∞, −6) ∪ (−6,0) ∪ (0,6) ∪ (6, ∞) 82 −52 −42 52 −82 −42 42 −52 −82 30. 𝐴 = cos −1 ( ),𝐵 = cos −1 ( ),𝐶 = cos−1 ( ) −2(5)(4) −2(8)(4) −2(5)(8) 𝐴 = 125°, 𝐵 = 31°, 𝐶 = 24° 31. -4 32. (−∞, −3) ∪ (−3,3) ∪ (3, ∞) 33. 𝑥 = 2  230 34. sin(20°) Calculator answer= 672.5 feet 𝜋 35. 3 36. (−∞, −3] ∪ [0,5] 37. Domain =(−∞, ∞) ; range = =(−∞, −3); asymptote: y = -3 38. a. 2 b.2 c.1 16 39. − 65  𝜋 11𝜋 40. 6 , 6 41. (√3, 1) 42. a. For large values of |x|, the graph of f(x) will resemble the graph of 𝑦 = −𝑥 4.The graph will fall on the left and the right b. x-intercepts: (-3, 0) , (0, 0), and (1, 0) c. The graph of f crosses the x-axis at (1, 0) and (-3, 0) and touches the x-axis at (0, 0) d. 43. a. 0 b. Undefined c. 1 d. 0 𝜋 𝜋 44. a. Amplitude=2; Period=𝜋; Phase shift = − 8 (8 to the left) Final key points and graph  𝜋 𝜋 𝜋 b. a. Amplitude=2; Period= 2 ; Phase shift = 4 (4 to the right) Final key points and graph 45. a. 𝑎𝑛 = 7𝑛 − 26 b. 114