University:
University of North Georgia
Course:
MATH 1113 | Precalculus
Academic year:

2021
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295

Pages:

25
Author:

5CelestialChronicle

S1 Final Exam Review Pre-Calculus Name:___________________________________ CHAPTER 1 - FUNDAMENTALS Express the interval in terms of inequalities. 1. (2, 8] 2. (5, ) Express the inequality in interval notation. 4. x < 5 5. -4 < x Express each set in interval notation. 7. -1 3. (- 6 6. x 8. 5 0 4 Simplify the expression. 9. 12. 2 3 1 6 − + 1 2 3 12 10. (3𝑎𝑏 𝑐)( 2 2 2𝑎 𝑏 3 𝑐 −2 ) 13. 2 7 1 6 11. 1 8 − ( ) 4 16𝑥 𝑦 −8 𝑦 1 2 , 5] 14. -4 15. (4x – 3y)(2x + 5y) 16. (2x + 1)(3x – 2) – 5(4x – 1) Factor the expression. 17. 18. 19. 20. 21. 22. 2 23. 𝑠𝑖𝑛 𝑥 + 3 sin 𝑠𝑖𝑛 𝑥 − 10 5 2 3 2 𝑥 − 6𝑥 + 8𝑥 4 2 3 2 24. 36𝑥 − 15𝑥 + 84𝑥 − 35 25. 1 2 26. 𝑥 − 2𝑥 + 1 2 27. 5(𝑥 − 2) + 4(𝑥 − 2) − 1 2 28. 𝑡𝑎𝑛 𝑥 − 81 2𝑥 𝑥 29. 𝑒 − 5𝑒 − 6 Simplify the expression. 30. 33. 31. 𝑥 2 𝑥 −9 + 1 𝑥−3 36. 34. 1 𝑥−2 𝑥 𝑦 1 𝑥 37. 2 − − + 𝑦 𝑥 1 𝑦 35. 2 3 2+ 6 Solve the equation. 38. 32. 3 𝑥+1 39. 1 𝑥+3 − 5 𝑥+2 40. (2𝑥 + 5)(2𝑥 + 4) = 30 41. (4𝑥 + 25)(2𝑥 + 21) = 1239 42. 43. 5 𝑥−2 − 2 𝑥+2 = 30 2 𝑥 −4 CHAPTER 2 - FUNCTIONS 2 Use the function 𝑓(𝑥) = 2𝑥 − 1 to evaluate the following. 1. f(-2) 2. f(x – 2) 3. f(a + h) 4. Evaluate f(0), f(3), and f(-4) for each function. 5. f(x) = 3x +x 6. f(x) = x + 7. f(x) = 𝑓(𝑎+ℎ)−𝑓(𝑎) ℎ Sketch the graph of the function. 8. 9. f(x) = 10. The graph of a function g is given. a) Find g(-3), g(0), and g(4) b) Identify the domain and range of g. 11. Determine the intervals on which the function is increasing and decreasing. A function is given. Determine the average rate of change of the function between the given values. 2 12. f(x) = 3x – 2; x = 2, x = 3 13. 𝑓(𝑥) = 𝑥 − 3𝑥; 𝑥 =− 2, 𝑥 =− 2 + ℎ Suppose the graph of f is given. Describe how the graph of g could be obtained from f. 14. g(x) = f(x) – 5 15. g(x) = f(x – 5) + 2 16. g(x) = 3f(x) 17. g(x) = -f(2x) 18. g(x) = f(x + 3) 19. g(x) = f( 20. The graph of f is given. Sketch the graph of y = –f(x + 1) – 3 Write the equation for the graph of f with the given transformations. 21. f(x) = x ; shift left 3 units, reflect over the x-axis, and shift down 1 unit. x) + 1 22. f(x) = ; shift right 2 units, stretch vertically by a factor of 3, and shift up 2 units. 23. f(x) = ; reflect over the y-axis and stretch horizontally by a factor of 2. 24. Given a) Find the axis of symmetry. b) Find the vertex. c) Find the maximum or minimum value. 25. Express in standard form. Describe the transformations on the parent function. Find the local maximum and minimum values of the function. 26. 27. f(x) = x – 2x – 11x 28. The profit P (in dollars) generated by selling x units of a certain commodity is given by P(x) = –1500 + 12x – 0.0004x . What is the maximum profit? How many units must be sold to generate it? 29. If the perimeter of a rectangular garden is to be 480 ft., find the dimensions of the garden that would give the largest area. What is the maximum area? 30. A rancher wants to build a rectangular pig pen with an area of 100 m2. a) Find a function that models the length of fencing required. b) Find the pen dimensions that would require the minimum amount of fencing. Given f(x) = x + 2x and g(x) = x – 4, evaluate the following. Then state the domain. 31. f + g 32. f – g 33. fg 34. 35. 36. Given f(x) = 1 + x and g(x) = 37. , evaluate the following. Then state the domain. 38. 39. Draw the graph of f and use it to determine if the function is one-to-one. 40. f(x) = x – x 41. f(x) = Find the inverse of function f. 42. f(x) = 3 – 5x 43. f(x) = 5 – 4x CHAPTER 3 - POLYNOMIAL + RATIONAL FUNCTIONS Evaluate the expression and write the result in a + bi form. 44. f(x) = 1. (2 – 6i) + (3 + 4i) 2. 4. (5 – 3i)(1 + i) 5. 6. 8. 9. 7. (–4 + i) – (2 – 5i) 3. 2i( (3 – – i) )(1 + ) For each polynomial function, identify the zeros and end behavior. Then graph the function. 2 2 10. 𝑃(𝑥) =− (𝑥 + 3) (𝑥 − 5) 11. 𝑃(𝑥) = (𝑥 − 4) (𝑥 + 3) Zeros: Zeros: End behavior: End behavior: 12. f(x) = –x – 2x + 8x 13. f(x) = x – 2x + 8x – 16 Zeros: Zeros: End behavior: End behavior: Use long division to find the quotient. 14. (x – 6x – 8) (x – 4) 15. (3x – 5x – 20x – 5) Use synthetic division to find the quotient. 16. (x – 5x + 4) (x – 1) 17. (x – 8x + 2) (x + 3) (x + x + 3) Find the indicated value of the polynomial using the Remainder Theorem. 18. P(x) = 2x – 9x – 7x + 13; P(5) 19. Q(x) = x + 4x + 7x + 10x + 15; Q(-3) List all possible rational zeros of the polynomial. 20. P(x) = x – 3x – 6x + 8 21. P(x) = 6x – x + 2x + 12 Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeros for the polynomial. 22. P(x) = 2x –x + 4x – 7 23. P(x) = x –x +x –x +x –x+1 Find all zeros of P. You may use your graphing calculator to help with synthetic solutions, but only to get started – work must be shown. 24. P(x) = 2x + 5x 3 – 6x – 9 2 26. 𝑃(𝑥) =− 𝑥 − 4𝑥 + 7𝑥 + 10 25. P(x) = x + 7x 3 + 9x 2 – 17x – 20 27. 𝑃(𝑥) = 𝑥 + 6𝑥 + 7𝑥 − 4 Find the complete factorization of the following polynomials 28. P(x) = x + 25 30. P(x) = x + 10x 32. P(x) = x – 81 29. P(x) = x + 25 – 8x + 17 31. P(x) = x + 2x + 4x + 8 33. P(x) = x + 15x + 54 34. Find a polynomial of least degree with integer coefficients and zeros 4 and 3i. 35. Find a polynomial of degree three with zeros -1, 1, and 3 and in which the constant coefficient is 15. Use Synthetic or Long Division to rewrite the following functions. Then describe their transformation from the parent function 𝑓(𝑥) = 36. 1 𝑥 37. 38. For each rational function, identify the x- and y-intercepts, asymptotes and sketch a graph of the function. 39. r(x) = 40. s(x) = x-intercept: x-intercept: y-intercept: y-intercept: vertical asymptote: vertical asymptote: horizontal asymptote: horizontal asymptote: slant asymptote: slant asymptote: D: D: CHAPTER 4 - EXPONENTIAL + LOGARITHMIC FUNCTIONS Find the domain of the function. 1. (𝑥 − 5) 2. 𝑦 = ln 𝑙𝑛 (3 − 7𝑥) 3. 𝑦 = 𝑥 − 4 Graph the following functions. 𝑥−2 4. 𝑓(𝑥) = 3 𝑥 5. 𝑓(𝑥) = − 2 − 3 + 1 D:______________ D:_____________ R:______________ R:_____________ HA:____________ HA:____________ 6. 𝑓(𝑥) = 𝑙𝑜𝑔2𝑥 7. 𝑓(𝑥) = 3 + 𝑙𝑜𝑔2(𝑥 + 4) D:______________ D:_____________ R:______________ R:_____________ VA:____________ VA:____________ 𝑥 8. Find the exponential function 𝑓(𝑥) = 𝑎 9. Find the logarithmic function 𝑓(𝑥) = 𝑙𝑜𝑔𝑎𝑥 whose graph is given. whose graph is given. Express the equation in exponential form. 10. 𝑙𝑜𝑔84 = 2 3 11. 𝑙𝑜𝑔0. 1 =− 1 12. ln 𝑙𝑛 (𝑥 − 1) = 4 Express the equation in logarithmic form. 3 1 2 𝑥+1 13. 7 = 343 14. 81 = 9 15. 𝑒 Evaluate the expression. 16. 𝑙𝑜𝑔464 17. 𝑙𝑜𝑔99 18. 𝑙𝑜𝑔61 19. 𝑙𝑜𝑔497 20. 𝑙𝑜𝑔9 3 21. 𝑒 𝑙𝑛π = 0. 5 22. 𝑙𝑜𝑔2160 − 𝑙𝑜𝑔25 33 23. 𝑙𝑜𝑔26 − 𝑙𝑜𝑔215+𝑙𝑜𝑔220 24. 𝑙𝑜𝑔28 Expand the expression. ( ) 25. 𝑙𝑜𝑔2 2 3 2 26. 𝑙𝑛 3𝑟 𝑠 𝑥 3 𝑦𝑧 Condense the expression. 27. 𝑙𝑜𝑔𝑎𝑏 + 𝑐𝑙𝑜𝑔𝑎𝑑 − 𝑟𝑙𝑜𝑔𝑎𝑠 28. 1 2 (𝑙𝑜𝑔5𝑥 + 2𝑙𝑜𝑔5𝑦 − 4𝑙𝑜𝑔5𝑧) 29. Use the change of base formula to evaluate 𝑙𝑜𝑔52. Solve the equation. Find exact answers whenever possible; otherwise, round answers to four decimal places. 𝑥 2𝑥−1 30. 4 = 32 1−𝑥 33. 10 31. 3 𝑥 =6 = 5 34. 𝑙𝑜𝑔42 = 𝑥 12𝑥 32. 2𝑒 = 17 35. ln 𝑙𝑛 (2 + 𝑥) = 1 2 36. 𝑙𝑜𝑔2 𝑥 − 𝑥 − 2 = 2 ( ) 38. 37. log 𝑙𝑜𝑔 𝑥 + log 𝑙𝑜𝑔 (𝑥 − 3) = 1 𝑙𝑜𝑔5𝑥 + 𝑙𝑜𝑔5(𝑥 + 1) = 𝑙𝑜𝑔520 39. Find the time required for an investment of $4000 to increase to $9000 if it is compounded quarterly at 6% annual interest. 40. Find the interest rate for an investment of $600 to triple if it is compounded continuously for 9 years? 41. A baseball card increased in value from $15 to $2000 in 25 years. Find its average annual rate of appreciation. 42. A cup of coffee has a temperature of 200°F and is placed in a room that has a temperature of 70°F. After 10 minutes, the coffee has cooled to 150°F. Find the temperature of the coffee after 15 minutes. 43. A man invests $6500 in an account that pays 6% interest per year, compounded continuously. a) What is the amount after 2 years? b) How long will it take for the amount to be $8000? 44. In 1990, a fish population in a man-made lake began with 150 fish. Since then, the population has had a relative growth rate of 9% per year. a) Write a function that models the amount of fish at time t, where t represents the number of years since 1990. b) What was the fish population in 2000? c) How long will it take the population to reach 2500 fish? 45. The half-life of radium-226 is 1600 years. How long will it take a 22-mg sample to decay to 18-mg? CHAPTER 5-6 - TRIGONOMETRIC FUNCTIONS of ANGLES 1. Convert − 300° to radians. 2. Convert 11π 3 to degrees. Find the value of x in the triangle. Number 7 must be exact. 3. 6. 4. 5. 7. 8. From the top of a 200 ft. lighthouse, the angle of depression to a ship in the ocean is 23º. How far is the ship from the base of the lighthouse? 9. To estimate the height of a mountain above a level plain, the angle of elevation to the top of the mountain is measured to be 32º. One thousand feet closer to the mountain along the plain, it is found that the angle of elevation is 35º. Estimate the height of the mountain. 10. Points 𝐴 and 𝐵 (on the same side of a tower) are 16 m apart. The angles of elevation of the top of a tower are 38° and 48° respectively. Find the tower’s height. 11. A ship leaves port at 2 p.m. and has a bearing of 𝑆 38° 𝑊. The ship sails at 35 knots. How many nautical miles south and how many nautical miles west will the ship have traveled by 9:30 p.m.? 12. An airplane flying at 390 miles per hour has a bearing of 67°. After flying for 5 hours, how far north and how far east has the plane traveled form its point of departure. 13. A plane is 200 miles south and 65 miles east of an airport. The pilot wants to fly directly to the airport. What bearing should be taken? Find the sign of the expression if the terminal point determined by θ is in the given quadrant. 14. cos 𝑐𝑜𝑠 θ cot 𝑐𝑜𝑡 θ ; QII 15. csc 𝑐𝑠𝑐 θ tan 𝑡𝑎𝑛 θ ; QIV From the information given, find the quadrant in which θ lies. 16. sec 𝑠𝑒𝑐 θ > 0 and < 0 17. tan 𝑡𝑎𝑛 θ > 0 and sec 𝑠𝑒𝑐 > 0 Find the value of the trigonometric functions of θ given the quadrant in which the terminal point lies. 18. cos 𝑐𝑜𝑠 θ =− 4 5 ; Quad II 19. tan 𝑡𝑎𝑛 θ = 1 4 ; Quad III Find the missing coordinate of P, using the fact that P lies on the unit circle in the given quadrant. ( 20. 𝑃 − 3 5 ) , 𝑦 ; Quad III ( 21. 𝑃 𝑥, 1 3 ); Quad II ( 22. 𝑃 𝑥, − 2 7 ); Quad IV Find a) the reference angle and b) the terminal point P(x, y) determined by the angle. 23. 3π 2 24. − 3π 4 25. 26. 960º 11π 6 Find the exact value of the trigonometric function. 27. cos 𝑐𝑜𝑠 29. 30. csc 𝑐𝑠𝑐 sec 𝑠𝑒𝑐 28. sin 𝑠𝑖𝑛 315° 5π 3 ( cot 𝑐𝑜𝑡 − 3π 2 ) 7π 6 31. tan 𝑡𝑎𝑛 2700° 32. 13π 4 33. Find the length of an arc that subtends a central angle of 45° in a circle with radius 10 m. 34. Find the area of a sector with central angle 60° in a circle with radius 3 mi. 35. A woman is riding a bicycle whose wheels are 28 in. in diameter. If the wheels rotate at 130 revolutions per minute (rpm), find the speed at which she is traveling in mi/h. 36. A boy rotates a stone in a 3 ft. long sling at the rate of 15 revolutions every 10 seconds. Find the linear and angular velocities of the stone. Find the amplitude, period, vertical shift, and phase shift of the function. Then graph the function. ( 37. 𝑦 = 2 sin 𝑠𝑖𝑛 2 𝑥 − π 4 ) −1 Amplitude:______________ Period:__________________ Phase Shift:______________ Vertical Shift:____________ Use the graph to find the amplitude, period, and phase shift of the curve. Then write an equation that represents the curve. 38. 39. Amplitude:__________________ Amplitude:__________________ Period:______________________ Period:______________________ Phase Shift:__________________ Phase Shift:__________________ Equation:____________________ Equation:____________________ Identify whether the graph represents 𝑦 = tan 𝑡𝑎𝑛 𝑥 , 𝑦 = cot 𝑐𝑜𝑡 𝑥 , 𝑦 = csc 𝑐𝑠𝑐 𝑥 , 𝑜𝑟 𝑦 = sec 𝑠𝑒𝑐 𝑥 . 40. 41. 42. 43.

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