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Mat 170 Homework problems
Introduction
Exercise Set 0.1
In problems 1-10, write down each of the following without absolute value sign (do not simplify), x is a real number:
1. [pic] 2. [pic] 3. [pic] 4. [pic]
5. [pic] 6. [pic] 7. [pic] 8. [pic]
9. [pic], if [pic] 10. [pic], if [pic]
In problems 11-13, use order operation to simplify the expressions (always perform division before multiplication). One may remember PEDMAS (P for parentheses, E for exponents, D for division, M for multiplication, A and S for addition and subtraction).
11. [pic] 12. [pic] 13. [pic]
In problems 14-17, evaluate the expressions
14. [pic] 15. [pic] 16. [pic] 17. [pic]
In problems 18-20, determine the value of the following expressions
18. [pic] 19. [pic] 20. [pic]+[pic]
21. Classify the following numbers as whole number, rational number, and/or irrational number:
[pic]
Simplify the expressions (22-24)
22. [pic] 23. [pic] 24. [pic]
Exercise Set 0.2
Simplify with positive exponents
1. [pic] 2. [pic] 3. [pic] 4. [pic]
5. [pic] 6. [pic] 7. [pic] 8. [pic]
9. [pic] 10. [pic] 11. [pic] 12. [pic]
13. [pic] 14. [pic] 15. [pic] 16. [pic]
17. [pic] 18. [pic] 19. [pic] 20. [pic]
Express the given numbers in scientific notations
21. 2860000000 22. 1220000 23. 0.0000000142 24. 0.00808
Simplify the numbers
25. [pic] 26. [pic] 27. [pic] 28. [pic]
Exercise Set 0.3
Simplify the expressions
1. [pic] 2. [pic] 3. [pic] 4. [pic] 5. [pic]
6. [pic] 7. [pic] 8. [pic] 9. [pic] 10. [pic]
Rationalize the denominator
11. [pic] 12. [pic] 13. [pic] 14. [pic]
15. [pic] 16. [pic] 17. [pic] 18. [pic]
19. [pic] 20. [pic]
Simplify the expressions
21. [pic] 22. [pic] 23. [pic] 24. [pic]
25. [pic] 26. [pic]
Exercise Set 0.4
1. Determine the polynomial and its degree
a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
g) [pic] h) [pic]
i) [pic] j) [pic]
k) [pic] l) [pic]
m) [pic] n) [pic]
2. Perform the indicated operations, write the result in standard form of a polynomial and indicate its degree.
a) [pic]
b) [pic]
c) [pic]
d) [pic]
e) [pic]
f) [pic]
Exercise Set 0.5
|Factor completely |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
Exercise Set 0.6
1. Find domain of the following functions
a) [pic] b) [pic] c) [pic] d) [pic]
e) [pic] f) [pic] g) [pic] h) [pic]
2. Simplify and determine the domain
a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
g) [pic] h) [pic]
3. Simplify and find domain
a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
Exercise Set 0.7 - 0.8
|Solve the following Inequalities and write answers in |[pic] |
|interval(s), use real line test: |[pic] |
| |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
| |[pic] |
| |[pic] |
|43. [pic] |[pic] |
|44. [pic] |[pic] |
|45. [pic] |[pic] |
|Hint: Do not cross multiply to solve. Use [pic] etc | |
|46. [pic] | |
|47. [pic] Hint: This inequality has no solution, as the left | |
|side is always positive. | |
| |[pic] |
| |[pic] |
| |[pic] |
| |[pic] |
| |[pic] |
| |Evaluate the following and write as [pic]: |
| |a) [pic] |
| |b) [pic] |
| |c) [pic] |
| |d) [pic] |
Exercise Set 1
|Plot the points in the same xy – plane | |
| |In exercises 28 – 70, discuss the transformation of the function |
|1. [pic] |using the techniques of shifting, stretching, shrinking and/or |
|2. [pic] |reflecting. Compare the given function with its basic function: |
|3. [pic] | |
|4. [pic] |In exercises 28 – 37 use the basic function [pic] |
|5. [pic] | |
| |28. [pic] 29. [pic] |
|Find the distance and mid point |30. [pic] 31. [pic] |
| |32. [pic] 33. [pic] |
|6. [pic] 7. [pic] |34. [pic] 35. [pic] |
|8. [pic] 9. [pic] |36. [pic] 37. [pic] |
|10. [pic] 11. [pic] |In exercises 38 – 49 use the basic function [pic] |
|12. [pic] 13. [pic] | |
|14. [pic] 15. [pic] |38. [pic] 39. [pic] |
| |40. [pic] 41. [pic] |
|Write in the standard form of [pic], find slope, y – intercept |42. [pic] 43. [pic] |
|and determine if the straight lines are parallel, perpendicular |44. [pic] 45. [pic] |
|or neither. Plot the straight lines. |46. [pic] 47. [pic] |
| |48. [pic] 49. [pic] |
|16. [pic] |In exercises 50 – 56 the basic function is [pic] |
|17. [pic] | |
|18. [pic] |50. [pic] 51. [pic] |
|19. [pic] |52. [pic] 53. [pic] |
|20. [pic] |54. [pic] |
|21. [pic] |55. [pic] |
|22. [pic] |56. [pic] |
| | |
|Plot the following functions: |In exercises 57 – 64 the basic function is [pic] |
| |57. [pic] 58. [pic] |
|23. [pic] | |
|24. [pic] |81. [pic] |
|25. [pic] |82. [pic] |
|26. [pic] |83. [pic] |
|27. [pic] |84. [pic] |
|59. [pic] 60. [pic] |85. [pic] |
|61. [pic] 62. [pic] |86. [pic] |
|63. [pic] 64. [pic] |87. [pic] |
| |88. The function [pic] is not one- |
|In exercises 65 – 70 the basic function is [pic] |to-one. Choose a largest possible |
|65. [pic] 66. [pic] |domain containing the number –30 so |
|67. [pic] 68. [pic] |that the function restricted to the |
|69. [pic] 70. [pic] |domain is one-to-one. Find the inverse |
| |on the restricted domain. |
|In exercises 71 – 76 perform the operation using the functions | |
|[pic]. Also determine their domain. |89. The function [pic] is not one- |
| |to-one. Choose a largest possible |
|71. [pic] |domain containing the number 100 so |
|72. [pic] |that the function restricted to the |
|73. [pic] |domain is one-to-one. Find the inverse |
|74. [pic] |on the restricted domain. |
|75. [pic] | |
|76. [pic] |90. Plot the following points for [pic]: |
| |[pic], [pic], [pic], [pic]. Then plot |
|In exercises 77 – 87 find inverse of the given function and |the points for [pic] |
|verify that [pic], where [pic] is the inverse of [pic] and [pic] | |
|is the inverse of [pic]respectively. |91. Plot the function [pic], choose |
| |several points on the function to plot the |
|77. [pic] 78. [pic] |inverse function. |
|79. [pic] 80. [pic] | |
Chapter 2
|In exercises 1 – 12 identify the polynomial as monomial, quadratic, |22. [pic] |
|cubic, etc., and as even, odd or neither. |23. [pic] |
|1. [pic] 2. [pic] |24. [pic] |
|3. [pic] 4. [pic] |25. [pic] |
|5. [pic] 6. [pic] |26. [pic] |
|7. [pic] 8. [pic] |27. [pic] |
|9. [pic] 10. [pic] |28. [pic] |
|11. [pic] 12. [pic] |29. [pic] |
|In exercises 13 – 20 consider the standard form for the quadratic |30. [pic] |
|polynomial [pic] to find maximum or minimum value of the given |31. [pic] |
|polynomials: |32. [pic] |
|13. [pic] |33. [pic] |
|14. [pic] |34. [pic] |
|15. [pic] |In exercises 35 – 41 solve for x using complete factors. |
|16. [pic] |35. [pic]are |
|17. [pic] |constants. |
|18. [pic] |36. [pic] |
|19. [pic] |37. [pic] |
|20. [pic] |38. [pic] |
|In exercises 21 – 34 discuss the end behavior |39. [pic] |
|21. [pic] |54. [pic] |
|40. [pic] |In exercises 55 – 64 use rational zero test to list a) all |
|41. [pic] |possible rational zeros, and |
|In exercises 42 – 44 divide the first polynomial by the second |b) find real zeros by factoring |
|polynomial. |c) use synthetic division to find zeros |
|42. [pic], [pic]. |55. [pic] |
|43. [pic], [pic] |56. [pic] |
|44. [pic], [pic] |57. [pic] |
|In exercises 45 – 54 find domain, vertical, horizontal, and/or slant |58. [pic] |
|Asymptote, hole (if any) |59. [pic] |
|45. [pic] |60. [pic] |
|46. [pic] |61. [pic] |
|47. [pic] |62. [pic] |
|48. [pic] |63. [pic] |
|49. [pic] |64. [pic] |
|50. [pic] |65. Find a third degree polynomial with |
|51. [pic] |zeros 1, 2, 3 and leading coefficient –2. |
|52. [pic] |66. Find all zeros of [pic] |
|53. [pic] |67. Find a polynomial with zeros 1, [pic] and |
| |leading coefficient 1. |
|asymptotes at [pic] and |68. Find a rational function having vertical |
|horizontal asymptote at [pic]. | |
|69. Find a polynomial of degree 3 with | |
|leading coefficient –1 and zeros at |77. The concentration of a drug t |
|4, and –5i. Simplify the polynomial |seconds after injection is given by |
|with real coefficients. |[pic] |
|70. Find a polynomial of degree 4 with |Estimate the time when will the |
|leading coefficient –1 and zeros of |concentration be maximum. |
|multiplicity 2 at 4, and –5i. |Determine the horizontal asymptote |
|Simplify the polynomial with real |(if any) and explain in this context. |
|coefficients. | |
|71. Find a rational function having |78. The population of a certain species |
|vertical asymptotes at [pic] |in millions is given by the rational |
|horizontal asymptote at [pic] |function [pic] , where t is |
|and x intercepts at (4, 0), and [pic] |in months after January 1st, 2000. |
|In exercises 72 – 76 use Descartes rule of sign to determine the | |
|nature of roots |Graph the polynomial |
| |Estimate the initial population |
|72. [pic] |Estimate population for January 1st, 2010 |
|73. [pic] |d) Estimate population in the long run. |
|74. [pic] | |
|75. [pic] | |
|76. [pic] | |
Exercise Set 3.1
|In Exercises 1 – 9, classify the angles as acute, right, obtuse |34. [pic] 35. [pic] 36. [pic] |
|or straight and reflex and draw each angle. If degree sign is not|37. [pic] 38. [pic] 39. [pic] |
|given the figure is in radian measure. |In Exercises 40 – 48, find an angle between 0 and 360 degrees |
|1. [pic] 2. [pic] 3. [pic] |that is coterminal with the given angle ([pic]) |
|4. [pic] 5. [pic] 6. [pic] |40. [pic] 41. [pic] 42. [pic] |
|7. [pic] 8. [pic] 9. [pic] |43. [pic] 44. [pic] 45. [pic]radian |
|In Exercises 10 – 21, convert each angle in degrees to radians, |46. [pic] 47. [pic] 48. [pic] |
|express your answer as multiple of [pic] |In Exercises 49 – 56, find the arc length, and area of the |
|10. [pic] 11. [pic] 12. [pic] |sector, where r is the radius in inches and [pic] is the central |
|13. [pic] 14. [pic] 15. [pic] |angle |
|16. [pic] 17. [pic] 18. [pic] |49. [pic] 50. [pic] |
|19. [pic] 20. [pic] 21. [pic] |51. [pic] 52. [pic] |
|In Exercises 22 – 30, convert each angle in radians to degrees |53. [pic] 54. [pic] |
|22. [pic] 23. [pic] 24. [pic] |55. [pic] 56. [pic] |
|25. [pic] 26. [pic] 27. [pic] |57. A wheel has a radius of 12 feet, and is rotating at 6 |
|28. [pic] 29. [pic] 30. [pic] |revolutions per minute. Find the angular speed and linear speed |
|In Exercises 31 – 39, find an angle between 0 and [pic] that is |in feet per minute. |
|coterminal with the given angle ([pic] radian). Find also the |58. The blades of a wind machine are 12 feet long and rotating at|
|reference angle if any. |5 revolutions per second. Find the angular and linear speed. |
|31. [pic] 32. [pic] 33. [pic] |59. A mountain bike with 26 inches wheels (13 inch radius) is |
|second. Find the angular and linear speed. |rotating at 10 revolutions per |
|60. The diameter of car wheel is 185 mm, it rotates at 30 |62. Find the radian measure of a central |
|revolutions per second, find its angular and linear speed. |angle that cuts off an arc of length 8 |
|61. Find the angle in radians formed by the |inches with a radius of 4 inches. |
|hands of a clock at 1:30. | |
Exercise Set 3.2
|In Exercises 1 – 12, show the approximate location of [pic] on |a) [pic] b) [pic] c) [pic] |
|the unit circle for the given value of t. |d) [pic] e) [pic] f) [pic] |
|1. [pic] 2. [pic] 3. [pic] |g) [pic] h) [pic] |
|4. [pic] 5. [pic] 6. [pic] |15. If [pic], find the coordinates of given point. |
|7. [pic] 8. [pic] 9. [pic] |a) [pic] b) [pic] c) [pic] |
|10. [pic] 11. [pic] 12. [pic] |d) [pic] e) [pic] f) [pic] |
|13. If [pic], find the coordinates of given point. |g) [pic] h) [pic] |
|a) [pic] b) [pic] c) [pic] |16. If the point [pic] lies on the unit circle find x. |
|d) [pic] e) [pic] f) [pic] |17. Find exact value of [pic] and |
|g) [pic] h) [pic] |[pic]for the given value of t. |
|14. If [pic], find the coordinates of given point. |a) [pic] b) [pic] c) [pic] |
| |d) [pic] e) [pic] f) [pic] |
|j) [pic] k) [pic] l) [pic] |g) [pic] h) [pic] i) [pic] |
|m) [pic] n) [pic] o) [pic] | |
|p) [pic] q) [pic] r) [pic] |21. Find the angle in radians formed by the |
|18. Find all vales of t in the interval [0, [pic]] |hands of a clock at 1:30. |
|satisfying the equation [pic]. |22. Find the radian measure of a central |
|19. Find all vales of t in the interval [0, [pic]] |angle that cuts off an arc of length 8 |
|satisfying [pic]. |inches with a radius of 4 inches. |
|20. Find all vales of t in the interval [0, [pic]] |23. Find the radian measure of a central |
|satisfying [pic]. |angle that cuts off an arc of length 10 |
| |inches with a radius of 6 inches. Also |
| |find the area of the sector. |
|Exercise Set 3.3 | |
|In exercises 1-10, find all trigonometric functions, from the |18. [pic] |
|given information. |19. [pic] |
| |20. [pic] |
|1. [pic]is in quadrant II | |
|2. [pic]is in quadrant I |In exercises 21-30, determine exact values |
|3. [pic]is in quadrant IV | |
|4. [pic]is in quadrant III |21. [pic] |
|5. [pic]is in quadrant IV |22. [pic] |
|6. [pic] |23. [pic] |
|7. [pic]is in quadrant II |24. [pic] |
|8. [pic]is in quadrant III |25. [pic] |
|9. [pic] |26. [pic] |
|10. [pic] |27. [pic] |
| |28. [pic] |
|In exercises 11- 20, determine all values of t in the interval |29. [pic] |
|[pic] |30. [pic], [pic] |
| |31. Simplify [pic] |
|11. [pic] 12. [pic] |32. Show that [pic] |
|13. [pic] 14. [pic] |33. Find exact value: [pic] |
|15. [pic] |34. Find exact value: [pic] |
|16. [pic] | |
|17. [pic] | |
Exercise Set 3.4
| |5. [pic] |
|In exercises 1-10 find amplitude, period, horizontal and vertical|6. [pic] |
|shift |7. [pic] |
| |8. [pic] |
|1. [pic] | |
|2. [pic] |22. Use calculator to find x so that |
|3. [pic] |[pic] |
|4. [pic] |In exercises 23-30 find all values of in the interval [pic] |
|9. [pic] |satisfying the given equations. |
|10. [pic] | |
|In exercises 11-18 use the graphs of the sine and cosine to |23. [pic] |
|sketch one period of the graph of the function. |24. [pic] |
| |25. [pic] |
|11. [pic] |26. [pic] |
|12. [pic] |27. [pic] |
|13. [pic] |28. [pic] |
|14. [pic] |29. [pic] |
|15. [pic] |30. [pic] |
|16. [pic] |31. Given that [pic]. |
|17. [pic] |Find [pic] |
|18. [pic] |32. For [pic], solve for x when |
|19. Find exact value of x so that |[pic] |
|[pic] | |
|20. Use calculator to find x so that |33. An object is thrown from the point A |
|[pic]. |on the inclined plane (see the figure). |
|21. Use calculator to find x so that |The object hits the inclined plane at B. |
|[pic] |Find the distance between A and B, if |
| |AC = 20 cm. |
| | |
| |B |
| | |
| |A [pic] C |
Exercise Set 4.1
|In exercises 1 – 8 find the exact vale(s) of the expression: |In exercises 16 – 20 find exact value(s) on [pic] |
|1. [pic] 2. [pic] |16. [pic] |
|3. [pic] 4. [pic] |17. [pic] |
|5. [pic] 6. [pic] |18. [pic] |
|7. [pic] 8. [pic] |19. [pic] |
|In exercises 9 – 15 find exact value(s ) |20. [pic] |
|9. [pic] |In exercises 21 – 25 find the value(s) in terms of x |
|10. [pic] |21. [pic] |
|11. [pic] |22. [pic] |
|12. [pic] |23. [pic] |
|13. [pic] |24. [pic] |
|14. [pic] |25. [pic] |
|15. [pic] |26. Rewrite the expression [pic] |
|In exercises 27 – 45 find all solutions of the equation on [pic] |as a expression of x. |
|27. [pic] |36. [pic] |
|28. [pic] |37. [pic] |
|29. [pic] |38 [pic] |
|30. [pic] |39. [pic] |
|31. [pic] |40. [pic] |
|32. [pic] |41. [pic] |
|33. [pic] |42. [pic] |
|34. [pic] |43. [pic] |
|35. [pic] |44. [pic] |
| |45. [pic] |
Exercise Set 4.2
|In exercises 1 – 20 prove the identities |9. [pic] |
|1. [pic] |10. [pic] |
|2. [pic] |11. [pic] |
|3. [pic] |12. [pic] |
|4. [pic] |13. [pic] |
|5. [pic] |14. [pic] |
|6. [pic] |15. [pic] |
|7. [pic] |16. [pic] |
|8. [pic] |22. It is given that [pic]. |
|17. [pic] |Determine the function [pic] |
|18. [pic] |23. It is given that [pic]. |
|19. [pic] |Determine the function [pic] |
|20. [pic] |25. It is given that [pic]. |
|21. A pole of 100 feet is supported by a |Determine the constant [pic]. |
|cable of length 230 feet. Find the angle |26. It is given that |
|of elevation from the top of the pole to |[pic]. |
|the point on the ground. |Determine the constants [pic]. |
Exercise Set 4.3
|In exercises 1 – 10 find exact value(s) |IV and [pic], y is in quadrant II. |
|1. [pic] 2. [pic] |Find |
|3. [pic] 4. [pic] |a) [pic] b) [pic] |
|5. [pic] 6. [pic] |c) [pic] d) [pic] |
|7. [pic] 8. [pic] |e) The quadrant where [pic] lies |
|9. [pic] 10. [pic] |f) The quadrant where [pic] lies |
|11. Given that [pic], x is in quadrant |12. Find the exact value of |
|[pic] |[pic] |
|14. Find the exact value of |13. Find the exact value of |
|[pic] | |
|15. Find the exact value of |In exercises 16 – 20 simplify |
|[pic] |16. [pic] |
| |17. [pic] |
| |18. [pic] |
| |19. Show that[pic] |
| |20. Use [pic] formula to prove |
| |[pic] |
Exercise Set 4.4
|Given that [pic], find |If [pic], find the function [pic]. |
|[pic] |If [pic], find the function [pic] |
|Given that [pic] | |
|[pic] |In exercises 10 – 15 find A and B |
|Given that [pic], find [pic] |[pic] |
|Given that [pic] |[pic] |
|[pic] |[pic] |
|Use half angle formula to write Given that [pic], then find x and|[pic] |
|y. |[pic] |
|If [pic], find |[pic] |
|The function [pic] |In exercises 16 – 20 find all values of x in [0, 2π] that satisfy the|
|Given that [pic] is in quadrant IV. Find [pic] |given equation |
|[pic] | |
|[pic] |Show that [pic] |
|[pic] |Use double angle formula to find exact value of [pic] |
|[pic] |Show that [pic] |
|[pic] | |
|Show that [pic] | |
|Show that [pic] | |
Exercise Set 4.5
|In exercises 1 – 10, rewrite each product as a sum or difference |8. [pic] 9. [pic] |
|1. [pic] |10. [pic] |
|2. [pic] |11. Solve for x if [pic] |
|3. [pic] |12. Solve for x if [pic] |
|4. [pic] |13. Solve for x if [pic] |
|5. [pic] |14. Solve for x if [pic] |
|6. [pic] |15. Show that |
|7. [pic] |[pic] |
Exercise Set 4.6
|In exercises 1 – 10, solve for x on [0, 2π] |5. [pic] |
|1. [pic] |6. [pic] |
|2. [pic] |7. [pic] |
|3. [pic] |8. [pic] |
|4. [pic] |9. [pic] |
|10. [pic] |14. Find all solutions for x when |
|11. Find all solutions for x when |[pic] |
|[pic] |15. Find all solutions for x when |
|12. Find all solutions for x when |[pic] |
|[pic] |16. Find all solutions for x when |
|13. Find all solutions for x when |[pic] |
|[pic] |17. Find all solutions for x when |
| |[pic] |
Exercise Set 4.7
|In exercises 1 – 10 draw the triangle and find all missing |[pic], angle [pic]. See the adjacent graph below |
|information |B C |
|Angle [pic], angle [pic], [pic] | |
|Angle [pic], [pic],[pic] | |
|Angle [pic], [pic],[pic] | |
|Angle [pic], [pic],[pic] |A |
|Angle [pic], [pic],[pic] | |
|Sides[pic] |Two ships leave the St. Martin Island at the same time, traveling|
|Side [pic], angles [pic], [pic] |on courses that have an angle of 100 degrees between them. The |
|Side [pic], angles [pic], [pic] |first ship travels at 25 miles per hour and the second ship |
|Side [pic], angles [pic], [pic] |travels at 40 miles per hour. Find how far apart the ships are |
|Sides[pic] |after 3 hour 30 minutes. |
|Find the distance AB across a river, a distance BC = 250 ft is |Two ships leave the St. Martin Island at the same time, traveling|
|laid off on one side of the river. Also given that angle |on courses that have an angle of x degrees between them. The |
| |first ship travels at 30 miles per hour and the second ship |
|14. The path of a satellite orbiting the earth causes it to pass |travels at 40 miles per hour. The distance between the ships |
|directly over two stations situated at A and B, which is 50 miles|after 2 hours is 100 miles, find x. |
|apart. The angle of elevation of the satellite at A is 85 degrees| |
|and the angle of elevation at B is 78 degrees. See the graph |Find how far is the satellite from |
|below |station A |
|C |Find how far is the satellite from station B |
| |Find the height of the satellite above the ground |
|85 78 |Round your answer to three decimal places. |
|A B | |
Exercise Set 5.1
|In exercises 1 – 10, sketch the graphs, showing any horizontal |14. The graphs of [pic], [pic] |
|asymptote. |and [pic] |
|1. [pic] 2. [pic] |15. The graphs of [pic], [pic] |
|3. [pic] 4. [pic] |and [pic] |
|5. [pic] 6. [pic] |16. Given that [pic], determine and simplify [pic] |
|7. [pic] 8. [pic] |17. Given that [pic], determine and simplify [pic] |
|9. [pic] 10. [pic] |18. Given that [pic], determine [pic] |
|In exercises 11 – 15, use a graphing device to plot and compare |Use a graphing device to compare to compare the rates of growth |
|11. The graphs of [pic], [pic], [pic], and [pic], [pic], [pic] |of the functions [pic] and [pic] by graphing the two functions in|
|12. The graphs of |the following viewing windows |
|[pic] and [pic] |[pic] |
|13. The graphs of [pic] |[pic] |
|for [pic] |[pic] |
| |Approximate the solutions to [pic] correct to three decimal |
|20. Determine the value of an investment of $5000 in 10 years at |places. |
|the interest rate of 5% compounded as indicated. |A population of bacteria doubles every hour. At noon the number |
|a) Annually b) Monthly |of bacteria was 1,000. Set up an exponential function to model |
|c) Semiannually d) Quarterly |the growth of the bacteria, and forecast the population at 5 p.m.|
|e) Weekly f) Biweekly |Also, estimate the population at 10 a.m., two hours before the |
|g) Daily h) Continuously |number of bacteria were counted. |
| | |
|What initial investment at 5% compounded semiannually for 10 |The number of bacteria in a culture is given by [pic], where |
|years will accumulate to $10000? |[pic] is measured in grams. |
| | |
| |Find the mass at time t = 0 |
| |How much of the mass remains after 18 years? |
Exercise Set 5.2
|In exercises 1 – 12, evaluate the expression. |In exercises 21 – 30, express the equation in exponential form |
|1. [pic] 2. [pic] 3. [pic] |21. [pic] 22. [pic] |
|4. [pic] 5. [pic] 6. [pic] |23. [pic] 24. [pic] |
|7. [pic] 8. [pic] 9. [pic] |25. [pic] 26. [pic] |
|10. [pic] 11. [pic] 12. [pic] |27. [pic] 28. [pic] |
|In exercises 13 – 20, determine domain, x intercept and vertical |29. [pic] 30. [pic] |
|asymptote if any. |In exercises 31 – 35, rewrite the expression as a single |
|13. [pic] 14. [pic] |logarithm |
|15. [pic] 16. [pic] |31. [pic] |
|17. [pic] 18. [pic] |32. [pic] |
|19. [pic] 20. [pic] |33. [pic] |
|34. [pic] |47. [pic] |
|35. [pic] |48. [pic] |
|In exercises 36 – 40, solve for x |49. [pic] |
|36. [pic] |50. [pic] |
|37. [pic] |51. An investment of $5000 will grow to $12500 at 5% interest |
|38. [pic] |compounded quarterly in t years. Find t. |
|39. [pic] |52. The number of bacteria in a culture is modeled by the |
|40. [pic] |exponential function [pic], where t is in hours. Find |
|41. Plot the graph of |The initial count of this bacterium |
|[pic]and |The relative rate of growth |
|solve algebraically for x. |After how many hours will the bacteria count reach 10000? |
|42. Use logarithm to solve the exponential equation [pic] |53. The count in a bacteria culture was 8000 after 5 hours and |
|43. Given [pic], simplify the |16500 after 7 hours. Assume the growth model by the function |
|expression [pic] |[pic], where t is in hours. Find the time when the count will be |
|44. Evaluate the expression |double to its initial size. |
|[pic] |54. The half-life of strontium-90 is 29 years. How long will it |
|In exercises 45 – 50, determine the constants |take a 56 mg sample to decay to a mass of 10 mg? |
|45. [pic] |55. A culture has initial bacteria count |
|46. [pic] |9000. After one hour the count is 4500. |
| |Find the relative growth rate and the |
|56. The radioactive isotope strontium 90 |number of bacteria after 2 hours. |
|has a half-life of 28.5 years. Find | |
| |How much strontium 90 will remain after 15 years from an initial |
| |amount of 450 kilograms. |
| |How long will it take for 75% of the original amount to decay? |
| |The time when the amount is 100 kilograms. |
Exercise Set 6
| | |
|In exercises 1 – 14 Find Cartesian form of the polar equation |In exercises 23 – 28, convert the rectangular equation to a polar|
| |equation. |
|1. [pic] 2. [pic] | |
|3. [pic] 4. [pic] |23. [pic] 24. [pic] |
|5. [pic] 6. [pic] |25. [pic] 26. [pic] |
|4. [pic] 8. [pic] |27. [pic] 28. [pic] |
|9. [pic] 10. [pic] | |
|11. [pic] 12. [pic] |29. Find the polar equation of the given Cartesian equation |
|13. [pic] 14. [pic] |[pic] |
| |where h, k are the constants representing center of the circle. |
|In exercises 15 – 20, convert the rectangular coordinates to | |
|polar coordinates. |30. Identify the conic and write its polar form: [pic] |
| | |
|15. [pic] 16. [pic] |31. Find the point(s) of intersection of the |
|17. [pic] 18. [pic] |curves [pic] |
|19. [pic] 20. [pic] | |
|21. [pic] 22. [pic] | |
Answers to odd number problems
Section 0.1
1. [pic] 3. 4 5. [pic]
7. [pic] 9. [pic] 11. 6/5
13. [pic] 15. [pic]
17. [pic]
19. 13
21. Rational numbers are: 1/3,
3.6666…,[pic]
Irrational number is [pic]
23. [pic]
Section 0.2
1. [pic] or -10000 3. [pic] 5. [pic]
7. [pic] 9. [pic] 11. [pic] 13. [pic]
15. [pic] 17. [pic] 19. [pic]
21. [pic] 23. [pic]
25. 123000 27. 0.004842520
Section 0.3
1. [pic] 3. [pic] 5. [pic]
7. [pic] 9. [pic]
11. [pic] 13. [pic]
15. [pic] 17. [pic]
19. [pic] 21. [pic]
23. [pic] 25. [pic]
Section 0.4
1. a) Second order b) Not a polynomial
` c) Not a polynomial d) Ninth degree
e) Not a polynomial f) Not a polynomial
g) Not a polynomial h) Not a polynomial
i) First degree j) Twelfth degree
k) Fifth degree l) Fourth degree
m) Third degree n) Ninth degree
Section 0.5
1. [pic]
3. [pic] 5. [pic]
7. [pic] 9. [pic]
11. [pic]
13. [pic]
15. [pic] 17. [pic]
Section 0.6
1. a) [pic] b) all real values of x
c) [pic] d) [pic]
e) [pic] f) all real values of x
g) [pic] h) [pic]
3. a) [pic]
b) [pic]
c) [pic]
d) [pic]
e) [pic]
f) [pic]
Section 0.7-0.8
1. [pic] 3. [pic]
5. [pic] 7. [pic]
9. [pic]
11. [pic]
13. [pic], [pic]
15. [pic], [pic]
17. [pic], [pic]
19. [pic] 21. [pic]
23. [pic]
25. [pic]
27. [pic]
29. [pic]
31. [pic] 33. [pic]
35. [pic]
37. [pic] 39. [pic]
41. [pic]
43. [pic] 45. [pic]
47. No solution
49. [pic]
51. [pic]
53. a) [pic] b) [pic] c) [pic]
d) [pic]
Chapter 1
For 1 – 5 follow example 1 on section 1.1.
7. [pic] 9. [pic]
11. [pic] 13. [pic]
15. [pic]
17. [pic], parallel
19. [pic], neither
21. [pic]
[pic], perpendicular
23 – 26 follow example 9-12 of section 1.5.
29. Vertically stretched by a factor 2 and shift 3 units upward.
31. Vertically stretched by a factor 2, horizontal shift by 1 unit to the right and 1 unit upward
33. Vertically stretched by a factor 3, horizontal shift by 2 units to the right, 5 units upward and reflection about x axis.
35. Vertically stretched by a factor 2, horizontal shift by 1 unit to the right and 3 units upward
37. Vertically compressed by a factor 3, and shift 3 units downward
39. Vertically stretched by a factor 2, and shift 3 units upward
41. Vertically stretched by a factor 2, horizontal shift by 1 unit to the right and 1 unit upward
43. Vertically stretched by a factor 3, horizontal shift by 2 units to the right and 5 unit upward and reflection about x axis
45. Vertically stretched by a factor 2, horizontal shift by 1 unit to the right and 3 units upward
47. Vertically compressed by a factor 3, and 3 units upward
49. Horizontally stretched by a factor 2, shift by 3 units upward then reflection about x axis
51. Vertically stretched by a factor 2, horizontal shift by 1 unit to the right and 1 unit upward then reflection about x axis
53. Vertically compressed by a factor 2, horizontal shift by 1 unit to the right and 1 unit upward then reflection about x axis
55. Horizontally stretched by a factor 2, shift by 7 units downward then reflection about y axis
57. Horizontal shift by 1 unit, shift 3 units upward then reflection about x axis
59 .Vertically compressed by a factor 2, shift by 1 unit to the right
61. Horizontally stretched by a factor 2, shift by 7 units downward
63. Horizontally compressed by a factor 3, vertically stretched by a factor 2, shift by 2 units to the left, and shift 2 units upward
65. Vertically compressed by a factor 2, shift by 1 unit to the right, and reflection about x axis
67. Horizontally stretched by a factor 2, shift 7 units downward and reflection about x axis
69. Vertically compressed by a factor 3, shift by 1 units to the right, and shift 2 units upward
71. [pic]
73. [pic]
75. [pic]
77. [pic]
79. [pic]
81. [pic]
83. [pic]
85. [pic]
87. [pic]
89. [pic]
Chapter 2
1. Monomial, odd, linear
3. Quadratic, neither
5. Neither, linear
7. Cubic, neither
9. Cubic, odd
11. Cubic, neither
13. [pic], max
15. [pic], max
17. [pic], max
19. [pic], min
21. [pic]
23. [pic]
25. [pic]
27. [pic]
29. [pic]
31. [pic]
33. [pic]
35. [pic] 37. [pic]
39. [pic] 41. [pic]
43. [pic]
45. [pic]
47. [pic]
49. [pic]
51. [pic]
53. [pic]
55. a) [pic]
[pic]
[pic]
b) Irrational zeros are 0.176, 0.824 (correct to three decimal places)
57. a) [pic] b) One rational zero 4, two irrational zeros are [pic]
59. a) [pic]
b) One irrational zero -1.48 (correct to 2
decimal places)
61. a) [pic] b) Rational zero -3
63. a) [pic]
b) Rational zero -3 with multiplicity 3.
65. [pic]
67. [pic]
69. [pic]
71. [pic]
73. Two positive, one negative
75. Four negative zeros
77. [pic], horizontal asymptote is [pic], the meaning of zero is that the is no concentration of the drug in the long run.
Section 3.1
1. Obtuse 3. Obtuse 5. Acute 7. Acute
9. Reflexive 11. [pic] 13. [pic] 15. [pic]
17. [pic] 19. [pic] 21. [pic] 23. [pic]
25. [pic] 27. [pic] 29. [pic]
31. [pic] 33. [pic] 35. [pic]
37. [pic] 39. [pic] 41. [pic]
43. [pic] 45. [pic]
47. [pic] 49. 12.22 inches, 12.22 sq. inch
51. 2.09 inches, 4.19 sq. inch
53. 8.17 inches, 8.17 sq. inch
55. 9.42 inches, 14.14 sq. inch
57. [pic] 59. [pic] 61. [pic]
Section 3.2
1. Quadrant I 3. Quadrant I 5. Quadrant I
7. Quadrant II 9. Quadrant I 11. Quadrant IV
13. a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
g) [pic] h) [pic]
15. a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
g) [pic] h) [pic]
17. a) [pic], [pic]
[pic]
b) [pic], [pic]
[pic]
c) [pic], [pic]
[pic]
d) [pic],
[pic]
[pic]
e) [pic],
[pic]
[pic]
f) [pic], [pic]
[pic]
g) [pic], [pic]
[pic] does not exist
h) [pic], [pic]
[pic]
i) [pic],
[pic]
[pic]
j) [pic], [pic]
[pic] does not exist
k) [pic],
[pic]
[pic]
l) [pic], [pic]
[pic]
m) [pic],
[pic], [pic]
n) [pic],
[pic], [pic]
o) [pic],
[pic], [pic]
p) [pic],
[pic], [pic]
q) [pic],
[pic], [pic]
19. [pic] 21. [pic]
23. [pic] rad, 30 sq. inch
Section 3.3
1. [pic] [pic]
[pic] [pic]
[pic] [pic]
3. [pic] [pic]
[pic] [pic]
[pic] [pic]
5. [pic] [pic]
[pic] [pic]
[pic] [pic]
7. [pic] [pic]
[pic] [pic]
[pic] [pic]
9. [pic] [pic]
[pic] [pic]
[pic] [pic]
11. [pic]
13. [pic]
15. [pic]
17. [pic]
19. [pic] 21. [pic]
23. Does not exist 25. [pic]
27. 1 29. [pic]
31. 5 33. 1
Section 3.4
1. [pic] 3. [pic]
5. [pic] 7. [pic]
9. [pic]
11.
3
0 [pic]
-1
13.
3
0 1
15.
3
0 2
17.
3
0 [pic]
19. [pic]
21. [pic]
23. [pic]
25. [pic] 27. [pic]
29. [pic]
31. [pic] 33. [pic]
Section 4.1
1. [pic] 3. [pic] 5. 0 7. [pic]
9. [pic] 11. [pic] 13. [pic] 15. 12/5
17. [pic] 19. [pic] 21. [pic]
23. [pic] 25. [pic]
27. [pic]
29. [pic] 31. [pic]
33. [pic]
35. [pic] 37. [pic] 39. [pic]
41. [pic] 43. [pic]
45. [pic]
Section 4.2
21. [pic] 23. [pic] 25. [pic]
Section 4.3
1. [pic] 3. [pic] 5. [pic]
7. [pic] 9. [pic]
11. a) [pic] b) [pic] c) [pic]
d) [pic]
e) [pic] is in quadrant I and [pic] is in quadrant II.
13. [pic] 15. [pic] 17. 0
Section 4.4
[pic]
[pic]
5. [pic] 7. 0.2117 9. [pic]
11. A = 1 13. A = 1 15. [pic]
17. [pic] 19. [pic]
Section 4.5
1. [pic] 3. [pic]
5. [pic]
7. [pic]
9. [pic]
11. [pic] or [pic]
13. [pic] or [pic]
Section 4.6
Assume [pic]
1. [pic] 3. [pic]
5. [pic] 7. [pic]
9. [pic] 11. [pic]
13. [pic]
15. [pic] 17. [pic]
Section 4.7
1. [pic]
3. [pic]
5. [pic]
7. [pic]
9. [pic]
11. 250 ft
13. [pic]
Section 5.1
1.
Horizontal asymptote [pic]
3.
Horizontal asymptote [pic]
5.
Horizontal asymptote [pic]
7.
Horizontal asymptote [pic]
9. No horizontal asymptote
11.
[pic] [pic]
[pic] [pic]
[pic]
[pic]
13. [pic]
[pic]
[pic]
[pic]
[pic]
[pic]
15.
[pic]
[pic]
17. [pic]
19. a) 8144.47 b) 8235.05 c) 8193.08
d) 8218.10 e) 8241.63 f) 8239.65
g) 8243.32 h) 8243.61
21. 6102.71 23. a) 150 b) [pic] years
Section 5.2
1. 2 3. 4 5.[pic] 7. [pic] 9. [pic] 11. 10
13.[pic]; [pic]
15.[pic],[pic]; [pic], [pic]
17.[pic]; [pic], [pic]
19.[pic],[pic]; [pic],
[pic]
21.[pic] 23. [pic] 25. [pic]
27. [pic] 29. [pic] 31. [pic]
33. [pic] 35. [pic]
37. [pic] 39. [pic]
41. [pic] 43. [pic]
45. [pic] 47. [pic]
49. [pic] 51. [pic] years
53. [pic]
55. [pic] years
57. [pic]
Chapter 6
1.[pic], a circle of radius of 3
3. [pic]
5. [pic]
7. [pic]
9. [pic]
11. [pic]
13. [pic]
15. [pic]
17. [pic]
19. [pic]
21. [pic]
23. [pic]
25. [pic]
27. [pic]
29. [pic]
where[pic]
31. [pic]
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