Summer Math Review Directions

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2 Summer Math Review Directions 1. From the table of contents page, click on the name of the course that corresponds with your Math level/course for the upcoming academic year. This will take you directly to your assignment. 2. If you wish to print the summer assignment, do not print the entire file. Print only the pages that correspond to your grade level. For example, 6 th grade will only print the pages for the 6 th grade review packet. 3. Complete all of the problems on a separate sheet of paper. When solving the problems, show all work. You will only receive credit if you show written step-by-step solutions. Be sure to put your name on every sheet. 4. This assignment is due the first day of school. It will be counted as a grade, be sure to try your best.

3 Grade 6 Math CP Grade 6 Math Honors Grade 7 Math CP Pre-Algebra CP and Honors Algebra 1 CP & Honors Geometry CP and Honors Algebra 2 CP Algebra 2 Honors College Algebra & Trigonometry Pre-Calculus Honors Calculus AP Calculus AB AP Calculus BC Analysis of Functions

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5 Summer Review 2018 For students entering the Mathematics 6 CP and Honors Class. Please complete all problems, showing all your work. Remember, the dot ( ) you see starting with problem #5 is another symbol for multiplication just like (x). OPERATIONS WITH WHOLE NUMBERS OPERATIONS WITH DECIMALS = The Greatest Common Factor (GCF) and the Least Common Multiple (LCM) 21. Find the GCF of the numbers 15 and Find the GCF of the numbers 6, 36, and Find the LCM of the numbers 9 and Find the LCM of the numbers 3, 4, and 15

6 OPERATIONS WITH FRACTIONS 25. Simplify the fraction: 26. Write the reciprocal of = = = = 8 1 = = = 7 8 MIXED NUMBERS 36. Change to fraction: Change to mixed number: 20 9 = = = = = = = = = =

7 THE ORDER OF OPERATIONS (PEMDAS) RATIOS AND PROPORTIONS (9 2) 59. Are the ratios Are the ratios 5 7 and 9 5 equivalent? and equivalent? (9 2) (7 5) n n n n 3 12 EQUATIONS x r z y 7 91 k

8 PERCENTS. Complete the table: Percent Decimal Fraction % WRITING DECIMALS AS FRACTIONS AND MIXED NUMBERS 68. Write the decimal 0.31 as fraction or mixed number. 69. Write the decimal 2.05 as fraction or mixed number. 70. Write the fraction as 71. Write the fraction decimal. as decimal. 72. Write the mixed number as decimal. EXPONENTS 73. Write as an exponent Evaluate the exponent Evaluate the exponent 2

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10 Summer Review 2018 For students entering the Mathematics 6 CP and Honors Class. Please complete all problems, showing all your work. Remember, the dot ( ) you see starting with problem #5 is another symbol for multiplication just like (x). OPERATIONS WITH WHOLE NUMBERS OPERATIONS WITH DECIMALS = The Greatest Common Factor (GCF) and the Least Common Multiple (LCM) 21. Find the GCF of the numbers 15 and Find the GCF of the numbers 6, 36, and Find the LCM of the numbers 9 and Find the LCM of the numbers 3, 4, and 15

11 OPERATIONS WITH FRACTIONS 25. Simplify the fraction: 26. Write the reciprocal of = = = = 8 1 = = = 7 8 MIXED NUMBERS 36. Change to fraction: Change to mixed number: 20 9 = = = = = = = = = =

12 THE ORDER OF OPERATIONS (PEMDAS) RATIOS AND PROPORTIONS (9 2) 59. Are the ratios Are the ratios 5 7 and 9 5 equivalent? and equivalent? (9 2) (7 5) n n n n 3 12 EQUATIONS x r z y 7 91 k

13 PERCENTS. Complete the table: Percent Decimal Fraction % WRITING DECIMALS AS FRACTIONS AND MIXED NUMBERS 68. Write the decimal 0.31 as fraction or mixed number. 69. Write the decimal 2.05 as fraction or mixed number. 70. Write the fraction as 71. Write the fraction decimal. as decimal. 72. Write the mixed number as decimal. EXPONENTS 73. Write as an exponent Evaluate the exponent Evaluate the exponent 2

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15 2018 Summer Review Review packet for students entering Mathematics 7 CP (109 problems) Please complete all problems, showing all your work. Remember, the dot ( ) you see starting with problem #5 is another symbol for multiplication just like (x). OPERATIONS WITH WHOLE NUMBERS OPERATIONS WITH DECIMALS The Greatest Common Factor (GCF) and the Least Common Multiple (LCM) 21. Find the GCF of the numbers 15 and Find the GCF of the numbers 6, 36, and Find the LCM of the numbers 9 and Find the LCM of the numbers 3, 4, and 15

16 OPERATIONS WITH FRACTIONS 25. Simplify the fraction: Write the reciprocal of MIXED NUMBERS 36. Change to fraction: Change to mixed number:

17 THE ORDER OF OPERATIONS (9 2) (9 2) (7 5) (14 4 ) (15 9) 2 OPERATIONS WITH INTEGERS ( 17) ( 5) ( 12) ( 5) ( 17) ( 14) EQUATIONS x r z y a n y 7.86 p k

18 x s p b r RATIOS AND PROPORTIONS 80. Are the ratios equivalent? 81. Are the ratios 82. PERCENTS equivalent? 2 3 n 24 Complete the table: and 9 5 and n n n 3 12 Percent Decimal Fraction % 89. What percent of 140 is 28? is 45% of what number? 91. What number is 5% of 340?

19 EXPONENTS 92. Write as an exponent xxxxxxx 93. Write as an exponent Evaluate the exponent 95. Evaluate the exponent 96. Evaluate the exponent THE COORDINATE PLANE 97. Graph each point: A(2,0), B(-3,-4), C(2,-5), D(-1,4), E(1,2), F(0,-3) 98. Write the coordinates of the points:

20 WRITING DECIMALS AS FRACTIONS AND MIXED NUMBERS 99. Write the decimal 0.31 as fraction or mixed number Write the decimal 2.05 as fraction or mixed number Write the fraction 3 as decimal Write the fraction as 9 20 decimal Write the fraction 5 12 as decimal Write the mixed number as decimal Write the mixed number as decimal ORDERING NUMBERS 106. Order the numbers from least to greatest: 246.8,248.6, 244.9, Order the numbers from least to greatest: 9, 6.7, 7.24, Order the numbers from least to greatest: -17.8, 3 4, 0.8, -15, Order the numbers from least to greatest: , 2,, 0.12, 0.102, 9.5, 7 3

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26 Review packet for students entering Algebra 1 (CP and Honors) class. PLEASE COMPLETE ALL PROBLEMS, SHOWING ALL YOUR WORK. REMEMBER, NO CREDIT WILL BE GIVEN UNLESS ALL WORK IS SHOWN. OPERATIONS WITH WHOLE NUMBERS OPERATIONS WITH INTEGERS ( 17) ( 5) ( 3) ( 12) ( 3) ( 17) ( 5) ( 17) ( 13) OPERATIONS WITH DECIMALS OPERATIONS WITH FRACTIONS. 30. Simplify the fraction:

27 MIXED NUMBERS. 41. Change to fraction: Change to mixed number: THE ORDER OF OPERATIONS (9 2) (9 2) (7 5) (15 9) (17 3) (5 7) SIMPLIFYING AND EVALUATING EXPRESSIONS. EQUATIONS. 63. Evaluate: 35 4x y for x 7, y Evaluate: 12x 7 y ( z 6) for x 3, y 4, z Simplify: 2( c 4) 3 c 66. Simplify: 3(2x 5) 3( x 4) 67. Simplify: 4m6m 7m z 0 r 13 5 k INEQUALITIES c 48 x r 41 5r ( t 1) z18 45z 76. 6( x 8) 5x x Solve and graph: 5 p 2 a Solve and graph: Solve and graph: 95d Solve and graph: 3y y

28 EXPONENTS. RULES OF EXPONENTS. NEGATIVE AND ZERO EXPONENT. SCIENTIFIC NOTATION. 82. Write the answer using exponents: 83. Write the answer using exponents: 84. Simplify: a a xx 4 12x m a m 85. Simplify: am 86. Write using only positive exponents: 2 5w 87. Write using only positive 3 0 exponents: st 88. Write in scientific notation: = 89. Write in scientific notation: 16,000,000= 90. Write in standard form: Write in standard form: RATIOS. RATES. PROPORTIONS. 92. Find the unit rate: 58 mi 4h 93. Write equivalent ratio: 94. Solve the proportion: 95. Solve the proportion: 4? 5 35 b s THE GCF AND THE LCM. PERCENTS. 96. Find the GCF of the numbers 15 and Find the GCF of the numbers 6, 36, and Find the GCF of the monomials 16x 2 and 68x 99. Find the GCF of the monomials 3y, 9y 2, and 12y Find the LCM of the numbers 9 and Find the LCM of the numbers 3, 4, and Find the LCM of the monomials 4s 3 and 36s Find the LCM of the monomials 10n 2 p and 16np 104. Write as percent: 0.045= 105. Write as percent: Write as fraction in simplest form: 15%= 107. Write as decimal: 3.7%= 108. What percent of 140 is 28? is 45% of what number? 110. What number is 0.5% of 3400? 111. Find the percent of change from 40 to Write the formula for simple interest Write the formula for compound interest.

29 COORDINATE PLANE Name each quadrant s number Graph each point: A(2,0), B(-3,-4), C(2,-5), D(-1,4), E(1,2), F(0,-3) 116. Write the coordinates of the points: SQUARE ROOTS Find the two square roots of each number: a) 49 b) 2500 c) Between which two integers is the square root: a) 123 b) 27

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31 Name: Class: Date: ID: A Geometry Summer Review Packet Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which angle is a right angle? A B C D 2. Which statement is true? A All rectangles are quadrilaterals. B All quadrilaterals are parallelograms. 3. Solve by substitution: 3x + 2y = 4 y = 4x 2 A Ê Ë Á 0, 2 ˆ B no solution C Ê Ë Á 2, 6 ˆ D ( 1, 1 2 ) Solve by elimination: 4. 3x + 6y = 9 x 6y = 11 A (5, 1) B (0, 3 2 ) C (10, 1 ) D no solution 6 1

32 Name: ID: A 5. Solve the system using the addition method: 2x 4y = 12 3x + 4y = 8 A Ê Ë Á 4, 1 ˆ B no solution C Ê Ë Á20, 1ˆ D Ê Ë Á 0, 3 ˆ Find the sum. Ê 6. 2a 7 + 3a 3 ˆ 6 Ë Á + Ê 2a a 7 ˆ Ë Á A 8a 7 + a 3 2 C a 7 + 8a B a 7 + 8a 3 2 D 8a 7 + a Simplify the expression. Ê 7. 5q 5 ˆ + 4 Ë Á Ê Ë Á 2q3 + 9 ˆ + Ê 6q5 q 3 ˆ Ë Á A 11q 5 3q 3 5 C 11q 3 + 3q B 3q q 3 5 D 11q 5 + 3q

33 Name: ID: A Find the difference. Ê 8. 6b 3 + 3b 2 ˆ + 8 Ë Á Ê 2b3 8b 2 Ë Á + 6b 5 ˆ A 4b b 2 6b + 13 C 11b 3 4b 2 6b + 3 B 4b b 2 + 6b 13 D 11b 3 + 4b 2 6b 3 Short Answer 1. Approximate the square root to the nearest integer Approximate the square root to the nearest integer Approximate the square root to the nearest integer Evaluate the expression. 16 3

34 Name: ID: A 5. Evaluate the expression Evaluate the expression. ± 4 7. Evaluate the expression. x when x = Order the numbers from least to greatest: 1.6, 4, 0, 3.1, 5 Solve the equation. 9. a 2 = w 2 = n 2 = 72 4

35 Name: ID: A Simplify the expression Simplify the expression Simplify Evaluate the expression. 3x x 1 when x = 3 5

36 Name: ID: A 18. Evaluate the expression. 2m + 9 m when m = Evaluate the expression. 3( 5m 4) when m = Find the sum or difference. Ê 5x 2 ˆ Ê ˆ 11x + 9 Ë Á + 7x 13 3x2 Ë Á 21. Find the sum or difference. Ê 17y 2 ˆ 6y + 5 Ë Á Ê Ë Á 11y2 2y + 8 ˆ 22. Find the product. ( 3a 5b) Evaluate the expression. 9x 2 4 when x = 3 6

37 Name: ID: A 24. Evaluate the expression Evaluate the expression. È 4 32 ( 17 12) 2 ÎÍ 26. Evaluate the expression Check whether the given number is a solution of the equation or inequality. 5c 13 = 12; Solve the equation. 17 = 4x Evaluate the expression

38 Name: ID: A 30. Solve the equation. 2( x + 3) = 3 8x 12 4 ( ) 31. Solve the equation. 28 = 10w 3w 32. Solve the equation. 17 = 5x 6x Solve the equation. 9 n 3 = Solve the equation. 16w 10w+ 13 = Solve the equation. 2 3 t = 18 8

39 Name: ID: A 36. Solve the equation. 1 2 = 4( 5x 3) 37. Solve the equation. k 7 9 = Solve the equation, if possible. 7( h + 3) + 4 = Solve the equation. d = Solve the equation, if possible. 2y + 5 = 3Ê Ë Á4y 5ˆ 41. Solve the equation, if possible. 24 = 13z 4z Solve the equation, if possible. 6 11x = 7x 12 9

40 Name: ID: A 43. Solve the equation. 112 = 7n 44. Solve the equation. 4h 13 = 7h Solve the equation, if possible. 12( x + 3) = x 46. Solve the equation. 2 5 ( 25z 30 ) = 3 4 ( 12z+ 16 ) 47. Solve the proportion. x 8 = Solve the proportion. 12 3w =

41 Name: ID: A 49. Solve the proportion. Check your solution. 2m = 5m Solve the proportion. Check your solution. 13 w = 26 w Solve the proportion. t 65 = Solve the proportion. m + 18 m = Solve the proportion. j = Solve the proportion = 3k

42 Name: ID: A 55. Write the equation so that y is a function of x. 12x + 3y = Write the equation so that y is a function of x. 5x = 10y Write the equation so that y is a function of x. 8x 4y = Identify the slope and y-intercept of the line with the given equation. y = 4 5 x Graph the equation. y = 1 4 x 5 12

43 Name: ID: A 60. Find the slope of the line that passes through the points. Ê Ë Á 7,3 ˆ and Ê Ë Á 3,8 ˆ 61. Write an equation in slope-intercept form of the line with the given characteristics. slope 3; y-intercept Write an equation in slope-intercept form of the line with the given characteristics. m = 4; passes through Ê Ë Á 3, 2ˆ Solve by elimination: 63. 3x 4y = 5 5x + 4y = Solve the system. y = 4x + 4 y = x Use elimination to solve the linear system. 3x 4y = 21 4x + 2y = 6 13

44 Name: ID: A 66. Solve the system. 2x 6y = 18 3x + 7y = 37 Problem 1. Find the x-intercept and the y-intercept of the graph of the equation. y = 7x 3 2. Find the slope of the line that passes through the points. Ê Ë Á 4,3 ˆ and Ê Ë Á 7,5 ˆ 3. Find the slope of the line that passes through the points. Ê Ë Á 1,2 ˆ and Ê Ë Á 2,2 ˆ 4. Graph the equation. y = 7 2 x 1 14

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46 Algebra II CP Summer Review (111 problems) Please write all answers on the provided answer sheets. Simplify each expression. Part 1 Equations and Inequalities 1. 25x x 2. 6y + 12x 12y 9x 3. 5(2u + 3w) 2(5u 7w) 4. 10m 4(3m + 7) + 6m 5. 9t t + 6t 8t (2b + 3) + 8(b 6) 7. 3g + 9g 2 12g 2 + g 8. 7t 4 + 7t 2 2t 2 9t (n 2) 8n A New York City taxi charges $2.50, plus $.40 for each fifth of a mile if it is not delayed by traffic. Write an expression for the cost of the ride if you travel x miles in the taxi with no traffic delays. Solve each equation x + 16 = y + 15 = (q 5) = m + 38 = 5m j + 25 = 12j (2n 5) = 3(6n 2) n + 3(8 + 8n) = 6(n 4) p p = 2(2p + 4) 3(2p 2) (x + 3) ( 9x 4) = x (2k + 11) = 12(2k + 12) (x 12) = 9(1 + 7x) (p + 10) = 4 5(2p + 11) Complete each question. 23. At a vegetable stand, you bought 3 pounds of peppers for $4.50. Green peppers cost $1 per pound and orange peppers cost $4 per pound. Find how many pounds of each kind of pepper you bought. (Pounds don t have to be in whole numbers.) 24. You can wash one window in 15 minutes and your sister can wash one window in 20 minutes. How many minutes will it take to wash 12 windows if you work together?

47 Solve for y. Then find the value of y for the given value of x x + y = 7; x = y 3x = 18; x = xy 6y = 15; x = x = 6y + 9; x = x 2y = 10; x = x 3xy = 1; x = While on vacation, your family rented a car for $293. The car rental cost $180, plus $.25 for every mile driven over 150 miles. How many miles did you drive while on vacation? 32. The formula S = 2πrh + 2πr 2 gives the surface area of a cylinder with height h and radius r. Solve the formula for h. Find h if r = 5 centimeters and S = 400 square centimeters. 33. The formula 1 3 πr2 h gives the volume of a cone with height h and base radius r. Solve the formula for h. Then find h when r = 2 inches and V = 45 cubic inches. Solve the inequality. Then graph the solution x 3 < x x + 8 > 9x < 10 x < x x x b < 7 or 7 5b < n > 44 or 10 12n > A triangle has sides of length 10, 2x, and 3x. As you learned in Geometry, the sum of the lengths of any two sides must be greater than the length of the third side. Write and solve three inequalities to find the possible values of x.

48 Solve the equation. Check for extraneous solutions p + 2 = q 5 = 2q 45. 8r + 1 = 3r Solve the inequality. Then graph the solution on a number line. 46. x y > z x y + 4 > z 5 < The circumference of a volleyball should be 26 inches, with a tolerance of 0.5 inches. Write and solve an absolute value inequality that describes the acceptable circumference of a volleyball. (Tolerance means it can be that much larger (+) or smaller (-).)

49 Part 2 Linear Equations and Functions Find Slope and Rate of Change Find the slope of the line passing through the given points. 53. ( 2, 1), (4, 3) 54. (1, 5), (1, 2) 55. (5, 3), (1, 7) 56. (6, 2), ( 8, 2) 57. A skateboard ramp has a rise of 15 inches and a run of 54 inches. What is its slope? 58. A new set of car tires has a tread depth of 8 millimeters. The tread depth decreases by 0.12 millimeter per thousand miles driven. Write an equation that gives the tread depth as a function of the distance driven. Then determine at what distance the tread depth will be 2 millimeters. Graph equations of lines Write the equation in slope-intercept form and then graph the equation. 59. y = 5 x 60. y 5x = x = x 4y = y = y = 3 x x + 2y = y = 2x x + 5y = x 2y = x = y = 4 Write Equations of Lines Write an equation of the line that meets the following criteria. 71. m = 2, b = m = 5, b = m = 5, b = (0, 2), m = Passes through: (3, -1) Slope: Passes Through: ( 4, 1), (3, 6) 76. Passes through: (-4, 3) Slope: Passes Through: ( 3, 4), (2, 6) 80. Passes through: (9, -1) Parallel to: y = 1 3 x Passes through: (9, -1) Perpendicular to: y = 5x Passes Through: ( 4, 5), (12, 7)

50 Graph Linear Inequalities in Two Variables Tell whether the given ordered pair is a solution of the inequality. 82. y 5x; (0,1) 83. y > 3x 7; ( 4, 6) 84. 3x 4y < 8; ( 2, 0) Graph the inequality in a coordinate plane (not a number line!) y < y < 2x x 8y 24

51 Part 3 Linear Systems Solve Linear Systems by Graphing Graph the system and estimate the solution. Check the solution algebraically x y = 9 x + 3y = x 3y = 2 x + y = x + y = 6 x + 2y = x y = 2 4x + y = x + 2y = 6 6x 2y = x 3y = 15 x 3 2 y = x y = 12 x + 8y = 4 Solve Linear Systems Algebraically Solve the system using any algebraic method x + y = 9 x 2y = x + 3y = 2 4x + 7y = x + 4y = 26 5x 2y = x + 4y = 6 4x + 5y = x + 2y = 5 2x + 3y = x 7y = 36 x 3y = x + 5y = 5 2x 3y = x + 3y = 5 9x 6y = x + 3y = 9 3x + y = The cost of 14 gallons of regular gasoline and 10 gallons of premium gasoline is $ Premium costs $.40 more per gallon than regular. What is the cost per gallon of each type of gasoline? 105. A total of $15,000 is invested in two m b bonds that pay 5% and 7% simple annual interest. The investor wants to earn $880 in interest per year from the bonds. How much should be invested in each bond? 106. For the opening day of a carnival, 800 admission tickets were sold. The receipts totaled $3775. Tickets for children cost $3 each, tickets for adults for $8 each, and tickets for senior citizens cost $5 each. There were twice as many children s tickets sold as adult tickets. How many of each type of ticket were sold? 107. On a certain river, a motorboat can travel 34 miles per hour with the current and 28 miles per hour against it. Find the speed of the motorboat in still water and the speed of the current. Systems of Linear Inequalities Graph the system of linear inequalities x + y < 1 x + 2y x + 3y > 6 2x y x + 3y 5 x + 2y < x + y < 6 y > 2

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59 Algebra II Honors Summer Review (150 problems) Simplify each expression. Part 1 Equations and Inequalities 1. 25x x 2. 6y + 12x 12y 9x 3. 5(2u + 3w) 2(5u 7w) 4. 10m 4(3m + 7) + 6m 5. 9t t + 6t 8t (2b + 3) + 8(b 6) 7. 3g + 9g 2 12g 2 + g 8. 7t 4 + 7t 2 2t 2 9t (n 2) 8n A New York City taxi charges $2.50, plus $.40 for each fifth of a mile if it is not delayed by traffic. Write an expression for the cost of the ride if you travel x miles in the taxi with no traffic delays. Solve each equation x + 16 = y + 15 = (q 5) = m + 38 = 5m j + 25 = 12j (2n 5) = 3(6n 2) n + 3(8 + 8n) = 6(n 4) p p = 2(2p + 4) 3(2p 2) (x + 3) ( 9x 4) = x (2k + 11) = 12(2k + 12) (x 12) = 9(1 + 7x) (p + 10) = 4 5(2p + 11) Complete each question. 23. At a vegetable stand, you bought 3 pounds of peppers for $4.50. Green peppers cost $1 per pound and orange peppers cost $4 per pound. Find how many pounds of each kind of pepper you bought. (Pounds don t have to be in whole numbers.) 24. You can wash one window in 15 minutes and your sister can wash one window in 20 minutes. How many minutes will it take to wash 12 windows if you work together?

60 Solve for y. Then find the value of y for the given value of x x + y = 7; x = y 3x = 18; x = xy 6y = 15; x = x = 6y + 9; x = x 2y = 10; x = x 3xy = 1; x = While on vacation, your family rented a car for $293. The car rental cost $180, plus $.25 for every mile driven over 150 miles. How many miles did you drive while on vacation? 32. The formula S = 2rrrh + 2rrr 2 gives the surface area of a cylinder with height h and radius r. Solve the formula for h. Find h if r = 5 centimeters and S = 400 square centimeters. 33. The formula 1 rrr 2 h gives the volume of a cone with height h and base radius r. 3 Solve the formula for h. Then find h when r = 2 inches and V = 45 cubic inches. Solve the inequality. Then graph the solution x 3 < x x + 8 > 9x < 10 x < x x x b < 7 or 7 5b < n > 44 or 10 12n > A triangle has sides of length 10, 2x, and 3x. As you learned in Geometry, the sum of the lengths of any two sides must be greater than the length of the third side. Write and solve three inequalities to find the possible values of x.

61 Solve the equation. Check for extraneous solutions p + 2 = q 5 = 2q 45. 8r + 1 = 3r Solve the inequality. Then graph the solution on a number line. 46. x y > z x y + 4 > z 51 < The circumference of a volleyball should be 26 inches, with a tolerance of 0.5 inches. Write and solve an absolute value inequality that describes the acceptable circumference of a volleyball. (Tolerance means it can be that much larger (+) or smaller (-).)

62 Part 2 Linear Equations and Functions Find Slope and Rate of Change Find the slope of the line passing through the given points. 53. ( 2, 1), (4, 3) 54. (1, 5), (1, 2) 55. (5, 3), (1, 7) 56. (6, 2), ( 8, 2) 57. A skateboard ramp has a rise of 15 inches and a run of 54 inches. What is its slope? 58. A new set of car tires has a tread depth of 8 millimeters. The tread depth decreases by 0.12 millimeter per thousand miles driven. Write an equation that gives the tread depth as a function of the distance driven. Then determine at what distance the tread depth will be 2 millimeters. Graph equations of lines Write the equation in slope-intercept form and then graph the equation. 59. y = 5 x 60. y 5x = x = x 4y = y = y = 3 x x + 2y = y = 2x x + 5y = x 2y = x = y = 4 Write Equations of Lines Write an equation of the line that meets the following criteria. 71. m = 2, b = m = 5, b = m = 5, b = (0, 2), m = Passes through: (3, -1) Slope: Passes through: (-4, 3) Slope: Passes Through: ( 3, 4), (2, 6) 78. Passes Through: ( 4, 5), (12, 7) 79. Passes Through: ( 4, 1), (3, 6) 80. Passes through: (9, -1) Parallel to: y = 1 x Passes through: (9, -1) Perpendicular to: y = 5x + 7

63 Graph Linear Inequalities in Two Variables Tell whether the given ordered pair is a solution of the inequality. 82. y 5x; (0,1) 83. y > 3x 7; ( 4, 6) 84. 3x 4y < 8; ( 2, 0) Graph the inequality in a coordinate plane (not a number line!) y < y 2x > x 8y 24

64 Part 3 Linear Systems Solve Linear Systems by Graphing Graph the system and estimate the solution. Check the solution algebraically x y = 9 x + 3y = x 3y = 2 x + y = x + y = 6 x + 2y = x y = 2 4x + y = x + 2y = 6 6x 2y = x 3y = 15 x 3 y = x y = 12 x + 8y = 4 Solve Linear Systems Algebraically Solve the system using any algebraic method x + y = 9 x 2y = x + 4y = 6 4x + 5y = x + 2y = 5 2x + 3y = x + 3y = 2 4x + 7y = x 7y = 36 x 3y = x + 5y = 5 2x 3y = x + 4y = 26 5x 2y = x + 3y = 5 9x 6y = x + 3y = 9 3x + y = The cost of 14 gallons of regular gasoline and 10 gallons of premium gasoline is $ Premium costs $.40 more per gallon than regular. What is the cost per gallon of each type of gasoline? 105. A total of $15,000 is invested in two corporate bonds that pay 5% and 7% simple annual interest. The investor wants to earn $880 in interest per year from the bonds. How much should be invested in each bond? 106. For the opening day of a carnival, 800 admission tickets were sold. The receipts totaled $3775. Tickets for children cost $3 each, tickets for adults for $8 each, and tickets for senior citizens cost $5 each. There were twice as many children s tickets sold as adult tickets. How many of each type of ticket were sold? 107. On a certain river, a motorboat can travel 34 miles per hour with the current and 28 miles per hour against it. Find the speed of the motorboat in still water and the speed of the current.

65 Systems of Linear Inequalities Graph the system of linear inequalities x + y < 1 x + 2y x + 3y > 6 2x y x + 3y 5 x + 2y < x 3y 9 1 x y x + y < 6 y > 2

66 Part 4 Polynomials Classifying polynomials Write each polynomial in standard form, then name each polynomial by degree and number of terms a + 8 7a p 4 + 5p x x x 2 3x + 15x Operations with polynomials Add, subtract, or multiply the polynomials. Write your answer in standard form (13a + 6a 4 6) (13a a) 120. (7x 5y)( 4x 2 + 4xy 5) 121. (11x 3 + 8x 3 y 4 ) + (x 3 y 4 9x xy 3 ) ( 5xy 3 + 5x 3 y 4 ) 122. (6a 2 8b) ( 3 n ( x + 9y)( x 9y) 125. (2x 2 3y)(2x 2 + 3y)

67 Factoring polynomials Factor the polynomial completely x 2 + 8x n 2 11n t 2 + 7t 129. y 3 + 2y 2 81y n 3 121n a a a 2 Solve the equation a = 10a t 2 + 7t = x 2 = 22x y 2 10y + 25 = n 2 49 = a 3 + a 2 = 64a + 64

68 Part 5 Exponents and Radicals Simplify the expression Properties of Exponents k -7 m 5 b ( k 3 m 6 4 9y 2k ( 3 9n (3g 3 h 4 ) 2 (2g 4 h 9 ) 3 Simplifying Radical Expressions Simplify the radical expression h 6 k J14q 2J4q w ( ) x 15 y 9 7xy 11

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70 Evaluating and Simplifying Algebraic Expressions EXPONENTS AND RADICALS

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87 Evaluating and Simplifying Algebraic Expressions EXPONENTS AND RADICALS

88 8. Sim plif y the exponential expression. 9. Sru.mplify the exponential expre sion. 28x 4 y 4 21x 7 y Simplify the exponential e pre s1on. ] 1. Simpl ify the exp onential e xpression. Assume that variables represent nonzero real numbers. 12. S n. m p lify the exponential expre sion. Assume that the variables repre ent nonzero real number.. x5y2z6 ) - 3 ( - 5 X y Z _ 13. Use the product rule to simplify the follow i ng express ion. Assume that varia ble represent non negative real numbers. 14. Use the product rule to simplify the following expre sion. A sume that variab]e represent nonnega ti.ve real numbers.

89 15. Use the quotient rule to simplify the fouowing expression. Assume that x > 0. J6. Add the terms if possible. V98+ 6V2 17. Add and ubtract the following term, if pos ible. 2V8- 'VY2 + 2V72- V Ratio naml ze the denominator fu 19. Rationa ml ze the denom inator. Simplify the answer. 20. Evaluate the following expres ion or indicate that the root is not a real number. 21. Simplify. 22. Simplify by factoring. Assume that the variab le in the radicand represents a positive real number and that the radicand does not involve negative quantities raisedto even powers.

90 23. Simplify the radic:a1 e xpress ion. 24. Add or subtract tenns whenever possible. { { 6 \J 16 V Evaluate the express ion without using a calculator / Simphfy using propertie of exponents. (5x1 12 ) (2x 11s) 27. SimpHfy by reducing the index of the radical.,19 V Simplify by reduc ing the in dex of the radic al. 29. Simplify the given expression. Assume that all variables represent pos itive numbers. () x2-5y 10-) 1/5(xy 11 5) 30. SimpUfy tbe given expression. Assume tbat all varia bles repr-esent positivenumbers. X- 5 / Sy 1/ 3 ) - 6.( X - l / 8

91 POLYNOMIALS

92 39. Find the product. (x - y) (x 2 + 2xy + y 2 ) 40.. Find the product. (x y + 11)(x + y - 11) 41. Perform the indicated operations. (8x + 9y) 2 - (8x - 9y) ' Factor the po lynom ial using the greatest commo11 factor. If iliere is no common factor other than 1 and the polynom ial can not be factored, so state. 15x Factor the greatest common factor from thepolynomial. 18 x x 44.. Factor by grouping. X 3-6x X F a ctor the following by grouping. 3-9x 2 4x Factor the expression by grouping. 3x 3-5x 2-15x Factor the given polynomial. x x + 77

93 48. Facto r the trinorrual, or state that the trinornrual is prime. X 2-4!,x Factor the given po]ynomial. x-? - 9x Factor the trinomial co1npletdy. 5x 2-14x Factor the following trrnomial, or state that the trinom ial is prime Factor the trinomial, or state that the trinomia l isprime. 6y 2 + l l y Factor the trin omialcompletely. 7a ab + 2b Facto r the difference of two squares. I6x Factor the difference of two quares. 81x 2 - l69y Factor the following difference ofnvo squares. y 4-256

94 57. Factor the difference of two quares. 256x Facto r the fojlowing polynomial using the formula for the diffe re nce of two cubes. y Facto r using the fonnula for the sum or difference of twocubes. 27x Factor the trinomial completely. 3x 2 + 1Sx Factor completely, or state that the polynomial is prime. x 3-5x 2-16x Factor the expression completely or state that the polynom ial is prime. 7x 2-7x Factor completely, or state that the polynomial isprime. x Fac torco1npletely. 14x 2 (x - 3) - 9x(x - 3) - 8(x - 3)

95 RATIONAL EXPRESSIONS (FRACTIONS)

96 71. Multiply the following rational ex.pressions. x-? - 4x 4 x Multiply the following rational expre ions. x 2 + 8x x x 'ḷ J. Divide as indicated. x 1-9 x + 3 X x Divide the following rational expression. x 2-25 X +5 X - 5 5x Multiply and divide as indicat ed. x 2 + x - 72 x 2 + IOx + 24 x +4 x? - - 7x- 8 -:- x- ') + 7x Add as ind icated. 9x + I 3x x + 5 6x Add. x 2-9x x 2 +8x x- ) + 1O. x + x-? + 1Ox

97 SOLVING EQUATIONS

98 85. Solve and check the Linear equation. 5(X - 2) + }9 = 4(x + 4) 86. Solve and check the linear equation. X x = Complete parts a and b for the following equation = x + 7 X a. Write the - alue or aiue of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 88. Solve the following formula for the specified variable. A = Skw for S 89. Solve the formula for q. B=W+qrn 90. Solve the following formula for the specified varia ble. 1 C = - m(r + k) for r Solve the formula for f. 1 I I - +- =- a t f 92. Find the solution set for the equation. lx - 61= 9

99 93. Find the solution set for theequation. 3l2x - l l= Solve the absolute value equation or indicate that the equation has no solution. 95. Find thesolution set for the equation. l6x s = Use factoring to solve the quadratic equation. Check by substitution or by using a graphing utility and identifying x-inte rcepts. x 2-5x - l4 = Solve. 98. Solve the following quadratic equation by the square rootproperty. 2x = Solve the equation by the square root property. 2(x + 4)2 = 90 LOO. Solve the quadratic equation by completing the square. x 2 + 4x = 5 10 I. Solve the following equation using the quadra tic formufa. x 2 + 6x + 8 = 0

100 W2. Solve the equation using the quadratic formula. x 2 + 7x + 3 = Solve the equation u jng the quadratic formu la. I04. Solve the equat ion by the method of your choice. 2x 2 - x = 6 l05. Solve the equation using any method = 36x I06. Solve the equation by the met hod of your choice. (5x + 2) (x + 1) = So lve the following equation. 4x.2-48x = 0 1O.. Solve the equation by the method of yourchoice x X Find the real solut ions of the equation. V28-3x =x 11O. Solve the radical equation. V 2x + l 4 = x - 5

101 LINEAR EQUATIONS

102 Logarithms and Exponentials

103 Use the compound interest formulas A = l_.{ l+nj rnl t anda = Pert to solve. 6) Find l:he accu mu la ted value of an in ves tm en t of 17,000 at 6 % com pounded ann ua ll y f r 13 years. Write the equation in its equivalent exponential form. 7) log = 3 8) log 2 16 = x Write the equation in its equivalent logarithmic fonn. 9) 63 = ) 6 2 = X 11) 4x = 64 Eva luate the expression with out usii1g a calculator. 12) l g ) log 1 25

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105 Name Calculus Honors: Required Problems Solve all problems on notebook and/or graph paper. Show ALL your work, skipping a line between each problem. Functions For problems 1 6, perform the indicated function evaluations. 1. f(x) = 10x 3 a. f( 5) b. f(0) c. f(7) d. f(t 2 + 2) e. f(12 x) f. f(x + h) 4. f(x) = 4x + 5 a. f(0) b. f( 4) c. f( 2) d. f(5 12x) e. f(2x 2 + 8) f. f(x + h) 2. f(x) = 4x 2 7x + 1 b. f( 5) b. f(0) c. f(7) d. f(6z) e. f(1 3y) f. f(x + h) 5. f(x) = x2 +9 4x+8 a. f( 4) b. f(4) c. f(1) d. f(2 7x) e. f( 3x + 4) f. f(x + h) 3. f(x) = x+5 1 x c. f(4) b. f(0) c. f( 7) d. f(x 2 5) e. f(4 x + 9) f. f(x + h) 6. f(x) = 3 x x 2x+5 a. f(7) b. f(0) c. f( 4) d. f(x 2 10) e. f(5 x) f. f(6x x 2 )

106 Difference Quotient The difference quotient of a function f(x) is defined by: quotient of the following functions. f(x+h) f(x) h, compute the difference 7. f(x) = 4 7x 8. f(x) = f(x) = 2x f(x) = 3 8x x f(x) = 4 + 3x 12. f(x) = 4 1 2x Finding Roots For the following problems, determine all the roots of the given function. 13. f(x) = x x f(x) = 6x 4 5x 3 4x f(x) = x 2 11x f(x) = 4x x 5 + x f(x) = x 7 + 6x 4 16x 18. f(x) = x 1 2 8x

107 Finding Domain and Range For the following problems, determine the domain and range of the given function. 19. f(x) = x 2 8x f(x) = 4 7x x f(x) = 5 2 x 22. f(x) = x f(x) = x 24. f(x) = 6 5 x 25. f(x) = x f(x) = 8x2 12x+4 16x f(x) = x x 28. f(x) = x3 x 2 +x 1 35x 3 +2x 4 x f(x) = x2 +x x 3 9x 2 +2x 30. f(x) = 3 x4 4x 2 +10x f(x) = x f(x) = 36 9x f(x) = 4x 3 4x 2 + x Function Composition For the following problems, compute (fog)(x) and(gof)(x). 34. f(x) = 2x + 5 g(x) = 8 23x 35. f(x) = 2 x g(x) = 2x f(x) = 2x 2 + x 4 g(x) = 7x x f(x) = x 2x+3 g(x) = 5x + 8

108 Inverse Functions For each of the following functions, find the inverse of the function. Verify your inverse by computing (fog)(x) or (gof)(x), which will equal x if they are indeed inverses of eachother. 38. f(x) = 11x f(x) = 4 10x 40. f(x) = 7 + (2x + 1) f(x) = 2x+14 6x f(x) = 2x f(x) = f(x) = 6 18x 15x f(x) = 1 x 9 12x Trig Functions Without using a calculator, determine the exact value of each of the following. 46. tan ( 3π 4 ) 47. sin(7π 6 ) 48. sin( 3π 4 ) 49. cos (4π 3 ) 50. cot( 5π ) sin ( 5π ) 52. sec ( π ) 53. cos (11π ) cot ( 4π 3 ) 55. cos ( π 4 ) 56. csc (2π 3 ) 57. cos (5π 4 ) 58. tan ( 31π 6 15π 23π ) 59. cos ( ) 60. sec ( ) 61. cot (11π )

109 Solving Trig Equations Without using a calculator find the solution(s) to the following equations. If an interval is given then find only those solutions that are in the interval. If no interval is given then find all solutions to the equation cos(8x) = cos(8x) = 5; [ π, π ] sin ( 2x ) + 2 = 0; [0, 5π] sin ( x 4 ) = sin (x 4 ) = 3 ; [0,16π] = 8 cos(3x) ; [0, 5π 3 ] With the aid of a calculator find the solution(s) to the following equations. If an interval is given then find only those solutions that are in the interval. If no interval is given then find all solutions to the equation sin ( t 3 ) = cos(4x) + 8 = 10 cos (4x) = cos ( w 2 ) ; [ 20,5] tan(3w) + 3 = sin ( 3x ) 7 = 1 ; [0,15] sin ( x ) 9 = 7 sin (x ) + 17; [ 10, 20] 2 With the aid of a calculator find all of the solutions to the following equations. You may have to do crazy things, like factor, in order to get the solution. Use at least 4 decimal places tan ( x ) sin(2x) tan 3 (x ) = tan(4x) sec(2x 1) + sec(2x 1) = cos 2 (x) 4 cos(x) = sin(2x) = 3 sin 2 (2x)

110 Exponential Functions Sketch the graphs of each of the following functions. 78. f(x) = 7 3 x f(x) = 3 5 4x f(x) = 6e 2x f(x) = 7 + 9e 2 3t 5 Logarithmic Functions Without using a calculator, determine the exact value of each of the following. 82. log log log log ln log e Write each of the following in terms of simpler logarithms. 88. log 7 (10a 7 b 3 c 8 ) 89. log[z 2 (x 2 + 4) 3 ] 90. ln ( w2 t 3 4 t+x )

111 Combine each of the following into a single logarithm with a coefficient of one ln t 6 ln s + 5 ln w log(z + 1) 2 log x 4 log y 3 log z log 3(x + y) + 6 log 3 x 1 3 Use the change of base formula and a calculator to find the value of each of the following log log5 7 8 Exponential and Logarithm Equations Find all the solutions to the given equations. If there is no solution to the equation clearly explain why = e 10x e 2x+x2 7 = e 7x 12e 8x+5 = log(w) log(3w + 7) = xlog (6x + 1) 3x 2 log(6x + 1) = ln(3x + 1) ln(x) = z2 2 = 5

112 Compound Interest 103. We have $2,500 to invest for 80 months. How much money will we have if we put the money into an account that has an annual interest rate of 9% and interest is compounded: (a) quarterly (b) monthly (c) continuously 104. Starting with $60,000 and putting it into an account that earns an annual interest rate of 7.5%. How long will it take for the money to reach $100,000 if the interest is compounded: (a) quarterly (b) monthly (c) continuously Exponential Growth/Decay 105. A population of bacteria initially has 90,000 present and in 2 weeks there will be 200,000 bacteria present. (a) Determine the exponential growth equation for this population. Q = Q 0 e kt (b) How long will it take for the population to grow from its initial population of 90,000 to a population of 150,000? 106. We initially have 2kg of some radioactive element and in 7250 years there will be 1.5kg left. (a) Determine the exponential decay equation for this element. Q = Q 0 e kt (b) How long will it take for half the element to decay? (c) How long will it take until 250 grams of the element left?

113 Common Graphs Without using a graphing calculator or utility, sketch the graph of each of the following y = 2x f(x) = x f(x) = x f(x) = x f(x) = x f(x) = x 2 + 8x f(x) = (x + 5) f(x) = ln x 115. f(x) = 2 x 116. x 2 6x + y 2 + 8y + 24 = (x+4) (y+2)2 25 = f(x) = tan (x + π 3 ) 119. f(x) = sec(x) f(x) = 2 x 121. f(x) = 2 sin (2x π 6 ) 1

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115 Name AP Calculus AB: Required Problems Solve all problems on notebook and/or graph paper. Show ALL your work, skipping a line between each problem. Functions For problems 1 6, perform the indicated function evaluations. 1. f(x) = 10x 3 a. f( 5) b. f(0) c. f(7) d. f(t 2 + 2) e. f(12 x) f. f(x + h) 4. f(x) = 4x + 5 a. f(0) b. f( 4) c. f( 2) d. f(5 12x) e. f(2x 2 + 8) f. f(x + h) 2. f(x) = 4x 2 7x + 1 b. f( 5) b. f(0) c. f(7) d. f(6z) e. f(1 3y) f. f(x + h) 5. f(x) = x2 +9 4x+8 a. f( 4) b. f(4) c. f(1) d. f(2 7x) e. f( 3x + 4) f. f(x + h) 3. f(x) = x+5 1 x c. f(4) b. f(0) c. f( 7) d. f(x 2 5) e. f(4 x + 9) f. f(x + h) 6. f(x) = 3 x x 2x+5 a. f(7) b. f(0) c. f( 4) d. f(x 2 10) e. f(5 x) f. f(6x x 2 )

116 Difference Quotient The difference quotient of a function f(x) is defined by: quotient of the following functions. f(x+h) f(x) h, compute the difference 7. f(x) = 4 7x 8. f(x) = f(x) = 2x f(x) = 3 8x x f(x) = 4 + 3x 12. f(x) = 4 1 2x Finding Roots For the following problems, determine all the roots of the given function. 13. f(x) = x x f(x) = 6x 4 5x 3 4x f(x) = x 2 11x f(x) = 4x x 5 + x f(x) = x 7 + 6x 4 16x 18. f(x) = x 1 2 8x

117 Finding Domain and Range For the following problems, determine the domain and range of the given function. 19. f(x) = x 2 8x f(x) = 4 7x x f(x) = 5 2 x 22. f(x) = x f(x) = x 24. f(x) = 6 5 x 25. f(x) = x f(x) = 8x2 12x+4 16x f(x) = x x 28. f(x) = x3 x 2 +x 1 35x 3 +2x 4 x f(x) = x2 +x x 3 9x 2 +2x 30. f(x) = 3 x4 4x 2 +10x f(x) = x f(x) = 36 9x f(x) = 4x 3 4x 2 + x Function Composition For the following problems, compute (fog)(x) and(gof)(x). 34. f(x) = 2x + 5 g(x) = 8 23x 35. f(x) = 2 x g(x) = 2x f(x) = 2x 2 + x 4 g(x) = 7x x f(x) = x 2x+3 g(x) = 5x + 8

118 Inverse Functions For each of the following functions, find the inverse of the function. Verify your inverse by computing (fog)(x) or (gof)(x), which will equal x if they are indeed inverses of eachother. 38. f(x) = 11x f(x) = 4 10x 40. f(x) = 7 + (2x + 1) f(x) = 2x+14 6x f(x) = 2x f(x) = f(x) = 6 18x 15x f(x) = 1 x 9 12x Trig Functions Without using a calculator, determine the exact value of each of the following. 46. tan ( 3π 4 ) 47. sin(7π 6 ) 48. sin( 3π 4 ) 49. cos (4π 3 ) 50. cot( 5π ) sin ( 5π ) 52. sec ( π ) 53. cos (11π ) cot ( 4π 3 ) 55. cos ( π 4 ) 56. csc (2π 3 ) 57. cos (5π 4 ) 58. tan ( 31π 6 15π 23π ) 59. cos ( ) 60. sec ( ) 61. cot (11π )

119 Solving Trig Equations Without using a calculator find the solution(s) to the following equations. If an interval is given then find only those solutions that are in the interval. If no interval is given then find all solutions to the equation cos(8x) = cos(8x) = 5; [ π, π ] sin ( 2x ) + 2 = 0; [0, 5π] sin ( x 4 ) = sin (x 4 ) = 3 ; [0,16π] = 8 cos(3x) ; [0, 5π 3 ] With the aid of a calculator find the solution(s) to the following equations. If an interval is given then find only those solutions that are in the interval. If no interval is given then find all solutions to the equation sin ( t 3 ) = cos(4x) + 8 = 10 cos (4x) = cos ( w 2 ) ; [ 20,5] tan(3w) + 3 = sin ( 3x ) 7 = 1 ; [0,15] sin ( x ) 9 = 7 sin (x ) + 17; [ 10, 20] 2 With the aid of a calculator find all of the solutions to the following equations. You may have to do crazy things, like factor, in order to get the solution. Use at least 4 decimal places tan ( x ) sin(2x) tan 3 (x ) = tan(4x) sec(2x 1) + sec(2x 1) = cos 2 (x) 4 cos(x) = sin(2x) = 3 sin 2 (2x)

120 Exponential Functions Sketch the graphs of each of the following functions. 78. f(x) = 7 3 x f(x) = 3 5 4x f(x) = 6e 2x f(x) = 7 + 9e 2 3t 5 Logarithmic Functions Without using a calculator, determine the exact value of each of the following. 82. log log log log ln log e Write each of the following in terms of simpler logarithms. 88. log 7 (10a 7 b 3 c 8 ) 89. log[z 2 (x 2 + 4) 3 ] 90. ln ( w2 t 3 4 t+x )

121 Combine each of the following into a single logarithm with a coefficient of one ln t 6 ln s + 5 ln w log(z + 1) 2 log x 4 log y 3 log z log 3(x + y) + 6 log 3 x 1 3 Use the change of base formula and a calculator to find the value of each of the following log log5 7 8 Exponential and Logarithm Equations Find all the solutions to the given equations. If there is no solution to the equation clearly explain why = e 10x e 2x+x2 7 = e 7x 12e 8x+5 = log(w) log(3w + 7) = xlog (6x + 1) 3x 2 log(6x + 1) = ln(3x + 1) ln(x) = z2 2 = 5

122 Compound Interest 103. We have $2,500 to invest for 80 months. How much money will we have if we put the money into an account that has an annual interest rate of 9% and interest is compounded: (a) quarterly (b) monthly (c) continuously 104. Starting with $60,000 and putting it into an account that earns an annual interest rate of 7.5%. How long will it take for the money to reach $100,000 if the interest is compounded: (a) quarterly (b) monthly (c) continuously Exponential Growth/Decay 105. A population of bacteria initially has 90,000 present and in 2 weeks there will be 200,000 bacteria present. (a) Determine the exponential growth equation for this population. Q = Q 0 e kt (b) How long will it take for the population to grow from its initial population of 90,000 to a population of 150,000? 106. We initially have 2kg of some radioactive element and in 7250 years there will be 1.5kg left. (a) Determine the exponential decay equation for this element. Q = Q 0 e kt (b) How long will it take for half the element to decay? (c) How long will it take until 250 grams of the element left?

123 Common Graphs Without using a graphing calculator or utility, sketch the graph of each of the following y = 2x f(x) = x f(x) = x f(x) = x f(x) = x f(x) = x 2 + 8x f(x) = (x + 5) f(x) = ln x 115. f(x) = 2 x 116. x 2 6x + y 2 + 8y + 24 = (x+4) (y+2)2 25 = f(x) = tan (x + π 3 ) 119. f(x) = sec(x) f(x) = 2 x 121. f(x) = 2 sin (2x π 6 ) 1

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125 2017 AP CALCULUS BC - SUMMER REVIEW PROBLEMS NAME: DUE MONDAY AUGU2T 21, 2017 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the limit by substitution. 1) lim x2 (x3 + 5x2-7x + 1) 1) A) Does not exist B) 29 C) 0 D) 15 2) lim x3 x2 + 12x ) A) ± 9 B) 81 C) 9 D) Does not exist Complete the table and state the 3) f(x) = x 2 + 4x x lim x0 f(x). Round table values to six decimal places when necessary. x f(x)???????? 3) A) x f(x) lim x0 = 4 B) none of these C) x f(x) lim x0 = 3 D) x f(x) lim x0 = 4 Determine the limit algebraically, if it exists. 4) lim x -4 x2-16 x + 4 A) 1 B) Does not exist C) -8 D) -4 4) 5) lim x2 x2 + 3x - 10 x - 2 A) 3 B) Does not exist C) 0 D) 7 5) 1

126 Determine the limit graphically, if it exists. 6) lim f(x) x0 6) A) 1 B) 0 C) Does not exist D) -1 7) lim f(x) 7) x2- A) 2.3 B) -1 C) 4 D) 1.3 Find the limit. 8) lim x 6x x - 7 8) A) 0 B) C) 3 8 D) 9) Let lim f(x) = -5 and x -6 lim g(x) = 3. Find x -6 lim [f(x) + g(x)]2. 9) x -6 A) 34 B) -8 C) 4 D) -2 Find the intervals on which the function is continuous. x ) y = x2-4x + 3 A) (-, 1), (1, 3), (3, ) B) (-, 1), (1, ) C) (-, -1), (-1, 3), (3, ) D) (-, -3), (-3, 1), (1, ) 10) 11) y = e1/x 11) A) (-, -1), (-1, ) B) (-, 1), (1, ) C) (-, ) D) (-, 0), (0, ) 2

127 Find the points of discontinuity. Identify each type of discontinuity. x ) y = x2-7x + 6 A) x = -1, x = -6, both infinite discontinuities B) x = 1, x = 6, both infinite discontinuities C) x = 1, infinite discontinuity D) x = 6, infinite discontinuity 12) 13) y = e1/x 13) A) x = -1, infinite discontinuity B) None C) x = 0, infinite discontinuity D) x = 1, infinite discontinuity Find all points where the function is discontinuous. 0, x< 0 14) f(x) = x2-4x, 0 x 4 4, x > 4 A) x = 4 B) x = 0 C) x = 0 and x = 4 D) Nowhere 14) Find the slope of the line tangent to the curve at the given value of x. 15) f(x) = -4 at x = 11 15) x A) B) 4 11 C) D) 11 4 Find the equation of the tangent line. 16) y = x2 - x at x = -4 16) A) y = -9x + 12 B) y = -9x + 16 C) y = -9x - 12 D) y = -9x - 16 Find the equation of the normal line. 17) y = 5x2 at (4, 80) 17) A) y = x B) y = x C) y = 1 40 x D) y = x Find the instantaneous rate of change of the position function y = f(t) in feet at the given time t in seconds. 18) f(t) = t + 2, t = 2 18) t A) -2 ft/sec B) ft/sec C) 1 2 ft/sec D) - 1 ft/sec Solve the problem. 19) Find the points where the graph of the function has horizontal tangents. f(x) = x3-15x A) (- 5, 30 5), (0,0), ( 5, -30 5) B) ( 5, -30 5) C) (- 5, 10 5), ( 5, -10 5) D) (-5, -50), ( 5, 50) 19) 3

128 20) Find f'(x) if f(x) = 12x ) A) f'(x) = 12 B) f'(x) = 12x C) f'(x) = -12 D) f'(x) = 5 21) Find d dx (2x 2-3). 21) A) 4x2-3 B) 4x C) 2x D) 4x - 3 The graph of a function is given. Choose the answer that represents the graph of its derivative. 22) 22) A) B) C) D) Solve the problem. 23) If y = x2-3, find an equation of the tangent line to the graph of y at x =4. 23) A) y = 8x - 19 B) y = 8x - 38 C) y = 8x - 35 D) y = 4x

129 24) Find an equation of the tangent line to the graph of y = x - x2 at the point (4, -12). 24) A) y = 9x - 16 B) y = 9x + 16 C) y = -7x + 16 D) y = 7x ) Find an equation of the tangent line to the graph of y = 10 x - x + 7 at the point (100, 7). 25) A) y = x + 57 B) y = 7 C) y = 1 2 x - 57 D) y = x ) Find the equation of the normal line to the curve y = 4x - 2x2 at the point (2, 0). 26) A) x + 12y - 2 = 0 B) x - 4y = 0 C) x + 12y = 0 D) x - 4y - 2 = 0 27) Find the equation of the normal line to the graph of y = 3x2 at x = 4. 27) A) x + 24y = 0 B) x - 24y = 0 C) x + 24y = 0 D) x + 24y = 0 Find the values where the function is not differentiable. 28) 28) A) x = -2, 0, 2 B) x = -3, 3 C) x = -3, 0, 3 D) x = -2, 2 If the function is not differentiable at the given value of x, tell whether the problem is a corner, cusp, vertical tangent, or a discontinuity. 29) y = -7 x + 8, at x = 0 29) A) cusp B) corner C) vertical tangent D) function is differentiable at x = 0 30) y = -2, at x = 1 30) x - 1 A) discontinuity B) corner C) cusp D) function is differentiable at x = 1 31) y = -6-3 x, at x = 0 31) A) vertical tangent B) discontinuity C) cusp D) function is differentiable at x = 0 5

130 32) y = 3 x + 4, at x = -4 32) A) corner B) vertical tangent C) cusp D) function is differentiable at x = -4 Find dy/dx. 33) y = 11-5x2 33) A) 11-5x B) 11-10x C) -10 D) -10x 34) y = 1 2 x x 3 34) A) 3x6 - x3 B) 1 2 x x 2 C) 3x5 - x2 D) 3x7 - x4 Find the horizontal tangents of the curve. 35) y = x4-32x ) A) At x = 0 B) At x = 0, 4, -4 C) At x = 0, 4 D) At x = 4, -4, Find dy/dx using the Product or Quotient Rules. 36) y = (2x - 3)(6x + 1) 36) A) 24x - 8 B) 24x - 16 C) 12x - 16 D) 24x ) y = (3x - 2)(5x3 - x2 + 1) 37) A) 60x3-13x2 + 39x + 3 B) 45x3 + 39x2-13x + 3 C) 60x3-39x2 + 4x + 3 D) 15x3 + 13x2-39x + 3 Find dy/dx. 38) y = x + 2 x ) A) -4 (x - 2)2 B) -2 (x - 2)2 C) 2 x - 2 D) -4 (x + 2)2 39) y = x2 5-7x 39) A) 5x (5-7x)2 B) -21x x (5-7x)2 C) 7x 3-14x2 + 10x (5-7x)2 D) -7x x (5-7x)2 Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 40) u(1) = 4, u (1) = -7, v(1) = 6, v (1) = ) d (uv) at x = 1 dx A) 34 B) 50 C) -50 D) -40 6

131 41) u(2) = 6, u (2) = 3, v(2) = -2, v (2) = -4. d (uv) at x = 2 dx A) -18 B) 26 C) 30 D) ) Find the slope of the line tangent to the curve at the given value of x. 42) y = 3x2 + 6x; x = 2 42) A) 21 B) 18 C) -12 D) 24 43) y = x3 + 6; x = 3 43) A) 27 B) 28 C) -27 D) 33 Find dy/dx. 44) y = 5x2 + 10x + 4x-3 44) A) 5x + 4x-4 B) 10x x-4 C) 10x - 12x-4 D) 10x x-4 45) y = x - 2 x ) A) 4 (x + 2) x - 4 B) - 2 x( x + 2) 2 C) 2 x( x + 2) 2 D) 2 x ) y = 46) A) B) - (x + 3) x - 9 x( x + 3) 2 C) x( x + 3) 2 D) x + 3 Find the fourth derivative of the function. 47) y = 5x6-7x4 + 3x2 47) A) 1200x2-84x B) 1800x2-168x C) 1800x2-168 D) 1200x ) y = x - 2 x 48) A) - 24 x4 B) 48 x4 C) 24 x5 D) - 48 x5 Solve the problem. 49) Find the points on the curve y = 5x2 + 6x + 3 where the tangent is parallel to the x-axis. 49) A) - 3 5, 6 3 B) (0, -3) C) (-16, 1907) D) 5 5, ) Find an equation of the tangent to the curve y = 2x2-2x + 1 that has slope 2. 50) A) y = 2x + 1 B) y = 2x - 1 C) y = 2x D) y = 2x + 2 7

132 51) Find an equation of the line perpendicular to the tangent to the curve y = x3 + 7x - 6 at the point (2, 3). A) y = 1 19 x B) y = -5x + 59 C) y = 3x2 + 7 D) y = x ) 52) Find the x- and y-intercepts of the line that is tangent to the curve y = x3 at the point (-2, -8). 52) A) x-intercept = 8 3, y-intercept = -32 B) x-intercept = - 4, y-intercept = 16 3 C) x-intercept = - 4, y-intercept = -16 D) x-intercept = 0, y-intercept = 0 53) The population P, in thousands, of a small city is given by P(t) = 900t, where t is the time, in 2t2 + 1 months. Find the growth rate, dp dt. A) dp dt = 900(2t 2-1) (2t2 + 1)2 C) dp dt = 900(1-2t 2) (2t2 + 1)2 B) dp dt = 900(1 + 6t 2) (2t2 + 1)2 D) dp dt = 900(1-2t 2) 2t ) 54) A charged particle of mass m and charge q moving in an electric field E has an acceleration a given by 54) a = qe m, where q and E are constants. Find d 2a dm2. A) d 2a dm2 = qe m3 B) d 2a dm2 = - qe m2 C) d 2a dm2 = qe 2m D) d 2a dm2 = 2qE m3 55) The function V = s3 describes the volume of a cube, V, in cubic inches, whose length, width, and height each measure s inches. Find the (instantaneous) rate of change of the volume with respect to s when s = 3 inches. A) 27 in3/in. B) in3/in. C) 9 in3/in. D) 81 in3/in. 55) 56) Suppose that the dollar cost of producing x radios is c(x) = x - 0.2x2. Find the marginal cost when 40 radios are produced. A) $4 B) $880 C) -$880 D) $36 56) 57) The dollar profit from the expenditure of x thousand dollars on advertising is given by P(x) = x - 2x2. Find the marginal profit when the expenditure is x = 9. A) 800 thousand dollars B) -11 thousand dollars C) 189 thousand dollars D) 225 thousand dollars 57) 8

133 The figure shows the velocity v of a body moving along a coordinate line as a function of time t. Use the figure to answer the question. 58) v (ft/sec) 58) t (sec) When is the body's acceleration equal to zero? A) 2 < t < 3, 5 < t < 6 B) t = 2, t = 3, t = 5, t = 6 C) t = 0, t = 4, t = 7 D) 0 < t < 2, 6 < t < 7 Find dy/dx. 59) y = 5 tan2x 59) A) 10 tan2x sec x B) 10 tan3x C) 10 tan1x sec2x D) 10 tan1x 60) s = t7 tan t 60) A) - t7 sec2t + 7t6tan t B) t7 sec2t + 7t6tan t C) 7t6 sec2t D) t7 sec t tan t + 7t6tan t 61) y = 3 sin x 61) A) - 3 csc x cot x B) 3 csc x cot x C) 3 csc x cot x - sec2 x D) 3 cos x 62) y = x7 cos x - 13x sin x - 13 cos x 62) A) - 7x6 sin x - 13 cos x + 13 sin x B) - x7 sin x + 7x6 cos x - 13x cos x C) x7 sin x - 7x6 cos x + 13x cos x D) - x7 sin x + 7x6 cos x - 13x cos x - 26 sin x 9

134 Solve the problem. 63) Find the equations for the lines that are tangent and normal to y = cos x at x = 2. 63) A) y = - x - 2 ; normal: y = x + 2 B) y = 1; normal: y = 2 C) tangent: y = - x + 2 ; D) y = x + 2 ; normal: y = x - 2 normal: y = - x ) Find the equations for the lines that are tangent and normal to y = 2 - sin x at x =. 64) A) tangent: y = x - 2; B) tangent: y = x - + 2; normal: y = - x + 2 normal: y = - x C) tangent: y = - x + 2; D) tangent: y = - x + - 2; normal: y = x - 2 normal: y = x Find the indicated derivative. 65) Find y if y = 8 sin x. 65) A) y = 8 sin x B) y = - 8 sin x C) y = 8 cos x D) y = 64 sin x 66) Find y if y = -5 cos x. 66) A) y = -5 cos x B) y = 5 sin x C) y = 5 cos x D) y = -5 sin x Use the given substitution and the Chain Rule to find dy/dx. 67) y = 4, u = 2x ) u2 A) - 16 (2x - 3)3 B) - 8 2x - 3 C) x - 3 D) 16x 2x ) y = tan u, u = -7x ) A) -7 sec (-7x + 4) tan (-7x + 4) B) -7 sec2(-7x + 4) C) - sec2(-7x + 4) D) sec2(-7x + 4) Find dy/dx by implicit differentiation. If applicable, express the result in terms of x and y. 69) 8y2-9x2-15 = 0 69) A) 9x 2 18x + 15 B) C) 9x D) 9x 16y 16y 8y 8 70) 5y + -6xy - 4 = 0 70) 6(x + y) 6y 6y 6y(x + 1) A) B) C) D) x xy 5 10

135 Solve the problem. 71) Given x2 + y2 = 25, find dy/dx and the slope of the curve at the point (-4, -3). 71) A) dy dx = x y ; 4 B) dy 3 dx = - y x ; - 3 C) dy 4 dx = y x ; 3 D) dy 4 dx = - x y ; ) Given 2x2y - cos y = 3, find the slope of the curve at (1, ). 72) A) 0 B) -2 C) D) ) Given x2 + y2-2x + 4y = 8, find the line that is tangent to the curve at (4, 0). 73) A) y = 0 B) y = 1 2 (x - 4) C) y = - 3 (x - 4) D) x = 4 2 Find dy/dx. 74) y = x8/5 74) A) dy dx = 8 5 x -3/5 B) dy dx = x 3/5 C) dy dx = 8 5 x 7/5 D) dy dx = 8 5 x 3/5 75) y = 6 x ) A) dy dx = 3 B) dy x + 14 dx = - 3 (x + 14)3/2 C) dy dx = - 3 x + 14 D) dy dx = 1 2 x + 14 Solve the problem. 76) The position of a body moving along a coordinate line at time t is s = (9 + 4t)3/2, with s in meters and t in seconds. Find the body's acceleration when t = 4 seconds. A) 60 m/sec2 B) 15 4 m/sec 2 3 C) 20 m/sec 2 D) 12 5 m/sec 2 76) Find the derivative of y with respect to the appropriate variable. 77) y = 3 sin-1 (5x4) 77) 60x3 60x3 60x3 3 A) B) C) D) 1-25x8 1-25x8 1-25x4 1-25x 8 78) y = 3.5 cos-1 3t 78) A) - 1-9t 2 B) - 1-9t 2 C) D) 1 + 9t2 1-9t 2 79) y = tan-1 3x 79) A) B) C) D) 1-3x 2(1 + 3x) 3x 6 3x(1 + 3x) 1 + 3x 11

136 80) y = csc-1 4x 80) 1 1 A) - 4 x 16x 2 B) - 1 x 16x C) - x 16x 2 D) x 16x ) y = sec-1 x 5 81) A) 1 x 25x 2-1 B) 5 C) x 25 - x 2 25 x x 2-25 D) 5 x x ) y = cot-1 t ) A) - B) C) D) - t - 4 2(t - 4) t - 5 2(t - 5) t - 5 2(t - 4) t - 5 Find the value of df-1/dx at x = f(a). 83) f(x) = 2x + 6, a = 4 83) A) 1 2 B) 1 6 C) 6 D) 2 84) f(x) = 5x2, x 0, a = 2 84) A) B) C) 20 D) Find dy/dx. 85) f(x) = e4x 85) A) 4e4x B) 4ex C) e4x D) 1 4 e 4x 86) y = 5x 86) A) x ln 5 B) 5x ln x C) 5x D) 5x ln 5 87) y = ln 3x 87) A) 1 1 B) C) - 1 D) - 1 x 3x 3x x 88) y = ln 3x2 88) A) 2 2x B) C) 6 1 D) x x2 + 3 x 2x ) y = ln x 8 89) A) 1 x - ln 8 B) 1 x C) 8 x D) 1 8x 12

137 90) y = log (3x - 1) 90) A) B) C) D) 3x - 1 (3x - 1) ln 10 ln 10 (3x - 1) ln 10 3 ln 10 Use logarithmic differentiation to find dy/dx. 91) y = (cos x)x 91) A) (cos x)x (ln cos x - x tan x) B) (cos x)x (ln cos x + x cot x) C) ln x(cos x)x - 1 D) ln cos x - x tan x 92) y = 26x 92) A) 12 (ln 2) 26x B) 6 (ln 2) 26x C) 2 (ln 6) 26x D) 12 (ln 6) 26x Find the location of the indicated absolute extremum for the function. 93) Maximum 93) A) x = 4 B) x = -4 C) x = 1 D) No maximum 94) Minimum 94) A) x = -2 B) x = 2 C) x = 1 D) x = -1 13

138 Find the extreme values of the function on the interval and where they occur. Identify any critical points that are not stationary points. 95) g(x) = -x2 + 12x - 32, 4 x 8 95) A) Local maximum at 6, 4 ; minimum value is -32 at x = 0 B) Local maximum at 6, 68 ; minimum value is -32 at x = 0 C) Local maximum at 6, 4 ; minimum value is 0 at x = 8 and at x = 4 D) Local maximum at 7, 4 ; minimum value is 0 at x = 8 and at x = 4 96) f(x) = e-x, -8 x 3 96) A) Maximum value is e8 at x = -8; minimum value is 1 e3 at x = 3 B) Maximum value is e3 at x = 3; minimum value is 1 e8 at x = -8 C) Maximum value is e8 at x = -8; local minimum at (0, 1) D) Local maximum at (0, 1), minimum value is 1 e3 at x = 3 Find the extreme values of the function and where they occur. 97) f(x) = -3x4 + 20x3-36x ) A) The minimum is 5 at x = 0. B) The maximum is 0 at x = 0. C) The maximum is 5 at x = 0. D) There are none. 98) y = x3-3x2 + 7x ) A) The maximum is 6 at x = 2. B) The maximum is 6 at x = 1. C) The minimum is 6 at x = -1. D) There are none. Find the local extrema. 99) g(x) = -4x2-40x ) A) Local maximum at (5, -2) B) Local maximum at (-2, 5) C) Local maximum at (-5, 2) D) Local maximum at (2, -5) Find the intervals on which the function is increasing and the intervals on which the function is decreasing. 100) f(x) = 48x - x3 100) A) Increasing on (-, -4), decreasing on (-4, 4) B) Increasing on (-4, 4), decreasing on (-, -4) and (4, ) C) Increasing on (-16, 16), decreasing on (-, -16) and (16, ) D) Increasing on (-, 4), decreasing on (4, ) 101) y = e-2x 101) A) Decreasing on (, ) B) Increasing on (, 0), decreasing on (0, ) C) Increasing on (, ) D) Decreasing on (, 0), increasing on (0, ) 102) f(x) = 5 - x ) A) Decreasing on (-5, ) B) Increasing on (-5, ) C) Decreasing on (5, ) D) Increasing on (-5, 0), decreasing on (0, ) 14

139 Find all possible functions with the given derivative. 103) f'(x) = 24x2-4x ) A) 8x3-2x2-4x + C B) 8x3-2x2-3x + C C) 8x3-1x2-4x + C D) 9x3-2x2-4x + C 104) f'(x) = 6 104) A) 6 - x + C B) 6x + C C) x C D) x6 + C 105) f'(x) = 168 x A) 168 ln x + C B) 84x-2 + C C) ln x C D) 168x + C 105) 106) f'(x) = 3 cos 3x 106) A) sin 3x + C B) cos x + C C) cos 3x + C D) sin x + C Find the function with the given derivative whose graph passes through the point P. 107) f'(x) = x - 8, P(1, 7) 107) A) f(x) = x 2 2 C) f(x) = x 2 2-8x x B) f(x) = x2-8x + 14 D) f(x) = x2-8x 108) f'(x) = 1, x > -7, P(-6, 7) 108) x + 7 A) ln(x + 7) + 7 B) - 1 (x + 7)2 + 8 C) ln(x + 7) + 6 D) - 1 (x + 7)2 + 6 Solve the problem analytically. 109) Of all numbers whose difference is 4, find the two that have the minimum product. 109) A) 2 and -2 B) 8 and 4 C) 1 and 5 D) 0 and 4 Solve the problem. 110) A company wishes to manufacture a box with a volume of 40 cubic feet that is open on top and is twice as long as it is wide. Find the width of the box that can be produced using the minimum amount of material. Round to the nearest tenth, if necessary. A) 6.4 ft B) 3.6 ft C) 3.2 ft D) 7.2 ft 110) Find the linearization L(x) of f(x) at x = a. 111) f(x) = 8x + 9, a = 0 111) A) L(x) = 8 3 x + 3 B) L(x) = 4 3 x - 3 C) L(x) = 8 3 x - 3 D) L(x) = 4 3 x

140 Use the linearization (1 + x)k 1 + kx to approximate the value. Give your answer in the form indicated. 112) Give your answer as a decimal. A) B) 1.01 C) D) ) Solve the problem. 113) The radius of a right circular cylinder is increasing at the rate of 7 in./s, while the height is decreasing at the rate of 3 in./s. At what rate is the volume of the cylinder changing when the radius is 17 in. and the height is 5 in.? A) -19 in.3/s B) -272 in.3/s C) -272 in.3/s D) 323 in.3/s 113) 114) A spherical balloon is inflated with helium at a rate of 120 ft3/min. How fast is the balloon's radius increasing when the radius is 8 ft? A) 1.88 ft/min B) 1.41 ft/min C) 0.06 ft/min D) 0.47 ft/min 114) Use a finite approximation to estimate the area of the region enclosed between the graph of f and the x-axis for a x b. 115) f(x) = x2, a = 1, b = 5 115) Use LRAM with four rectangles of equal width. A) 54 B) 30 C) 41 D) ) f(x) = x2, a = 3, b = 7 Use RRAM with four rectangles of equal width. A) 86 B) 126 C) 105 D) ) 117) f(x) = x2, a = 3, b = 7 Use MRAM with four rectangles of equal width. A) 105 B) 86 C) 117 D) ) Evaluate the integral ) 4 dx 118) -1 A) 20 B) 7 C) 14 D) 28 Graph the integrand and use areas to evaluate the integral ) (-2x + 8) dx 119) -8 A) 48 B) 192 C) 144 D) 288 Solve the problem ) Suppose that f(x) dx = -4. Find f(x) dx and 5 3 f(x) dx. 120) A) 0; -4 B) -4; 4 C) 0; 4 D) 5; -4 16

141 USE NINT to find the average value of the function on the interval. At what point in the interval does the function assume its average value? 121) y = -6x2-1, [0, ] 121) A) -73, at x = B) 73, at x = C) -25, at x = 2 D) 25, at x = 2 Find the average value of the function without integrating, by appealing to the geometry region between the graph and the x-axis. 122) f(x) = x + 6, -6 x ) -x + 4, -1 < x 4 A) 2 B) 3 C) 5 D) 5 2 Interpret the integrand as the rate of change of a quantity and evaluate the integral using the antiderivative of the quantity. 123) 9 sin x dx 123) 0 A) 9 B) 2 C) 18 D) ) 3x-4 dx 124) -2 A) 7 24 B) 7 8 C) 21 D) 1 8 Find the average value over the given interval. 125) y = 6x + 6; [3, 7] 125) A) 144 B) 6 C) 66 D) 36 17

142 Construct a function of the form y = a x f(t) dt + C that satisfies the given conditions. 126) dy dx = 1, and y = 4 when x = 0 126) (4 + x) 2 x A) y = 0 x C) y = 4 x dt + B) y = 4 + t 4 4 x 1 (4 + t) 2 dt + 4 D) y = t dt dt + (4 + t) ) dy = cot x, and y = 8 when x = ) dx x A) y = 8-3 C) y = x cot t dt + -3 B) y = - cot t dt + 8 D) y = x csc 2 t dt x cot t dt Evaluate the integral ) 2x dx 128) 2 A) -3 2 ln 2 B) 5 2 ln 2 C) -9 2 ln 2 D) -7 2 ln 2 /2 129) 20 sin x dx 129) 0 A) 20 B) 0 C) -20 D) ) x-1/2 dx 130) 1 A) 0 B) 2 C) 3 D) 1 18

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