Chapter 2: Rocket Launch

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1 Chapter 2: Rocket Launch Lesson Domain:!" # x # " Range: 2! y! " y-intercept! y = 2 no x-intercepts 2-2. a. Time Hours sitting Amount Earned 8PM 1 $4 9PM 2 $4*2hrs = $8 10PM 3 $4*3hrs = $12 11:30PM 4.5 $4*4.5hrs = $18 12:00 5 $4*5 = $20 12:30AM 5.5 $4*5+$6*½ = $23 1AM 6 $4*5+$6*1 = $26 2AM 7 $4*5+$6*2 = $32 b. f (x) = 4t 0! t! 5 6(t " 5) + 20 t > a. { For x 1 For x > 1 x y = x x y = 2x y = (!5) = = 27 1 y = 2!1+ 7 = 9 4 y = (!4) = = 18 2 y = 2! = 11 3 y = (!3) = = 11 3 y = 2! = 13 2 y = (!2) = = 6 4 y = 2! = 15 1 y = (!1) = = 3 5 y = 2! = 17 0 y = (0) = = 2 6 y = 2! = 19 1 y = (1) = = 3 7 y = 2! = 21 b. c. D = (!", "); R = [2, ") ; CPM Educational Program 2012 Chapter 2: Page 1 Pre-Calculus with Trigonometry

2 Closed at (1, 3), open at (1, 9). CPM Educational Program 2012 Chapter 2: Page 2 Pre-Calculus with Trigonometry

3 2-5. x = a. D = (!", "); R = (!", 5) b. No, jump at x = 2. Review and Preview a. $8.00 per hour * 5 hours = $40.00 Rate of pay = $8.00 per hour b. Sample graph at right. c. 5 d. Each rectangle represents eight dollars of Kristof s earnings. ($8/hour)! (hours) = $ f (x + 2) = 3(x + 2) 2 + 5(x + 2)! 2 = 3(x 2 + 4x + 4) + 5x + 10! 2 = 3x x x + 10! 2 = 3x x + 20 f (x + h) = 3(x + h) 2 + 5(x + h)! 2 = 3(x 2 + 2xh + h 2 ) + 5x + 5h! 2 = 3x 2 + 6xh + 3h 2 + 5x + 5h! 2 = 3x 2 + 6xh + 5x + 3h 2 + 5h! 2 = 3x 2 + x(6h + 5) + (3h 2 + 5h! 2) a. (x + 2)(x + 2)! 3(x + 2) = (x + 2)(x + 2! 3) = (x + 2)(x! 1) b. (x + 2)(x + 2)(x + 2)! 4(x + 2) = (x + 2)(x 2 + 4x + 4! 4) = (x + 2)(x 2 + 4x) = x(x + 2)(x + 4) c. (x + 2)(x + 2) + 6(x + 2) = (x + 2)(2(x + 2) + 6) = (x + 2)(2x ) = (x + 2)(2x + 10) = 2(x + 2)(x + 5) CPM Educational Program 2012 Chapter 2: Page 3 Pre-Calculus with Trigonometry

4 CPM Educational Program 2012 Chapter 2: Page 4 Pre-Calculus with Trigonometry

5 2-11. a. 5x 2! 15x = 0 5x(x! 3) = 0 Either 5x = 0 or x! 3 = 0 x = 0 or x = a. (3 2 )! 3 2 = 3!3 = = 1 27 c ( ) 2 3 = 5 " 4! ( ) 3 # $ 2 3 = ( 5 4 ) 2 = b. 2(x! 6) 2 = 18 (x! 6) 2 = 9 x! 6 = ±3 Either x! 6 = 3 or x! 6 =!3 x = 9 or x = 3 b. (3 3 ) 2 3 = 3 2 = 9 d. ( 3!2 )!1 2 = 3 1 = a. d = 5! (!2) ( ) 2 +!2! 5 ( ) 2 = (!7) 2 = = 98 = 7 2 b. slope =!2! 5 5! (!2) =!7 7 =!1 point-slope!form:!!y! 5 =!(x + 2) y + 2 =!(x! 5) slope! intercept form:!!y! 5 =!x! 2 y =!x! y =!x ! a. 6 b.! 3" 4 Lesson a. Shift to the left two units. b. q(x) = (x + 2) 2 + (x + 2)! g(x) should be shifted to the right one unit and up three units. = x 2 + 4x x + 2! 2 = x 2 + 5x + 4 CPM Educational Program 2012 Chapter 2: Page 5 Pre-Calculus with Trigonometry

6 2-17. a and b. c. At x = 3 d. # h(x) = (x! 1)2!!!for!x < 3 $ %& (x + 1)!!!!!for!x " a. Shifts left 2 and down 3. c. b. # k(x) = (x + 2)2! 3!!for x < 0 $ %& x! 1!!!!!!!!!!!!!for x " Any multiple of 5 will work; 5n where n is any integer. a. D = (!", "); R = [2, 4] b. (0, 2) c. No asymptotes translational symmetry is present. d. The function is made up of line segments but it does not belong to any previously studied family a. Yes, because the difference in the x-values, 17 2 = 15 which is a multiple of the period, 5. b. Yes, shifting the function horizontally by its period will result in a graph that is identical to the original. c.!2, 1 2, 3, 5 1 2, 8,10 1 2,13, a. p = 7 b. The values visible on the graph are 2.5,.5, 4.5, and 7.5. We want students to find two more solutions using the idea that n and n in which n is an integer are also solutions. c. g(53) = g(4 + 49) = g(4 + 7p) = g(4) = 2 CPM Educational Program 2012 Chapter 2: Page 6 Pre-Calculus with Trigonometry

7 Review and Preview See graph at right. a. f (x) = x 3! 3x g(x) = y = (x + 2) 3! 3(x + 2) b. If f (x) = x 3! 3x f (x + 2) = (x + 2) 3! 3(x + 2) g(x) = f (x + 2). f (x) = 2 x! a. f (x) = x f (x) = x! 3 (right 3 units) f (x) = 2 x! 3 (change slope to 2) f (x) = 2 x! (up 1 unit) b. f (x) = 2 x! f (x + 5) = 2 x! f (x + 5)! 3 = 2 x ! 3 g(x) = 2 x + 2! 2 c y = x 5! 2x y = (x! 2) 5! 2(x! 2) (right 2 units) y = (x! 2) 5! 2(x! 2) + 3 (3 units up) a. The graphs are both cubic functions, but are in different places on the axes. b. The second graph is the first graph shifted to the left 2 units. c. f (x) = x f (x + 2) = (x + 2) 3 (shifted left 2 units) CPM Educational Program 2012 Chapter 2: Page 7 Pre-Calculus with Trigonometry

8 a. (2x + 3)(2x + 3) = 4x 2 + 6x + 6x + 9 = 4x x + 9 b. (x! a)(x! b) = x 2! ax! bx + ab = x 2! x(a + b) + ab CPM Educational Program 2012 Chapter 2: Page 8 Pre-Calculus with Trigonometry

9 2-27. a. c 2 = a 2 + b 2! 2 " a " b " cos(x) 6 2 = ! 2 " 8 "10 " cos(x) 36 = 164! 160 " cos(x) 128 = 160 " cos(x) 0.8 = cos(x) cos!1 (0.8) = x 36.9! = x a. 2x x! 16 = 2(x 2 + 7x! 8) = 2(x + 8)(x! 1) b. a sin A = b sin B x sin 60 = 28 sin 70 x = x = x = 25.8! b. 2x 3! 128x = 2x(x 2! 64) = 2x(x + 8)(x! 8) a. a 2 + a 2 = a 2 = 100 a 2 = 50 a = , 9, 16,, = 5 2cm b. cos(60! ) = x = x 8 4cm = x c. tan(30! ) = 3 x 1 3 = 3 x x = 3 3cm Lesson a = = 120 b. (4! ) + (4! ) + (4! ) = = 206 c ( ) ( 1 2) ( 1 2) ( 1 2) ( 1 2) 5 = ( ) + 16 ( 1 4 ) + 16 ( 1 8 ) + 16 ( 1 16 ) + 16 ( 1 32 ) = = d. (3! 2 + 1) + (3! 3 + 1) + (3! 4 + 1) + (3! 5 + 1) = = 46 e. 1 1(1+1) + 1 2(2+1) + 1 3(3+1) + 1 4(4+1) = = = = = 4 5 f. Argument is 4n The index is n CPM Educational Program 2012 Chapter 2: Page 9 Pre-Calculus with Trigonometry

10 Did not use integer values for p. Need to have p = 5, 6, 7. CPM Educational Program 2012 Chapter 2: Page 10 Pre-Calculus with Trigonometry

11 2-34. two coordinates = (1, 2.2) and (5, 3) slope = 3!2.2 5!1 = = 8 10 " 1 4 = 0.2 point-slope y! 2.2 = 0.2(x! 1) or y! 3 = 0.2(x! 5) slope-intercept y = 0.2x! 1+ 3 y = 0.2x = (1! ) + (2! ) + (3! ) + (4! ) + (5! ) = 0.2k " k= a = (1! ) + (2! ) + (3! ) + (4! ) = 0.4k b. Start the index at 1 and end it at a. The difference between each term is 0.4 and the first term is 3.6. b. 0.4! = = 23.6, so k = ! c. 0.4k k=0 d ( ) + (2! ) + (3! ) + (4! ) + + (100! ) = 100 " k=0 0.2k " k=0 Review and Preview y =! 2(x + 1) 3! 5 y =!2((x + 4) + 1) 3! 5 =!2(x + 5) 3! 5 (left 4 units) y =!2(x + 5) 3! 5! 7 =! 2(x + 5) 3! 12 (7 units down) x + 2y = 7 6x! 2y = 10 2(3x! y = 5) " 4x + 2y = 7 10x = 17 x = = 1.7 3(1.7)! y = 5 5.1! y = 5!y =!0.1 y = 0.1 CPM Educational Program 2012 Chapter 2: Page 11 Pre-Calculus with Trigonometry

12 2-40. a. x 4! 81y 4 = (x 2 + 9y 2 )(x 2! 9y 2 ) = (x 2 + 9y 2 )(x + 3y)(x! 3y) b. 8x 3 + 2x 7 = 2x 3 (4 + x 4 ) f (x) = 5x! 1 " y = 5x! 1 x = 5y! 1 x + 1 = 5y x+1 5 = y " f!1 (x) = x a = = 30 b. (6(2)! 2 2 ) + (6(3)! 3 2 ) + (6(4)! 4 2 ) + (6(5)! 5 2 ) + (6(6)! 6 2 ) = = 30 c = = a ! = k 2 = 5 2 c. c. 7 3 = 7 3 k=1 ( ) 1/3 = 5 2/3 4 b. ( 2 ) 5 = ( 2 1/4 ) 5 = 2 5/4 ( ) 1/2 = 7 3/2 d. ( 7 ) 3 = ( 7 1/2 ) 3 = 7 3/2 e. 7! 7 = 7 1! 7 1/2 = 7 3/2 f. 9! 3 = 3 2! 3 1/3 = 3 7/ a. x 2 + x 2 = 5 2 2x 2 = 25 x 2 = 25 2 x = 25 2 = 5 = sin(30! ) = x = x 8 4 = x cos(30! ) = y = y = 2y 4 3 = y CPM Educational Program 2012 Chapter 2: Page 12 Pre-Calculus with Trigonometry

13 Lesson ! ( 0.01j + 1.7) = j=2!!!!!!!!!!!!!!!!!!!!!!!!!!!! = a b. Index Expression Cumulative Sum (k) (k 2 ) = = =55 c. k = 1 and k = 5 d. At the start the sum is zero and at the end the sum is a. Initializes the sum to 0. d. The loop would go on forever because k would always be In step 2, change 1 to 7; in step 5, change 5 to In step 3, change k 2 to 10 k b. Program keeps looping through these commands. c. Since we are using Y1, we need the independent variable X. d. Change the test value of k from 5 to she was changing the 1! x to!4 " x and If x! 5 to If x! They need to change the line 1 X to B X and the line if X 5 to X E Change Y1 to 10 x. CPM Educational Program 2012 Chapter 2: Page 13 Pre-Calculus with Trigonometry

14 : Sum is same as in problem 2-46, but should take less time to calculate. CPM Educational Program 2012 Chapter 2: Page 14 Pre-Calculus with Trigonometry

15 Review and Preview = i! 2 (i + 1) 2 i= " (k! 2) 2 = (2! 2) 2 + (3! 2) 2 + (4! 2) 2 + (5! 2) 2 + (6! 2) 2 k=2!!!!!!!!!!!!!!!!!!!= = = ! 1 = 3 (length of interval)!!!!!!!! = 0.6 (length of each piece) 5 pieces x 0 = 1 x 1 = 1+.6 = 1.6!!!!!!!!!!!!!!x 2 = = 2.2 x 3 = = 2.8!!!!!!!!x 4 = = a. These summations are the same, just written in different ways. 7 j 2 j=3! = ;!!! k 2 = b. These summations are the same, just written in different ways j 2 j=3 8! = k=3! ( j " 1) 2 = (4 " 1) 2 + (5 " 1) 2 + (6 " 1) 2 + (7 " 1) 2 + (8 " 1) 2 = j=4 Change Y1 to (k + 1) 2, the starting index to B = 2 (or 2 X), and the end index to E = 7 (or X 7). a x x!3 5 = 5(x+2) 5(4) + 4(x!3) 4(5) = (5x+10)+(4 x!12) = 9x! b. 4 2!x + x 5 = 5(4) 5(2!x) + (2!x)(x) (2!x)(5) = 20+(2x!x2 ) =!x2 +2x+20 10!5x 10!5x c. 6 x+2! 4 x!2 = 6(x!2) (x+2)(x!2)! 4(x+2) (x!2)(x+2) = (6x!12)!(4x+8) x 2 = 2x!20!4 x 2!4 CPM Educational Program 2012 Chapter 2: Page 15 Pre-Calculus with Trigonometry

16 2-62. a. 2x!5 + x = 4(2x!5) + 3(x+1) = (8x!20)+(3x+3) = 11x!17 4(6) 3(8) b. x x x = +2x!8 x+4 (x+4)(x!2) + 2 x+4 CD = (x + 4)(x! 2) x (x+4)(x!2) + 2(x!2) (x+4)(x!2) = x+(2x!4) (x+4)(x!2) = 3x!4 (x+4)(x!2) Lesson a. g(x) b. It looks like steps. c = a. and b. See graph at right. d. Answer will likely be between 7.75 and Answers should range from 4.4 to Students should add the results of problems 2 and 3. CPM Educational Program 2012 Chapter 2: Page 16 Pre-Calculus with Trigonometry

17 2-67. a. See graph at right. hr = miles b. miles hr c. 25(0.5) + 25(0.25) + 1! 50! (0.25) + 75(1.25) 2 = = miles d. The distance Erin traveled from 30 minutes into her trip until 15 minutes later a. slope = 75! !0.5 = = 200 y! 25 = 200(x! 0.5) y! 25 = 200x! 100 y = 200x! 75 # 25 for 0 < x! 0.5 % b. E(x) = $ 200x " 75 for 0.5 < x! 0.75 % & 75 for 0.75 < x! 2 It is linear because Erin has a constant rate of change between 30 minutes and 45 minutes. Review and Preview y = = slope = 3!1 4!0 = 2 4 = 1 2 (y! 1) = 1 (x! 0) 2 y! 1 = 1 2 x y = 1 2 x + 1 The equation needs to be entered and the starting and ending values need to be adjusted. SUM = a. 4( ) =! 4k = 60 b =! 2 k = 63 k=1 5 k=0 CPM Educational Program 2012 Chapter 2: Page 17 Pre-Calculus with Trigonometry

18 100 c. 2( ) =! 2k = 10, a. See graph at right. b. The domain is time elapsed, and Lew walks for one hour. The range is 0 to 4 miles. c. The units are in miles since mi ihr = miles. hr d. Length of rectangle (1 hr 0 hrs) = 1 hr Height of rectangle (4 mph 0 mph) = 4 mph Area of rectangle = 1hr * 4mph = 4 miles a. (2x! 1)(2x! 1 + 3) = (2x! 1)(2x + 2) = 2(2x! 1)(x + 1) k=1 b. (2x! 1) 2 (2x! 1! 4) = (2x! 1) 2 (2x! 5) c. (2x! 1) 3 (2x! 1) 2 + 4x ( ) = (2x! 1) 3 (4x 2! 4x x) = (2x! 1) 3 (4x 2 + 1) x(x 2! 3x! 10) = 0 x(x + 2)(x! 5) = 0 Either x = 0, x + 2 = 0, or x! 5 = 0 x = 0,!2, a. b. " g(x) = x2 + 2 for x < 2 # $% for x! 2 " k(x) = (x & 3)2 for x < 5 # $% 4 for x! a. x+2 x!1 + x x+1 = (x+2)(x+1) (x!1)(x+1) + b. c a!b + c. x+2 3x c a+b = c(a+b) (a!b)(a+b) +! x+1 6x 2 = (x+2)(2x) (3x)(2x) x(x!1) (x+1)(x!1) = (x2 +1x+2x+2)+(x 2!x) x 2 +x!x!1 c(a!b) (a+b)(a!b) = (ca+cb)+(ca!cb) a 2 +ab!ab!b 2 = 2ca a 2!b 2! x+1 6x 2 = (2x2 +4x)!(x+1) 6x 2 = 2x2 +3x!1 6x 2 = 2x2 +2x+2 x 2!1 CPM Educational Program 2012 Chapter 2: Page 18 Pre-Calculus with Trigonometry

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20 Lesson a. (1, 1), (1.5, 2.25), (2, 4), (2.5, 6.25) b. The height. c. x 0 = 1 x 1 = 1.5 x 2 = 2 x 3 = 2.5 x 4 = 3 Width of each rectangle is 0.5. d. The height of the rectangle is the function value at the left endpoint of each interval. e. 0.5!1+ 0.5! ! ! 6.25 = = 6.75 This area is smaller than the true area a. The height. b. They start from the right endpoint of the interval. c. 0.5! ! ! ! 9 = = This area is larger then the true area Some possibilities are averaging the two values or using more rectangles a b. height rectangle 1 = 1 2 = 1 height rectangle 2 = = height rectangle 3 =1.5 2 = 2.25 height rectangle 4 = = height rectangle 5 =2 2 = 4 height rectangle 6 = = height rectangle 7 =2.5 2 = 6.25 height rectangle 8 = = c. They are the y-values of the left endpoints. d. The width of each rectangle. e. The 3 rd and the 6 th x k = k CPM Educational Program 2012 Chapter 2: Page 20 Pre-Calculus with Trigonometry

21 ! ( ) 2 a k k=1 b. Right-hand endpoints: x 1 = 1, x 8 = 8 8! k= ( 0.25k + 1) 2 Left-hand endpoints: x 0 = 0, x 7 = 7 7! k= ( 0.25k + 1) a. See graph at right. b. height rectangle 1 = 0.25 height rectangle 2 = 1 height rectangle 3 = 2.25 height rectangle 4 = 4 c. 0.5( ) d. 0.5( ) = 0.5(7.5) = 3.75 Review and Preview = = 19! k 2 k=4 Other answers are acceptable as well h 2 = h 2 = 144 h 2 = 108 h = 108 = 3! 36 = 6 3 A = 1 2!b! h = 1!12! 6 3 = cm CPM Educational Program 2012 Chapter 2: Page 21 Pre-Calculus with Trigonometry

22 2-87. a. 5! 2 = 3 (length of interval) 3 = 0.75 (length of each piece) 4 peices x 0 = 2 x 1 = = 2.75 x 2 = = 3.5 x 3 = = 4.25 x 4 = = 5 b. f (x) = x 2 f (x 2 ) = f (3.5) = (3.5) 2 = f (x 4 ) = f (5) = 5 2 = y =!x x 2! 2(!x 2 + 6) =!3 x 2 + 2x 2! 12 =!3 3x 2 = 9 " x 2 = 3 " x = ± 3 y =!(± 3) " y =!3 + 6 = a. x + 2! 0 " x! #2 b. 3! x " 0 #!x "!3 # x $ 3 c. x 2! 1 " 0 # (x + 1)(x! 1) " 0 # x + 1 " 0, x! 1 " 0 # x " 1,!1 d. All reals a. b. # f (x) + 2 = g(x) = 4! x2 if x " 1 $ %& x + 2 if x > a /2 = 100 1/2 ( ) 3 = (10) 3 = 1000 b. 27!2/3 = ( 27 1/3 )!2 = (3)!2 = = 1 9 CPM Educational Program 2012 Chapter 2: Page 22 Pre-Calculus with Trigonometry

23 c ( 27 ) 2 = 5 3 ( ) 2/3 = ( 125 ) 1/3 ( ) 2 = 25 9 d. a 6 27 ( )!2/3 " = a ( 6 27 ) 1/3 # $ % & '!2 ( )!2 = ( ) 3 2 = 9 = a2 3 a 2 a 4 CPM Educational Program 2012 Chapter 2: Page 23 Pre-Calculus with Trigonometry

24 2-92. a. x x!1 + x x+1 = x(x+1) (x!1)(x+1) + = (x2 +x)+(x 2!x) x 2 +x!x!1 x(x!1) (x+1)(x!1) = 2x2 x 2!1 b. 2x! 4 x+2 = 2x(x+2)! 4 (x+2) x+2 = 2x2 +4x!4 x+2 Lesson See graph at right. 4!2 a. = = 0.4 b. x 0 = 2.0 x 1 = = 2.4 x 2 = ! 0.4 = 2.8 x 3 = ! 0.4 = 3.2 x 4 = ! 0.4 = 3.6 x 5 = ! 0.4 = 4.0 d k 4! k=0 e. 0.4( ) = 0.4(37.558) = grams f. Lower because the rectangles lie below the curve See graph at right. 5 a. 0.4! 2 0.4k+2 Change the indices. k=1 b. 0.4( ) = 0.4(49.558) = grams c. Students should realize that the indices shift so that we use the same number of rectangles, but of different heights a. 60 miles! 2hours = 120 miles hour!1hours = 40 miles 40 miles hour = 160 miles b. See graph at right. c. Area under the curve = the distance traveled; i.e. hr! mi hr = mi CPM Educational Program 2012 Chapter 2: Page 24 Pre-Calculus with Trigonometry

25 CPM Educational Program 2012 Chapter 2: Page 25 Pre-Calculus with Trigonometry

26 2-96. a. Gain of about 40 million. b (!20) (!5) + 5 = 65! 25 = 40 c. They should be counted as negative See graph at right. a. 2 + ( 0.5) = 1.5 b. f(x) is positive or negative based on the equation. In this case f(x) is negative when x > 3. c = ( 2) = ( 6) = 3.5 d. There is more area below the x-axis than above. Review and Preview a. 5! 3 = 2 (length of interval) 2 4 rectangles = 0.5 (length of each sub-interval) x 0 = 3 x 1 = = 3.5 x 2 = = 4 x 3 = = 4.5 x 4 = = 5 ( ) L = ( ) = ( ) R = ( ) = b. left endpoints =! 0.5 ( k ) 2 + 2!!!!!right endpoints =! k k=0 4 k=1 ( ) c. The left endpoint indices go from k = 0 to k = 1 whereas the right endpoint indices go from k = 1!to k = CPM Educational Program 2012 Chapter 2: Page 26 Pre-Calculus with Trigonometry

27 a. 1 x! 1 y = 1y xy! 1x yx = y!x xy ( ) (x + y) = ( y!x xy ) (x+y) 1 b. 1 x! 1 y = xy+y2!x 2!xy = y2!x 2 xy xy CPM Educational Program 2012 Chapter 2: Page 27 Pre-Calculus with Trigonometry

28 slope = 5!3.5 4!1 = = 0.5 ( g(x)! 5) = 0.5(x! 4) g(x) = 0.5x! g(x) = 0.5x a.! i or! 0.5i + 3 i=1 4 i=1 b = a. 6! 4 = 2 (length of interval) 2 = 0.4 (length of each sub! interval) 5 rectangles b. x 0 = 4!x 1 = = 4.4 x 2 = = 4.8 x 3 = = 5.2 x 4 = = 5.6 x 5 = = 6 c. lower = upper = d. Use more rectangles. ( ) = ( ) = f (x) = 1 x!4 f (x + 2) = 1 (x+2)!4 = 1 x!2 f (x + 2)! 1 = g(x) = 1 x!2! a. b. c. 1 f(x) f(x+2) 1 f (x) CPM Educational Program 2012 Chapter 2: Page 28 Pre-Calculus with Trigonometry

29 Third angle = 180!! (102! + 26! ) = 52! Area = 62 sin(102! ) sin(52! ) sin(26! ) = 31.65cm 2 CPM Educational Program 2012 Chapter 2: Page 29 Pre-Calculus with Trigonometry

30 Lesson Students should recall, lower and upper a !1 = = b. The shift added a 2 by 1 rectangle with area 2. c. A(h, 2! x! 4) = = The shift added a 2 by 10 rectangle with area 20. d. 2a , the width is always 2 and the height added is a a. f(x) b. g(x) Shifts the graph up two units. A( f, 0! x! 3) = 0.5(3)(6) = 9 c. A(g, 0! x! 3) = = 15 d. h(x) (See graph below.) Area added is a 2 by 3 rectangle of area 6. A(h, 0! x! 3) = 0.5(2)(4) " 0.5(1)(2) a. No change b. Adding a 2 by 2 rectangle, which adds an area of 4 units. c. Nothing as long as the interval shifts also. d. Add a 2 by k rectangle Adds a 5 by 3 rectangle with an area of 15 units. A( j, 0! x! 5) = W Subtracts a 2 by 2 rectangle with an area of 4 units. A(n,1! x! 3) = " 4 = See graph at right. A(g, 0! x! 3) = " = = 4 " 1 = 3 CPM Educational Program 2012 Chapter 2: Page 30 Pre-Calculus with Trigonometry

31 Review and Preview a. 3 (length of interval) Width of the rectangles = (rectangles) b. 3! 2 = 6, = c. 3!1000 = 3000, = a. f (x) = x b. 6! 1 = 5 (length of interval) 5 = 1 (length of each sub! interval) 5 rectangles x 0 = 1 x 1 = 2 x 2 = 3 x 3 = 4 x 4 = 5 x 5 = 6 ( ) = ( ) = L = R = CJ 20m Amy ( ) = ( ) = m d John a.! t b.! t c. d 2 = (! t) 2 + (! t) 2 d = (40t! 60) 2 + (30t! 20) 2 d. t = 1.2 hours or at 1:12p.m; The distance between them is 20 miles. CPM Educational Program 2012 Chapter 2: Page 31 Pre-Calculus with Trigonometry

32 a. 4 x 2!8x x 2!9 b. (x 2! y 2 ) " 4y ( ) $ 2 a. 1 + x y " x2!x!12 2x 3!8x 2 = 4 x(x!2) (x+3)(x!3) " (x!4)(x+3) 2x 2 = 2(x!2) (x!4) x(x!3) # b. x 2!6x+8 x 2!16 4 x 2 +4 xy x 2 +2xy+y 2 = (x + y)(x! y) 4 x(x+y) = 4x(x! y) (x+y)(x+y) x 2!y 2 % & ' = y+x " 4y ( y ) $ 2 # (x+y)(x!y) x2!4 x+4 x+4 = (x!4)(x!2) (x+4)(x!4) " % & ' = 4 y (x!y) (x+4) (x!2)(x!2) = 1 x! a. b. # j(x) = x2! 3 for x "!2 $ %& 3x + 7 for x <! a. x sin(50! ) = 6 sin(35! ) x = x = 8.01 b. x 2 = ! 2 " 7 " 5 " cos(120! ) = 109 x = 109 = ! a. 3 " 180 = 4 " 60 = 240!! 18! b. 5 " 180 = 18 " 36 = 648!! 4 c. 1! 180 " = 720 " = 229.2! a. BC = 5 tan 30! = 5 3 AC = 5 sin 30! = 10 b. cos(a) = 5 10 = 1 2 c. sin(a) = = 3 2 CPM Educational Program 2012 Chapter 2: Page 32 Pre-Calculus with Trigonometry

33 Lesson a. 1! ( ) = 17 b. 1!( ) = 32 c. 0.5!1!(2 + 5) + 0.5!1!(5 + 10) + 0.5!1!( ) = = 24.5 d. c, the sum in c is the average of the sums in a and b. e. It is too high b. Students answers should be between a. See graph at right. b. Increasing. c. (20)(2) + 0.5(40)(2) = = 80 d. The units are miles a. See graph at right. b. Increasing. c. Not as far, at there is less area underneath the curve. d. Example of area using trapezoidal rule with four rectangles with length ! 0.5!(20 + 2(22.5) + 2(30) + 2(42.5) + 60) =67.5 True area 66.7 miles See graph at right. a. 0.5! 0.5(5 + 2(4.75) + 2(4) + 2(2.75) + 1) = 0.25(29) = 7.25 b. It is too low a. 0.5( ) = 8.25 square units b. 0.5( ) = 6.25 square units c = 7.25 They are the same The area underneath the curve is 80. a. Half of 80 miles, which is 40 miles. b. Vertical distances are halved. CPM Educational Program 2012 Chapter 2: Page 33 Pre-Calculus with Trigonometry

34 c. Draw the picture again or use the fact that the first car always travels twice as fast, so goes twice as far. CPM Educational Program 2012 Chapter 2: Page 34 Pre-Calculus with Trigonometry

35 Review and Preview ! 1 = 3 (length of interval) 3 = 0.5 (length of each sub! interval) 6 rectangles x 0 = 1 x 1 = 1.5 x 2 = 2 x 3 = 2.5 x 4 = 3 x 5 = 3.5 x 6 = 4 ( ) ( ) a. left-hand approximation = b. right-hand approximation = c. trapezoids 1 ( ( )) ( ) ( ) ( ) = a. See graph at right. b. 3( ) = 3 45 c. It is the units. ( ) = 135 miles a. 5! 0 = 5 (length of interval) 5 = 0.25 (length of each sub-interval) 20 rectangles b. 10! 1 = 9 (length of interval)!!!!!!!!!! 9 = 0.36 (length of each sub-interval) 25 rectangles c. 5! (!3) = 8 (length of interval)!!!!!!!!!! 8 = 0.08 (length of each sub-interval) 100 rectangles d. E! B = E! B (length of interval)!!!!!!!!!! E!B N rectangles = E!B (length of each sub-interval) N ! 1 = 9 (length of interval)!!!!!!!!!! 9 = 0.36 (length of each sub-interval) 25 rectangles a. x 0 = 1! height of rectangle = f (1) b. x 1 = = 1.36! height of rectangle = f (1.36) c. x 25 = 10! height of rectangle = f (10) d. x 24 = 10! 0.36 = 9.64 " height of rectangle = f (9.64) CPM Educational Program 2012 Chapter 2: Page 35 Pre-Calculus with Trigonometry

36 e. x 0 = B! height of rectangle = f (B) x 1 = B + W! height of rectangle = f (B + W ) x last = E! height of rectangle = f (E) x one before last = E " W! height of rectangle = f (E " W ) CPM Educational Program 2012 Chapter 2: Page 36 Pre-Calculus with Trigonometry

37 a. 1 cos! = 1 c/b = 1" b c = b c b. 1 sin! = 1 a/b = 1" b a = b a c. 1 tan! = 1 a/c = 1" c a = c a ! 1 = 1 (length of interval) 1 = 0.25 (length of each sub-interval) 4 rectangles x 0 = 1 x 1 = 1.25 x 2 = 1.50 x 3 = 1.75 x 4 = 2.0 a. Upper = 0.25 ((! ) + (! ) + (! ) + (! )) = Lower = 0.25 ((! ) + (! ) + (! ) + (! )) = " b. 0.25!(0.25k + 1) k= a. The Hypotenuse Leg Postulate AD! AB, AF! AE c. side of equilateral triangle = 3 cos 15! = Area = ( 3.106)2 3 = 4.177cm 4 2 b. DC = BC (def square) DF = BE (corr. Parts of! " ' s are! ) FC = CE (subtraction) a. 3 x+3! 5 x"5 = 3!5 x 2 "5x+3x"15 = 15 x 2 "2x"15 " x ( ) 2 y $ 2 c. x y! y x # x 2 y!xy 2 % " & ' = x2!y 2 # $ xy % & ' " # $ x 2 y 2 xy(x! y) b. % & ' = x2!y 2 x!y x+y x! x!y y = (x+y)(x!y) x!y = y(x+y) xy! x(x!y) xy = xy+y2!x 2 +xy xy = x + y = y2 +2xy!x 2 xy xy = 12! y = 12 x 3x " 2 12 x ( ) = "6 3x " 24 x = "6 3x 2 " 24 = "6x 3x 2 + 6x " 24 = 0 x 2 + 2x! 8 = 0 (x + 4)(x! 2) = 0 Either x =!4 or x = 2! 4y = 12 2y = 12 y =!3 y = 6 Solution : (!4,!3), (2, 6) CPM Educational Program 2012 Chapter 2: Page 37 Pre-Calculus with Trigonometry

38 Lesson !1 a. 20 rectangles b. 20 = 5 20 = c. Left =! 0.25(0.25k + 1) 2 Right =! 0.25(0.25k + 1) 2 k= a. Except for the first left-hand rectangle and the last right-hand rectangle, the other rectangles are in both areas. 19! b. 0.25( 0.25k + 1) 2 k= W = E!B N Left-hand area = Right-hand area = Add: (R + L)/2 Τ and Disp TRAP= T The trapezoidal area should be best. Students may mention that there are fewer gaps between the trapezoids and the curve, or that the trapezoidal area is the average of areas that are too high and too low. To get a better estimate, increase the number of partitions. Note: Remind students of calculator limitations for rounding If students use n = 20, L = , R = , T = a. Using more partitions will improve your estimate. b. y = 4! x 2 y 2 = 4! x 2 x 2 + y 2 = 4, y " 0 c. Students should see this as one fourth of a circle with radius 2, or 4! 4 =!. 20 k=1 CPM Educational Program 2012 Chapter 2: Page 38 Pre-Calculus with Trigonometry

39 Review and Preview a. See graph at right. b. Too low, as rectangles will be under the curve. c. Too high, as rectangles will be above the curve. d. Left = 8.360, Right = a. Gallons of water that passed through the pipe in the interval from 20 minutes to 50 minutes. b. Left-hand approx.! 1.5 "10 + 1" "10 = = 40 gallons Other answers are acceptable; Between 40 and 50 gallons Find h: 12 2 = ! 2(10)(18) cos A 144 = 424! 360 cos A!280 =!360 cos A 7 9 = cos A A = 38.94º sin = h 10 h = 10 sin = Area: 1 (18)(6.285) = cm 12 cm h Aº 18 cm x =!2ax + 7 for x = a =!2a " a =!2a + 7 3a = 6 a = 2 1 x! 3 y = 5 " y! 3x = 5xy (for x, y # 0) y! 5xy = 3x y(1! 5xy) = 3x y = 3x 1!5x CPM Educational Program 2012 Chapter 2: Page 39 Pre-Calculus with Trigonometry

40 1 ( y! x y ) 1 x + 1 ( ) = 1!x ( y ) 1+x x ( ) = 1+x!x!x2 = 1!x2 xy xy CPM Educational Program 2012 Chapter 2: Page 40 Pre-Calculus with Trigonometry

41 a. Flipped over x-axis then up 2. b. Left 2 and compressed vertically. f(x) f(x+2) ! k= k + 2 Lesson The area under a velocity curve represents distance traveled. a. Students should understand that the area is found by multiplying miles/hr times hours to get miles. b. (30)(2) + (20)(1) = = 80 miles c. The second car had half the velocity of the first and the height of the rectangles has changed Area represents 80 miles. a. The height of the trapezoid is t. b. 20 and 20t + 20 c. Area of the trapezoid: d. D(2) = 10! ! 2 D(t) = 0.5(20 + (20t + 20))(t) 0.5t(20t + 40) = 10t t = 10! = 80 miles CPM Educational Program 2012 Chapter 2: Page 41 Pre-Calculus with Trigonometry

42 e. D(3) = 10! ! 3 = 10! = 150 V(3) = 20! = = 80 mph a. She used the midpoints of the sections. One might refer to them as midpoint rectangles. c. Each rectangle would have width 1. d. Rectangles are still one unit wide, change the 1 to 1.5. e. Left = 1( ) = ( ) = 30 Right = 1( ) = ( ) = 60 Trap = 0.5!1( ! ! ! ) = 0.5(90) = 45 Connie = 1( ) = a Review and Preview For the first six hours, Jack makes $5/hr, meaning at the end of six hours, he has earned $30. For anytime after six hours (where he has already earned $30) he adds to the $30 by making $8 times the number of hours worked over six hours # 5t for 0! t! 6 a. J(t) = $ % (t " 6) for t > 6 # 6t for 0! t! 8 b. C(t) = $ % (t " 8) for t > 8 c (t! 6) = t! 60 = 46! t = 46 10t = 76 t = 76 = 7.6 hours 10 d. When is (t! 6) > (t! 8) t! 60 > t! 96! t >! t t > 12t 18 > 2t t < 9 AND 6t < (t! 6) 6t < t! 60 6t < 10t! 30!4t <!30 t > 7.5 " 7.5 < t < 9 CPM Educational Program 2012 Chapter 2: Page 42 Pre-Calculus with Trigonometry

43 CPM Educational Program 2012 Chapter 2: Page 43 Pre-Calculus with Trigonometry

44 ! 1 = 2 (length of interval) 2 = 0.2 (length of each sub-interval) 10 rectangles a. L = 0.2 3(1) % ( ) ( 3(1.2) 3 + 1) ( 3(2.6) 3 + 1) ( 3(2.8) 3 + 1) = 0.2! " 3(0.2k + 1) 3 + 1# $ k=0 b. R = 0.2 3(1.2) % ( ) ( 3(1.4) 3 + 1) ( 3(2.8) 3 + 1) ( 3(3.0) 3 + 1) = 0.2! " 3(0.2k + 1) 3 + 1# $ k=1 9 % c. M = 0.2! " 3(0.2k + 1.1) 3 + 1# $ k= /2 = = 64 They are the same because 1.5 = /2 = 16 1/2 ( ) 3 = 4 3 = a. 2 2(x+1) = 2 x 2 2x+2 = 2 x/2 ( ) 1/2! 2x + 2 = x 2 4x + 4 = x 4 = "3x x = " 4 3 b. 3 1 (3 2x ) = 3 3(x+1) 3 1+2x = 3 3x+3! 1+ 2x = 3x + 3 "2 = x c. ( 5 x ) 1/3 = ( 5!2 ) x!2 5 x/3 = 5!2x+4 " x 3 =!2x + 4 x =!6x x = 12 x = CPM Educational Program 2012 Chapter 2: Page 44 Pre-Calculus with Trigonometry

45 ( q) 2 = ! 2 " 51" 34 cos 25! ( q) 2 = "(0.906) ( q) 2 = q = sin 25 = 34 sin #P $ = 34 sin #P $ sin #P = = #P = 35.4! $ #R = 180!! 25!! 35.4! = 119.6! CPM Educational Program 2012 Chapter 2: Page 45 Pre-Calculus with Trigonometry

46 ! a. 3 b.! 5" a. It shows that the number of customers increases approaching lunchtime and decreases after lunchtime. b. The total number of customers served from 11 a.m. to 1 p.m. c. From the graph, it can be seen that during each hour, more than eight people are served, so the total must be more than a. If k(x) is continuous,! 2x 2 " 4 = "3x + 10 for x = 2 2 # 2 2 " 4 = "3# " 4 = " = 4 So, k(x) is continuous at x = 2. # b. j(x) = 2(x + 1)2! 4 for x < 1 $ %&!3(x + 1) + 10!for x " 1 Lesson a ft (left endpoint) ft (right endpoint) ft (trapezoidal) b. See graph at right. c. Altitude = 0.5! 23.5x! (x) = 11.75x (40.32) 2 " (rounded) d. h = 11.75t 2 Students will draw the line in part b. and find the area of the triangle which has a base of t and a height of 23.5t. e. h(t) = 11.75t 2 h(t) = 11.75t 2 30, 000 = 11.75t = t = t v(50.53) = 23.5! = 1188 ft/sec when it enters the ozone layer. 150, 000 = 11.75t = t = t v(112.99) = 23.5! = 2655 ft/sec when it leaves the ozone layer. CPM Educational Program 2012 Chapter 2: Page 46 Pre-Calculus with Trigonometry

47 2.3.8 Review and Preview to 3.16 depending on left or right rectangles a. g(x) = 2(x 3 + 2) = 2x b. See graph at right. c. The area under g(x) is twice the area under f(x) The interval is 8 units long and you add a height of 4 units. Therefore the new area will be 9 + (8)(4) = !M = 180! " 63! " 50! = 67! 7 sin 67! = o sin 50! = n sin 63! o = and = 0.766!!!!!!!!!!!!o = 5.8 and n = 6.8 n a. g(3) = 11! 3 2 = 11! 9 = 2 b. g(!2) = 2(!2) =!4 c. g(!1) = 2(!1) =!2 d. g(4) = 11! 4 2 = 11! 16 =! a. At around 50 minutes the number of fish entering the hatchery decreases greatly. b. The total number of salmon entering the hatchery during the 60-minute interval a. log 4 16 = 2 log = = 16 b. log 6 ( 36!1 ) =!2 log 6 6!2 =!2 ( ) 6!2 = 1 36 c. log 1000 = 3 log = = a. log 7 49 = log = 2 b. log 2 16 = log = 4 c. log = 8 CPM Educational Program 2012 Chapter 2: Page 47 Pre-Calculus with Trigonometry

48 Closure Problems # 1 x! 3 a. R(x) = $ % 2 x"3 x > 3 b. She started taking the potion at (3,1). The area represents the total length of Rapunzel s hair after a certain period. We are multiplying inches per month by months. c. Trapezoidal approximation with height 1 unit. 0.5(1! ! ! ) = 0.5( ) = 10.5 inches True value 10.1 inches. d. From 3 to 9 months, hair growth is 90.9 inches so total is about 93.9 inches (don t forget the original 3 inches). e. About 8.34 months a. A(3x! 7, 2 " x " 7) 8! b. 2x 2 dx 8 c. Answers will vary but, < A < CL a. b. Time Distance # 60t for!!!!!!!!t! 0 c. d. d(t) = $ % (t " 2)!for!2 < t! 3 CPM Educational Program 2012 Chapter 2: Page 48 Pre-Calculus with Trigonometry

49 CL a. liters b. minute minute ( ) t! = liters c.! dt d. Yes, only about 41 liters have leaked CL ! 1 a. 0.4 = 2 = 5 5 sub-intervals of width 0.4 b. 1! x! c. S = 0.4! 2 0.4k+1 d. left-hand k=0 CL # 150 for 0! t! 3 C(t) = $ % (t " 3) for t > t3 CL a. (3! ) + (3! ) + (3! ) + (3! ) = b. 0.2( ! ! ! !4 ) = k+3 c. A(4 x, 3! x! 4) CL a. See graph at right. b. Yes, it is continuous c. See graph in part (a). d. # h(x) = (x! 1)2! 1 if x " 3 $ %& 9! 2x if x > 3 CL f (x + 2) = 2 x! (x + 2) 2 = 2 x+1 + 4(x + 2) 2 4 " k=0 CPM Educational Program 2012 Chapter 2: Page 49 Pre-Calculus with Trigonometry

50 CL a. b. ( ) + ( ) + ( 4 + 3) ( ) ( ) Left = 1! # 2 # " ' Right = 1! # 2 # " ' $ & & % ( ) + ( 4 + 3) + ( ) ( ) + ( 4+ 5 ) $ & & % CL a. f (x! 1) shifts right 1. b.! f (x) is a reflection over the x-axis. CL a. x+1 x! x x!1 = (x+1)(x!1) x(x!1)! x"x x(x!1) = x2 +x!x!1!x 2 x(x!1) =! 1 x(x!1) " x ( ) 2 y 2 % # $ y 3 x!x 3 y & = y!x " x ( y ) 2 y $ 2 b. 1! x y # ' = xy(y! x)(y+ x) y!x " x ( y ) 2 y 2 # $ xy(y 2!x 2 ) % & ' = x y+x % & ' CL a. 15 ft 10 ft b.! = 15ft = 1.5 radians 10ft 1.5 " 180! = 270! = # #! c. 15 radians is about 2.4 times around the circle. The central angle is then (15! 4" ) # 2.34 radians # º. CPM Educational Program 2012 Chapter 2: Page 50 Pre-Calculus with Trigonometry