course Mth 164
24-23-836Assignment 3
Section 6.4
3.The maximum value of y = sin x, 0 is less than or equal to x is less than or equal to 2pi, is 1 and occurs at x= …-3pi/2, pi/3, 5pi/2, 9pi/2…
9. What is the y-int. of y= sin x ? origin(0,0)
11. For what numbers x, pi is less than or equal to x is less than or equal to pi, is the graph of y = sin x increasing ? 5pi/6, pi/2( I don’t think I am supposed to include pi/6 because even though it still gives a positive value for y=sin x the curve is going back down towards 0 )
13. What is the largest value of y = sin x ? 1
15 . For what numbers x, -2pi is less than or equal to x is less than or equal to 2pi, does sin x = 1 ? pi/2
Where does sin x = -1 3pi/2
Determine amplitude and period of each function without graphing.
A = amplitude Period = T = 2pi/w y = Asin(wx)
21. y = -4cos(2x) amplitude = -4 Period = T = 2pi/2 = pi
23. y = 6sin(pi x) A = amplitude = 6 Period = T = 2pi/w = 2pi/pi = 2
27. y = 5/3sin(-2pi/3x) A = 5/3 Period = T = (2pi/w) = 2pi/(2pi/3) = 3
33. y = -3sin(2x) A = -3 Period = T = 2pi/w = 2pi/2 = pi
**** I looked for a definition for amplitude in the book and notes. If amplitude is the distance from the x-axis, how can distance be negative ?
39. y = 3sin (1/2x) A = 3 Period = T = 2pi/w = 2pi/(1/2) = 4pi
Graph each function. Label key points and show at least 2 cycles.
45. y = -4sinx A = -4 Period 2pi
Graph starts at(-2pi,0),curve goes down to (-3pi, -4), curve goes up to (-pi/2, 4) and back down to(0,0), then continues down to (pi/2, -4), up to cross the x-axis at (pi, 0) to the top of the curve at (3pi/2, 4) and back down to the x-axis at (2pi,0).
A = 1 P = -1
The graph starts at the origin, curves down to (pi/4, -1), up to the x-axis at(pi/2, 0) up to (3pi/4, 1) and back down to the x-axis at (0, pi)
51. y = 2sin(1/2x) A= 2 P = 4
The graph starts at( -2pi, 0) and curves down to (-pi, -2), then up to the origin, then up to (pi, 2) then down to ( 2pi, 0).
53. y = -1/2cos(2x) A = -1/2 P = 2pi/2 = 4pi
The graph starts at (-2pi, .5) then goes down to (-3pi/2, -.5), then up to (-pi/2, .5,) then down to (0, -.5) then up to (pi/2, .5), then down to (3pi/2, -.5)
57. y = 5 cos(pi x) -3 A = 5 P = 2pi/pi = 2 Vertical shift down 3 units
Starting at(-1.7, 0)then down to around -7 and back up to the origin(0,2), curves down to (pi/2, 0), continues down to around - 7 then back up to (1.7, 0)…
***63. y= 5/3sin(-2pi/3 x) A = 5/3 P = 2pi/(-2/3) = -3pi
Write an equation of a sine function that has the given characteristics:
67. Amplitude = 3 Period = pi y= 3sin(2x)
69. A = 3, P = 2 y = 3 sin(pi x)
Find an equation for each graph:
71. A = 5 P = 1/4pi y = 5cos(1/4pi x)
75. A = ¾ P = 1 y = ¾sin(2pi x)
81.y = +- 3sin(pi x)
87.Find the average rate of change of f(x) = sin(x/2) from 0 to pi/2.
**** I’m not sure what this means, but I worked backward from the answer in the book and figured it was y/x , which gave me roughly the decimal equivalent of sq. rt. of 2/pi =
.450158 y=.707/x = 1.57 = .450318
The average rate of change is
change in f(x) / change in x.
Change in f(x) is f(pi/2) - f(0) = -1 - 1 = -2.
Change in x is pi/2 - 0 = pi/2.
So the average rate of change is
change in f(x) / change in x = -2 / (pi/2) = -4 / pi, about -1.3.
The average rate of change between two values of x can be visualized as the slope of the graph of y = f(x) vs. x, between the two points.
93. I(t) = 220sin(60pi t) t is less than 0
P = 1/30 A = 220
99. a) Find w for each characteristic:
Y = 50sin(23 t) +50 w = 2 pi/23
Y = 50 sin(28 t) + 50 w = 2pi/28 = pi/14
Y = 50sin(33t) + 50 w = 2pi/33
c) the graph is hard to read, but apparently each characteristic is represented by a different width bar, and there does not appear to be a point where all 3 bars lie on top of each other.
d) ?
Sec. 6.5
3. The graph of y = tan x is symmetric with respect to the y-axis and has vertical asymptotes at origin,… -3pi/2, -pi/2, pi/2, 3pi/2…
7. What is the y-int. of y = tan x (0,0)
9. What is the y – int. of y = sec x ? (0, 1)
15. For what numbers x, -2pi is less than or equal to x is less than or equal to 2pi, does the graph of of y = tan x have vertical asymptotes ? -3pi/2, -pi/2, pi/2, 3pi/2
Graph each function
21. y = tan(pi/2 x) no amplitude, P = computed as ½ but does not make sense
Curve passes through the origin with point (-pi/2, -1) below the origin and point(pi/2, 1) above the origin.
23. y = cot(1/4x) P = 4pi With the y-axis as a vertical asymptote, the curves pass through the points (pi, 1) above the origin and (-pi, 1) below the origin.
27. y = -3csc x range : absolute value of y is greater than or equal to the absolute value of 3 vertical asymptote is the y-axis, points at (-pi/2, 3) (pi/2, -3)
Not sure about period on this – according to the formula it would be 2pi but this doesn’t make sense since the curves don’t touch the x-axis. The distance between the points is pi. The book didn’t provide much help here.
31. y = -2csc(pi x) range: absolute value of y is greater than or equal to the absolute value of 2 vertical asymptote is the y-axis, points at (-pi/2, -2), (-1/2, 2), (1/2, -2) and(pi/2, 2).
37.y = 1/2 tan(1/4x) – 2 magnitude of vertical stretch= ½, vertical shift = -2, P = 4pi
Curve passes through x-axis at 5pi/3 and continues to the left through the y-axis at -2
43. Find the average rate of change of f from 0 to pi/6
F(x) = tan(2x) y = 3.732/x = .6544 = 5.68
Refer to the graphs to answer each question, if necessary.
6. What is the smallest value of y = cos x? -1
10. for what numbers x, -2`pi <= x <= 2`pi, does cos x =1? What about cos x = -1?
X= 1 at …-2pi, 0, 2pi, 4pi, 6pi… x=-1 ar …-pi, pi, 3pi, 5pi…
12. What is the y- intercept of y = cot x? 1
18. for what numbers x, -2`pi <= x <= 2`pi, does the graph of y= csc x have vertical asymptotes?
-3.27, 3.27
20. For what numbers x, -2`pi <= x <= 2`pi,does the graph of y= cot x have vertical asymptotes?
-3.27, 3.27
Use transformations to graph each of the following functions:
36. y =Cos x + 1 A = 1, P = 2pi, vertical shift up 1 unit, points at (-pi, -1) (0, 2) (pi, 1)
40. y = cos`pi / 2x A = 1, P = 4 crosses x-axis at(-4, 0)goes below at(-2,-1)crosses y-axis at (0,1) goes below x-axis at (2,-1) above axis at (4,0)
42. y =3 cos x + 3 A = 3, P = 2pi, vertical shift up 3 units, points at (0,6)(pi, 0) (2pi, 6)(3pi, )
48. y = -cot x no A, curves on origin and multiples of pi, vertical asymptotes at multiples of pi/2
50. y = csc (x - `pi) range : absolute value of y is greater than the absolute value of 1; points at (pi/2, -1)(3pi/2, 1)
54. y = 4tan 1/2 x no A, P = 2pi, curves go through the origin and multiples of 2pi, vertical asymptotes at multiples of pi
60. y = -3 tan 2x curves at multiples of pi/2 starting at the origin; asymptotes at intervals of pi/2 starting at pi/2
63. On the same coordinate axes, graph y =2 sin x and y = sin 2x, 0 <= x <= 2 `pi.
Compare each graph’s maximum and minimum value
Y = 2 sinx max = 2, min = -2; y = sin 2x max = 1 min= -1.
Compare each graph’s period.
Y = 2 sin x A = 2 P = 2pi y = sin 2x A= 1 P = pi
66. Repeat problem 63 for y = 4 cos x and y = cos 4x, 0 <= x <= 2`pi.
Compare each graph’s max/min.
Y = 4 cos x max = 4 min = -4 y = cos 4x max = 1 min = -1
Compare each graph’s period
Y = 4 cos x A= 4 P = 2pi y = cos 4x A = 1 P = 1/2pi
70. Graph y = 2 sin x, y = 1/2 sin x and y = 8 sin x. What do you conclude about the graph of y =A sin x, A > 0 ? the greater the amplitude the steeper the curves
72. Graph y = sin x, y = sin [ x - ( `pi / 3 ) ], y = sin { x - ( `pi / 4 )}, and y = sin [ x - ( `pi / 6 ) ]. What d you conclude about the graph of y = sin ( x -`phi ), `phi > 0?
Graphs of y = sin(x-phi) phi is greater than 0, are the same as the graphs of the cosine function - they do not go through the origin.
Sec. 5.6 online problems
Determine the amplitude and period of each function without graphing.
6. y = -3 cos 3x A = -3 P = T = 2pi/w = 2pi/3
10. y = 9 / 5 cos ( - 3 `pi/ 2 x ) A = 9/5 P = T = (-3pi/2) = 2pi/w = 4/3
Match the given functions to one of the graphs (A) – (J). p. 397 in 8th ed.
12. y = 2 cos (`pi/ 2x) I
18. y = - 2 cos( `pi /2 x) G
20. y = -2 sin (1/2 x) D
Match the given function to one of the graphs (A)-(F) p. 398
24. y =3 sin 2 x A
Graph each function.
30. y = 2 sin `pi x A= 2, P = 2, starting at the origin, points are at (pi/2, 2) (pi/2, -2)(3pi/4, 2)…..
36. y =( 4/3) cos (-1 / 3 * x ) A = 4/3, P = -6pi ;curves up to cross x-axis at (-3pi/2, 0) crosses y-axis at(0, 4/3) curves down to cross x-axis at((3pi/2, 0)
Find the amplitude, period, and phase shift of each function. Use transformation to graph the function. Show at least one period.
54. y = 3 cos (2x + `pi)
Y = Acos(wx-phi) A = 3, w(period) = 2, phi = -pi phase shift = phi/w = -pi/2
60. y = 2 cos (2`pi (x - 4 ) ) A=2, w(period) = 2pi, phi = -4 phase shift = phi/w = -4/2pi = -2/pi
Write the equation of a sine function that has the given characteristics.
66. Amplitude: 4
Period: 1 y = 4 sin(2pi x)
70. Amplitude: 2
Period: `pi
Phase shift: -2 y = 2sin(2x-2)
72. Alternating current (AC) circuits. The current I, in amperes, flowing through an ac (alternating current) circuit at time t is
• I = 120 sin 30`pi * t, t = 0
What is the period? What is the amplitude? Graph the function over two periods.
Period = 2pi/30pi = w w = 1/15, but P on the graph appears to be pi
A = 120 Starting at the origin, curves intercept the x-axis at pi/2. then pi, then 3pi/2, then 2pi…
80. Biorhythms. In the theory of biorhythms a sine function of the form P = 50 sin ( t ) + 50 is used to measure the percent P of a person's potential at time t, where t is measured in days starting with the person's day of birth. Three characteristics are commonly measured:
• Physical potential with a period of 23 days.
• Emotional potential with a period of 28 days.
• Intellectual potential with a period of 33 days.
a) Find for each characteristic
b) Graph all three functions
c) Is there a time t when all three characteristics have 100% potential? When is this?
d) Suppose that you are 20 years old today. Describe your physical, emotional, and intellectual potential for the next 30 days. (same as #99 on Sec. 6.4)
80. Monthly temperature. The following data represent the average monthly temperatures for Washington, D.C.:
• January, Month 1, has average temperature 34.6 deg.
• February, Month 2, has average temperature 37.5 deg.
• March, Month 3, has average temperature 47.2 deg.
• April, Month 4, has average temperature 56.5 deg.
• May, Month 5, has average temperature 66.4 deg.
• June, Month 6, has average temperature 75.6 deg.
• July, Month 7, has average temperature 80.0 deg.
• August, Month 8, has average temperature 78.5 deg.
• September, Month 9, has average temperature 71.3 deg.
• October, Month 10, has average temperature 59.7 deg.
• November, Month 11, has average temperature 49.8 deg.
• December, Month 12, has average temperature 39.4 deg.
a) Draw a scatter diagram of the data for one period.
b) Find a sinusoidal function of the form y = A sin( `omega x - `phi ) + B that fits the data.
Y =3cos(1/2x)
c) Draw the sinusoidal function found in (b) on the scatter diagram.
d) Use a graphing utility to find the sinusoidal function of best fit.
e) Draw the sinusoidal function of best fit on the scatter diagram.
Starting at (-3pi/4, 0), curve intercepts y-axis at (0, 3) then curves down to (3pi/4, 0)….
&#This looks good. See my notes. Let me know if you have any questions. &#