Copyright © by Houghton Mifflin Company, All rights reserved. Calculus Concepts 2/e LaTorre,...

Copyright © by Houghton Mifflin Company, All rights reserved.Copyright © by Houghton Mifflin Company, All rights reserved.

Calculus Concepts 2/eCalculus Concepts 2/eLaTorre, Kenelly, Fetta, Harris, and CarpenterLaTorre, Kenelly, Fetta, Harris, and Carpenter

Chapter 8Chapter 8Repetitive Change: Cycles and TrigonometryRepetitive Change: Cycles and Trigonometry


Chapter # Key ConceptsChapter # Key Concepts• Sine and Cosine Functions Sine and Cosine Functions

• Finding Sine ModelsFinding Sine Models

• Derivatives of Sine and CosineDerivatives of Sine and Cosine

• Extrema and Inflection PointsExtrema and Inflection Points

• More Accumulation FunctionsMore Accumulation Functions


Sine and Cosine FunctionsSine and Cosine Functions• Sine FunctionSine Function

• Cosine FunctionCosine Function


Sine and Cosine FunctionsSine and Cosine Functions• Variations of the Sine FunctionVariations of the Sine Function

– The parameters of the sine function The parameters of the sine function f(x) = asin(bx + h) + k are interpreted as f(x) = asin(bx + h) + k are interpreted as follows (we assume b > 0):follows (we assume b > 0):

– |a| is the amplitude, a < 0 indicates reflection|a| is the amplitude, a < 0 indicates reflection– b is the frequencyb is the frequency– 22/b is the period/b is the period– |h|/b is the horizontal shift, right if h < 0, left if h > 0|h|/b is the horizontal shift, right if h < 0, left if h > 0– k is the vertical shift, up if k > 0, down if k < 0k is the vertical shift, up if k > 0, down if k < 0


Sine and Cosine Functions: ExampleSine and Cosine Functions: ExampleGraph each function:Graph each function:y = 3 sinx - 4y = 3 sinx - 4

y = 0.5 sin(0.2x - 1) + 2.5y = 0.5 sin(0.2x - 1) + 2.5


Sine Functions: Exercise 8.1 #27Sine Functions: Exercise 8.1 #27Graph y = 2 sin(-2x + 3)Graph y = 2 sin(-2x + 3)


Finding Sine ModelsFinding Sine Models• Data that is periodic can be modeled using a Data that is periodic can be modeled using a

sine (or cosine) functionsine (or cosine) function

• Sine regression is frequently used to Sine regression is frequently used to determine modeldetermine model

• Model may also be determined by data Model may also be determined by data analysisanalysis


Finding Sine Models: ExampleFinding Sine Models: Example

12/21

6/21

12/21

6/21

12/21

Hours of Daylight

Date Day of the Year-10

81.5

173

265

355

446.5

538

629

720

811.5

0

12

24

12

0

12

24

12

0

12

The hours of daylight on the The hours of daylight on the Arctic Circle are shown in the Arctic Circle are shown in the table and on the scatter plot. Find table and on the scatter plot. Find a model for the data.a model for the data.

Since the data is periodic we Since the data is periodic we want a sine model.want a sine model.


Finding Sine Models: ExampleFinding Sine Models: Example

12/21

6/21

12/21

6/21

12/21

Hours of Daylight

Date Day of the Year-10

81.5

173

265

355

446.5

538

629

720

811.5

0

12

24

12

0

12

24

12

0

12

The output ranges from 0 to 24 so The output ranges from 0 to 24 so a = 24/2 = 12 hours.a = 24/2 = 12 hours.

f(x) = asin(bx + h) + k f(x) = asin(bx + h) + k

Period = 2Period = 2/b = 365 so b /b = 365 so b 0.017210.01721Horizontal shift to the right = |h|/b Horizontal shift to the right = |h|/b = 81.5 so h = 81.5 so h -1.40296 -1.40296

Vertical shift = k = 12Vertical shift = k = 12

h(t) = 12 sin(0.01721t - 1.40296) + h(t) = 12 sin(0.01721t - 1.40296) + 12 hours12 hours


Finding Sine Models: ExampleFinding Sine Models: Exampleh(t) = 12 sin(0.01721t - 1.40296) + 12 hoursh(t) = 12 sin(0.01721t - 1.40296) + 12 hours


Finding Sine Models: Exercise 8.2 #19Finding Sine Models: Exercise 8.2 #19A model fit to the mean air temperature at Fairbanks, A model fit to the mean air temperature at Fairbanks, Alaska, is Alaska, is

f(x) = 37 sin[0.01721(x - 101)] + 25 f(x) = 37 sin[0.01721(x - 101)] + 25 degrees Fahrenheitdegrees Fahrenheit

where x is the number of days since the last day of the where x is the number of days since the last day of the previous year. Estimate the highest and lowest daily previous year. Estimate the highest and lowest daily mean temperature.mean temperature.

The amplitude is 37 so the highest and lowest The amplitude is 37 so the highest and lowest temperatures will vary by 2(37) = 74 degrees. temperatures will vary by 2(37) = 74 degrees. The vertical shift is 25 so the high temperature is The vertical shift is 25 so the high temperature is 25 + 37 = 62 degrees Fahrenheit and the low 25 + 37 = 62 degrees Fahrenheit and the low temperature is 25 - 37 = -12 degrees Fahrenheit.temperature is 25 - 37 = -12 degrees Fahrenheit.


Derivatives of Sine and CosineDerivatives of Sine and Cosine• When x is measured in radians, the rate of When x is measured in radians, the rate of

change of the sine function is the cosine change of the sine function is the cosine function. That is, function. That is,

xcos)x(sindx

d xcos)x(sin

dx

d

• When x is measured in radians, the rate of When x is measured in radians, the rate of change of the cosine function is the negative change of the cosine function is the negative of the sine function. That is, of the sine function. That is,

xsin)x(cosdx

d xsin)x(cos

dx

d


Derivatives of Sine and Cosine: ExampleDerivatives of Sine and Cosine: ExampleThe calls for service made to a county sheriff’s The calls for service made to a county sheriff’s department in a certain county can be modeled as department in a certain county can be modeled as c(t) = 2.8 sin(0.262 t + 2.5) + 5.38 calls during the tth c(t) = 2.8 sin(0.262 t + 2.5) + 5.38 calls during the tth hour after midnight. How quickly is the number of calls hour after midnight. How quickly is the number of calls received each hour changing at noon?received each hour changing at noon?

c'(t) c'(t) = 2.8 [cos (0.262 t +2.5)](0.262)= 2.8 [cos (0.262 t +2.5)](0.262)= 0.7336 cos(0.262 t + 2.5) calls per hour= 0.7336 cos(0.262 t + 2.5) calls per hour

c'(12) = 0.7336 cos(0.262 (12) + 2.5) calls per hourc'(12) = 0.7336 cos(0.262 (12) + 2.5) calls per hour 0.6 call per hour0.6 call per hour

At noon, the number of calls is increasing by 0.6 At noon, the number of calls is increasing by 0.6 call per hour.call per hour.


Derivative of Sine: Exercise 8.3 #9Derivative of Sine: Exercise 8.3 #9The normal daily high temperature in Boston can be The normal daily high temperature in Boston can be modeled by B(m) = 22.926 sin(0.510m - 2.151) + modeled by B(m) = 22.926 sin(0.510m - 2.151) + 58.502 °F at the end of the mth month of the year. 58.502 °F at the end of the mth month of the year. Estimate how rapidly the temperature is changing at Estimate how rapidly the temperature is changing at the end of November.the end of November.

B'(m) B'(m) = 11.692 cos(0.510m - 2.151) °F per month= 11.692 cos(0.510m - 2.151) °F per monthB'(11) = 11.692 cos(0.510(11) - 2.151) °F per monthB'(11) = 11.692 cos(0.510(11) - 2.151) °F per month

-11.108 °F per month-11.108 °F per month

At the end of November, the temperature is dropping At the end of November, the temperature is dropping by about 11°F per month.by about 11°F per month.


Extrema and Inflection PointsExtrema and Inflection Points• Extrema of sine and cosine functions occur Extrema of sine and cosine functions occur

where the derivative is equal to zero.where the derivative is equal to zero.

• Points of inflection of sine and cosine Points of inflection of sine and cosine functions occur where the second derivative is functions occur where the second derivative is equal to zero.equal to zero.


Extrema and Inflection Points: ExampleExtrema and Inflection Points: ExampleWeekly sales of heating oil for a certain company Weekly sales of heating oil for a certain company may be modeled by may be modeled by

f(t) = 2000 sin(0.1208 t + 1.5707) + 3000 gallons f(t) = 2000 sin(0.1208 t + 1.5707) + 3000 gallons t weeks after the beginning of the year. Find the t weeks after the beginning of the year. Find the maximum and minimum of a cycle with derivatives.maximum and minimum of a cycle with derivatives.

The change in weekly sales of heating oil may be The change in weekly sales of heating oil may be modeled by f '(t) = 241.6 cos(0.1208 t + 1.5707) modeled by f '(t) = 241.6 cos(0.1208 t + 1.5707) gallons per week t weeks after the start of the year. gallons per week t weeks after the start of the year.

0 = 241.6 cos(0.1208 t + 1.5707) 0 = 241.6 cos(0.1208 t + 1.5707) 0 = cos(0.1208 t + 1.5707)0 = cos(0.1208 t + 1.5707)/2 = 0.1208 t + 1.5707 so t = 0 /2 = 0.1208 t + 1.5707 so t = 0 f(0) = 5000 is a max f(0) = 5000 is a max33/2 = 0.1208 t + 1.5707 so t = 26 /2 = 0.1208 t + 1.5707 so t = 26 f(26) = 1000, min f(26) = 1000, min


Extrema/Inflection Points: Exercise 8.4 #3Extrema/Inflection Points: Exercise 8.4 #3The number of males per 100 females can be modeled The number of males per 100 females can be modeled by m(t) = 5.3365 sin(0.0425 t + 1.268) + 99.938 males by m(t) = 5.3365 sin(0.0425 t + 1.268) + 99.938 males per 100 females t years after 1900. Determine when per 100 females t years after 1900. Determine when the graph is decreasing most rapidly and how fast it is the graph is decreasing most rapidly and how fast it is decreasing at that time.decreasing at that time.

m'(t) = 0.2268 cos(0.0425 t + 1.268) m'(t) = 0.2268 cos(0.0425 t + 1.268) m''(t) = -0.009639 sin(0.0425 t + 1.268) m''(t) = -0.009639 sin(0.0425 t + 1.268) m''(t) = 0 at t = 44.085, 118.004m''(t) = 0 at t = 44.085, 118.004m'(44.085) = -0.2268 males per 100 females per yearm'(44.085) = -0.2268 males per 100 females per yearm'(118.004) = 0.2268 males per 100 females per year m'(118.004) = 0.2268 males per 100 females per year The number of males per 100 females was decreasing The number of males per 100 females was decreasing most rapidly in 1944.most rapidly in 1944.


Cxcosdxxsin Cxcosdxxsin

Cxsindxxcos Cxsindxxcos

Ckx)hbxsin(b

adx)hbxcos(a Ckx)hbxsin(

b

adx)hbxcos(a

Ckx)hbxcos(b

adx)hbxsin(a Ckx)hbxcos(

b

adx)hbxsin(a

More Accumulation FunctionsMore Accumulation Functions• When x is measured in radians,When x is measured in radians,


More Accumulation Functions: ExampleMore Accumulation Functions: ExampleThe rate of change of the temperature in Philadelphia The rate of change of the temperature in Philadelphia on August 27, 1993, can be modeled ason August 27, 1993, can be modeled as

t(x) = 2.733 cos(0.285x -2.93) °F per hour t(x) = 2.733 cos(0.285x -2.93) °F per hour x hours after midnight. Find the accumulated change x hours after midnight. Find the accumulated change in the temperature between 9 a.m. and 3 p.m.in the temperature between 9 a.m. and 3 p.m.

15

9dx)93.2x285.0cos(733.2

15

9dx)93.2x285.0cos(733.2

15

9)93.2x285.0sin(

285.0

733.2

15

9)93.2x285.0sin(

285.0

733.2

F769.12 F769.12 The temperature increased by about 13 °F The temperature increased by about 13 °F between 9 a.m. and 3 p.m.between 9 a.m. and 3 p.m.


Accumulation Function: Exercise 8.5 #9Accumulation Function: Exercise 8.5 #9The rate of change of product sales for an athletic The rate of change of product sales for an athletic store can be modeled bystore can be modeled by

T'(x) = 23.944 cos(0.987x -1.276)T'(x) = 23.944 cos(0.987x -1.276)thousand dollars per monththousand dollars per month

where x = 1 in January, x = 2 in February, etc. Find a where x = 1 in January, x = 2 in February, etc. Find a model for sales in month t if January sales are $54,000model for sales in month t if January sales are $54,000

dx)276.1x987.0cos(944.23 dx)276.1x987.0cos(944.23

C)276.1987.0sin(987.0

944.23 C)276.1987.0sin(

987.0

944.23

C)276.1x987.0sin(259.24 C)276.1x987.0sin(259.24


Accumulation Function: Exercise 8.5 #9Accumulation Function: Exercise 8.5 #9

194.60C 194.60C

194.60)27.2x524.0sin(265.80)x(T 194.60)27.2x524.0sin(265.80)x(T

C)276.1x987.0sin(259.24)x(T C)276.1x987.0sin(259.24)x(T

C)276.1)1(987.0sin(259.24)1(T C)276.1)1(987.0sin(259.24)1(T

C194.654 C194.654

Copyright © by Houghton Mifflin Company, All rights reserved. Calculus Concepts 2/e LaTorre,...

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