1. Copyright © by Houghton Mifflin Company, All rights reserved. Calculus Concepts 2/e LaTorre, Kenelly, Fetta, Harris, and Carpenter Chapter 8 Repetitive Change: Cycles and Trigonometry

2. Copyright © by Houghton Mifflin Company, All rights reserved. Chapter # Key Concepts Sine and Cosine FunctionsSine and Cosine FunctionsSine and Cosine FunctionsSine and Cosine Functions Finding Sine ModelsFinding Sine ModelsFinding Sine ModelsFinding Sine Models Derivatives of Sine and CosineDerivatives of Sine and CosineDerivatives of Sine and CosineDerivatives of Sine and Cosine Extrema and Inflection PointsExtrema and Inflection PointsExtrema and Inflection PointsExtrema and Inflection Points More Accumulation FunctionsMore Accumulation FunctionsMore Accumulation FunctionsMore Accumulation Functions

3. Copyright © by Houghton Mifflin Company, All rights reserved. Sine and Cosine Functions Sine FunctionSine Function Cosine FunctionCosine Function

4. Copyright © by Houghton Mifflin Company, All rights reserved. Sine and Cosine Functions Variations of the Sine FunctionVariations of the Sine Function –The parameters of the sine function f(x) = asin(bx + h) + k are interpreted as follows (we assume b > 0): –|a| is the amplitude, a < 0 indicates reflection –b is the frequency –2 /b is the period –|h|/b is the horizontal shift, right if h 0 –k is the vertical shift, up if k > 0, down if k 0, down if k < 0

5. Copyright © by Houghton Mifflin Company, All rights reserved. Sine and Cosine Functions: Example Graph each function: y = 3 sinx - 4 y = 0.5 sin(0.2x - 1) + 2.5

6. Copyright © by Houghton Mifflin Company, All rights reserved. Sine Functions: Exercise 8.1 #27 Graph y = 2 sin(-2x + 3)

7. Copyright © by Houghton Mifflin Company, All rights reserved. Finding Sine Models Data that is periodic can be modeled using a sine (or cosine) functionData that is periodic can be modeled using a sine (or cosine) function Sine regression is frequently used to determine modelSine regression is frequently used to determine model Model may also be determined by data analysisModel may also be determined by data analysis

8. Copyright © by Houghton Mifflin Company, All rights reserved. Finding Sine Models: Example 12/21 6/21 12/21 6/21 12/21 Hours of Daylight DateDay 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 Arctic Circle are shown in the table and on the scatter plot. Find a model for the data. Since the data is periodic we want a sine model.

9. Copyright © by Houghton Mifflin Company, All rights reserved. Finding Sine Models: Example 12/21 6/21 12/21 6/21 12/21 Hours of Daylight DateDay 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 a = 24/2 = 12 hours. f(x) = asin(bx + h) + k Period = 2 /b = 365 so b 0.01721 Horizontal shift to the right = |h|/b = 81.5 so h -1.40296 Vertical shift = k = 12 h(t) = 12 sin(0.01721t - 1.40296) + 12 hours

10. Copyright © by Houghton Mifflin Company, All rights reserved. Finding Sine Models: Example h(t) = 12 sin(0.01721t - 1.40296) + 12 hours

11. Copyright © by Houghton Mifflin Company, All rights reserved. Finding Sine Models: Exercise 8.2 #19 A model fit to the mean air temperature at Fairbanks, Alaska, is f(x) = 37 sin[0.01721(x - 101)] + 25 degrees Fahrenheit where x is the number of days since the last day of the previous year. Estimate the highest and lowest daily mean temperature. The amplitude is 37 so the highest and lowest temperatures will vary by 2(37) = 74 degrees. The vertical shift is 25 so the high temperature is 25 + 37 = 62 degrees Fahrenheit and the low temperature is 25 - 37 = -12 degrees Fahrenheit.

12. Copyright © by Houghton Mifflin Company, All rights reserved. Derivatives of Sine and Cosine When x is measured in radians, the rate of change of the sine function is the cosine function. That is,When x is measured in radians, the rate of change of the sine function is the cosine function. That is, When x is measured in radians, the rate of change of the cosine function is the negative of the sine function. That is,When x is measured in radians, the rate of change of the cosine function is the negative of the sine function. That is,

13. Copyright © by Houghton Mifflin Company, All rights reserved. Derivatives of Sine and Cosine: Example The calls for service made to a county sheriffs 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 hour after midnight. How quickly is the number of calls received each hour changing at noon? c'(t) = 2.8 [cos (0.262 t +2.5)](0.262) = 0.7336 cos(0.262 t + 2.5) calls per hour c'(12) = 0.7336 cos(0.262 (12) + 2.5) calls per hour 0.6 call per hour 0.6 call per hour At noon, the number of calls is increasing by 0.6 call per hour.

14. Copyright © by Houghton Mifflin Company, All rights reserved. Derivative of Sine: Exercise 8.3 #9 The normal daily high temperature in Boston can be modeled by B(m) = 22.926 sin(0.510m - 2.151) + 58.502 °F at the end of the mth month of the year. Estimate how rapidly the temperature is changing at the end of November. B'(m) = 11.692 cos(0.510m - 2.151) °F per month B'(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 by about 11°F per month.

15. Copyright © by Houghton Mifflin Company, All rights reserved. Extrema and Inflection Points Extrema of sine and cosine functions occur where the derivative is equal to zero.Extrema of sine and cosine functions occur where the derivative is equal to zero. Points of inflection of sine and cosine functions occur where the second derivative is equal to zero.Points of inflection of sine and cosine functions occur where the second derivative is equal to zero.

16. Copyright © by Houghton Mifflin Company, All rights reserved. Extrema and Inflection Points: Example Weekly sales of heating oil for a certain company may be modeled by f(t) = 2000 sin(0.1208 t + 1.5707) + 3000 gallons t weeks after the beginning of the year. Find the maximum and minimum of a cycle with derivatives. The change in weekly sales of heating oil may be modeled by f '(t) = 241.6 cos(0.1208 t + 1.5707) gallons per week t weeks after the start of the year. 0 = 241.6 cos(0.1208 t + 1.5707) 0 = cos(0.1208 t + 1.5707) /2 = 0.1208 t + 1.5707 so t = 0 f(0) = 5000 is a max /2 = 0.1208 t + 1.5707 so t = 0 f(0) = 5000 is a max 3 /2 = 0.1208 t + 1.5707 so t = 26 f(26) = 1000, min

17. Copyright © by Houghton Mifflin Company, All rights reserved. Extrema/Inflection Points: Exercise 8.4 #3 The number of males per 100 females can be modeled by m(t) = 5.3365 sin(0.0425 t + 1.268) + 99.938 males per 100 females t years after 1900. Determine when the graph is decreasing most rapidly and how fast it is decreasing at that time. m'(t) = 0.2268 cos(0.0425 t + 1.268) m''(t) = -0.009639 sin(0.0425 t + 1.268) m''(t) = 0 at t = 44.085, 118.004 m'(44.085) = -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 most rapidly in 1944.

18. Copyright © by Houghton Mifflin Company, All rights reserved. More Accumulation Functions When x is measured in radians,When x is measured in radians,

19. Copyright © by Houghton Mifflin Company, All rights reserved. More Accumulation Functions: Example The rate of change of the temperature in Philadelphia on August 27, 1993, can be modeled as t(x) = 2.733 cos(0.285x -2.93) °F per hour x hours after midnight. Find the accumulated change in the temperature between 9 a.m. and 3 p.m. The temperature increased by about 13 °F between 9 a.m. and 3 p.m.

20. Copyright © by Houghton Mifflin Company, All rights reserved. Accumulation Function: Exercise 8.5 #9 The rate of change of product sales for an athletic store can be modeled by T'(x) = 23.944 cos(0.987x -1.276) thousand dollars per month where x = 1 in January, x = 2 in February, etc. Find a model for sales in month t if January sales are $54,000

21. Copyright © by Houghton Mifflin Company, All rights reserved. Accumulation Function: Exercise 8.5 #9