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Section One

Section One. Precalc Review, Average Value, Exponential Growth & Decay, and Slope Fields. By: RE, Rusty, Matthew, and D Money. Unit Circle. Geometrical figure relating trig functions to specific angles and coordinates. Unit Circle.

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Section One

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  1. Section One Precalc Review, Average Value, Exponential Growth & Decay, and Slope Fields By: RE, Rusty, Matthew, and D Money

  2. Unit Circle • Geometrical figure relating trig functions to specific angles and coordinates

  3. Unit Circle Coordinates are always fractions of 1; 1 is the radius of the unit circle Angle is in radians and measured counterclockwise from 0 radians.

  4. Practice Problem 1 • Graph the following equation using only skills from pre-calc: • Y=4/(x^2-9) • Vertical asymptotes are found by setting denominator equal to 0, then solving for x • (x^2-9)=0 • (x+3)(x-3)=0 • X=3,-3

  5. Practice Problem 1 • Y=4/(x^2-9) • Vertical asymptotes at x=3,-3 • Parent graph of 1/(x^2)  • Stretched vertically fourfold • Test values on either side of each asymptote • X=0, y=?; x=-4, y=?; x=4, y=? • (0, -4/9) negative; (-4, 4/5) positive; (4, 4/5) positive

  6. Solution 1

  7. Practice Problem 2 • Ratio of donkeys to people in Kazakhstan – 1:3 • Donkey population of Kazakhstan in 1900 – 7 • Growth rate of people in Kazakhstan - .23857 • How many donkeys were in Kazakhstan in 1999? • Assume that the donkey-people ratio remains constant, as it generally does

  8. Practice Problem 2 • Determine human population of Kazakhstan in 1900 • Donkeys/People=1/3 • 7/P=1/3 • P=21 people • Use growth equation with variables • y=Ye^kt • y is people in 1999; Y is people in 1900; k is growth rate of people; t is time in years

  9. Practice Problem 2 • y=(21)e^(.23857*99) • y=379,815,876,900 people • They reproduce rather quickly • How many donkeys? • Donkeys/People=1/3 • D/379,815,876,900=1/3 • D=126,605,292,300 donkeys in Kazakhstan in 1999

  10. 126,605,292,300 DONKEYS!!!

  11. Practice Problem 3 • Find the average value of f(x) on the interval [7,20] • Given:

  12. Practice Problem 3 • Find the value of the integral of f(x) from 7 to 20 in terms of known values

  13. Practice Problem 3 • Find the value of the integral of f(x) from 7 to 20 in terms of known values • =-(-17)-(-12)=29

  14. Practice Problem 3 • Use average value formula: • Plug in variables: • Now, solve: =1/13(29)=29/13

  15. Practice Problem 4 • Graph the slope field of the differential equation dy/dx=y-x • Make a table of values • X: 0 0 0 0 0 0 0 • Y: 0 1 2 3 -1 -2 -3 • Dy/dx: 0 1 2 3 -1 -2 -3

  16. Practice Problem 4 • X: 1 1 1 1 1 1 1 • Y: 0 1 2 3 -1 -2 -3 • Dy/dx: -1 0 1 2 -2 -3 -4 • X: 2 2 2 2 2 2 2 • Y: 0 1 2 3 -1 -2 -3 • Dy/dx: -2 -1 0 1 -3 -4 -5 • X: 3 3 3 3 3 3 3 • Y: 0 1 2 3 -1 -2 -3 • Dy/dx: -3 -2 -1 0 -4 -5 -6

  17. Practice Problem 4 • X: -1 -1 -1 -1 -1 -1 -1 • Y: 0 1 2 3 -1 -2 -3 • Dy/dx: 1 2 3 4 0 -1 -2 • X: -2 -2 -2 -2 -2 -2 -2 • Y: 0 1 2 3 -1 -2 -3 • Dy/dx: 2 3 4 5 1 0 -1 • X: -3 -3 -3 -3 -3 -3 -3 • Y: 0 1 2 3 -1 -2 -3 • Dy/dx: 3 4 5 6 2 1 0

  18. Practice Problem 4 • Plotting all points with slope dy/dx on a graph with domain and range both of [-3,3] should look like this:

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