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

Angle is in radians and measured counterclockwise from 0 radians.

Coordinates are always fractions of 1; 1 is the radius of the unit circle

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

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 ofeach asymptote• X=0, y=?; x=-4, y=?; x=4, y=?• (0, -4/9) negative; (-4, 4/5) positive; (4, 4/5)

positive

Solution 1

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

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

0y

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

126,605,292,300 DONKEYS!!!

Practice Problem 3

• Find the average value of f(x) on the interval [7,20]

• Given:

20

17)(

3

3

12)(

7

dxxf

dxxf

Practice Problem 3

20

7

20

3

7

3

)()()( dxxfdxxfdxxf

• Find the value of the integral of f(x) from 7 to 20 in terms of known values

20

3

3

20

)()( dxxfdxxf

Practice Problem 3

20

7

3

20

7

3

)()()( dxxfdxxfdxxf

• Find the value of the integral of f(x) from 7 to 20 in terms of known values

=-(-17)-(-12)=29

Practice Problem 3

• Use average value formula:

• Plug in variables:

• Now, solve:

=1/13(29)=29/13

20

7

)(720

1dxxf

b

a

dxxfab

)(1

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

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

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

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: