Differential Equations – Separation of VariablesIn order to solve a differential equation, one technique to help integrate both sides is by separation of variables.*Separate the variables but do not integrate

Find the general solution to,

Find the equation of the curve that passes through the point and has a slope of

Whiteboards:

Solve x dy

dx=2 ( y−4 )

with initial condition y (1 )=6

1998: 21 2003: 19 2008: 22,23 2012: 23,25 2013: 25 2014: 18 2015: 18,24 2016:14,22,23

Test Yourself:

When the rate of change of y is proportional to y:Deriving the growth/decay formula, (solve the differential equation )

The half life of plutonium PU-239 is 24,360. If 10 grams of plutonium was released during the Chernobyl nuclear accident, how long will it take to decay to only 1 gram?1) Determine points that could be used to solve the problem

2) Create a model using

3) Use the model to determine the value when 1 gram is remaining.

Suppose a room is kept at a constant and a cup of coffee with temperature is placed in the room. Newton’s law of cooling dictates that the temperature will change with an exponential decay model. If the coffee takes 10 minute to cool to , how long will it take to cool to ?

*Hint: The difference in temperature implies , and

Whiteboards:

Radioactive radium has a half life of approximately 1620years. If the initial quantity is 5 grams, how much remains after 600 years?

A bottle of soda pop at room temperature (72◦F) is placed in a refrigerator where the temperature is 44◦F. After half an hour the soda pop has cooled to 61◦F. How long will it take to reach 50◦F? What equation could model this situation?

Consider the reverse scenario, the bottle of soda pop is taken out of the fridge (44◦F) and placed in the room (72◦F) and has warmed to 61◦F after half an hour. What equation could model this situation?

1998: 84 2013: 90

Test yourself:The rate of change in the number of coyotes N(t) in a population is directly proportional to 650 - N(t). There are 300 coyotes initially and 2 years later there are 500. Find the population after 3 years.

*Hint:

Slope Fields

Solve the differential equation 2dy x

dx

If we do not know an initial condition, we can still represent the graph using a method known as a slope field.

Create a slope field for 2dy x

dx

How does the slope field compare to your solution above?

If the initial condition F(0) = –4, what is the particular solution?

Where is the particular solution represented on the graph?

Draw the slope field for

41

dydx x

Draw the initial condition F(0) = 0

What about F(1) = 1?

Solve the equation for the particular solution above and state it’s domain and range.

Slope fields in implicit form:

dy x ydx

Use your graph to sketch an approximation for the

initial condition 0 0F

Use your sketch to approximate 1F

Whiteboards:

Draw the slope field for

dydx

= xy

Sketch the initial conditions for:F(4) = 5F(4) = –3 F(–4) = 3F(–4) = –5

Solve the differential equation, how does the slope field help you determine the domain and range?

2008:27 2014:28 2015: 8 2016: 81

Test yourself:

Sketch the slope field for y '=2 x

y and the solution to y (2 )=1 . What is the solution to the equation with y (2 )=1? What is the domain and range?

Match the slope fields with their differential equations.

dydx

=12

x+1 dydx

=x− y dydx

= ydydx

=− xy

Differential Equations Practice (HW Assignment – bold):

2002B – 5 2004 – 6 2004B – 5 2005 – 6 2005B – 6 2006 – 5 2006B – 5 2008 – 5 2010 – 5 2012 – 5 2014 – 6 2016 – 4

Area Between Two CurvesConsider two curves, f(x) and g(x):

Find the area between

from

Find the area between the graphs

Whiteboards:

Find the area bounded by the functions

Find the area between one intersection of

1997: 16,83 1998: 25 2008: 83 2012: 10 2014: 89 2015: 81 2016: 12

Test yourself:

Find the area between y=2 x2+10 and y=4 x+16 on [−2,5 ]

Volume: Disk and Washer Methodshttp://www.shodor.org/interactivate/activities/FunctionRevolution/

Sketch the graph If the graph were rotated about the x axis, what would be the volume of the figure produced ?

Find the volume obtained by rotating the graph about the x axis

Find the volume obtained by rotating the graph about and bounded by the line

http://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch7/LC7_2washer.htm

The Washer Method: Consider rotating the area formed between the two curves,

about the x axisWhat is the volume obtained?

A manufacturer drills a hole through the center of a metal sphere of radius 5 inches. The hole has radius 3 inches, what is the volume of the resulting metal ring?

Whiteboards:

Determine the volume of the solid obtained by rotating the region bounded by  and that lies in the first quadrant about the x-axis.

Find the volume obtained by revolving the region bounded by y=√x−1 and y= (x−1 )2 about the line y=2

Volume: The Shell MethodThe shell method is similar to the disk but the rotation of the rectangles is parallel rather than perpendicular. Horizontal Axis of Revolutions Vertical Axis of Revolution

r – h –

Find the volume of the solid of revolution formed by revolving the region bounded by and the x axis

about the y axis.

Compare the shell method with the disk method for the region formed by revolving the graph around the y axis, bounded by y = 0, x = 0, and x = 1

Here is an example where the shell method is clearly better than the disk method:

Find the volume obtained by revolving the region bounded , y = 1, x = 1 about the line x = 2.

Whiteboards

Using any method, find the volume obtained by revolving the graph about the x axis for

.

Test yourself:

Find the volume obtained by revolving the region bounded by y= (x−2 )2 and y=x about the y axis.

Volume of Solids With Known Cross Sections

THE way we find volumes: http://web.monroecc.edu/manila/webfiles/calcNSF/JavaCode/other/myXSection.htm

Find the volume of a solid which is bound by quadrant I and the function if every cross section taken perpendicular to the x-axis is a square:

A solid has its base is the region bounded by the lines x + 2y = 6, x = 0 and y = 0 and the cross sections taken perpendicular to x-axis are circles. Find the volume the solid.

A solid has its base is the region bounded by the lines x + y = 4, x = 0 and y = 0 and the cross section is perpendicular to the x-axis are equilateral triangles. Find its volume.

Find the volume of a solid which is bound by y = 5 and the function if every cross section taken perpendicular to the y-axis is a semi-circle:

Find the volume of a solid which is bound by quadrant I, y = 5, and the function if every cross section taken perpendicular to the y-axis is the leg of an isosceles right triangle:

Whiteboards:Find the volume of the figure whose base is the region in the second quadrant bounded by the line y=x−5if every cross section perpendicular to the y axis is a semi circle.

Test yourself:

Find the volume of the figure whose base is the region bounded by y=x+1 and y=x2−1if every cross section perpendicular to the x axis is a rectangle with height 5.

Volume Practice (HW Assignment – bold):1997: 23,84 1998: 86 2003: 86 2012: 92 2013: 89 2014: 79 2015: 21 2016: 782001 – 1 2002B – 1 2003 – 1 2004 – 2 2005 – 1 2005B – 1 2006 – 1 2007 – 12007B – 1 2008 – 1 2008B – 1 2009 – 4 2009B – 4 2010 – 4 2010B – 1 2012 – 22013 – 5 2014 – 2