6.1 Differential Equations and Slope Fields

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6.1 Differential Equations and Slope Fields
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First, a little review
It doesnt matter whether the constant was 3 or
-5, since when we take the derivative the
constant disappears.
However, when we try to reverse the operation
We dont know what the constant is, so we put C
in the answer to remind us that there might have
been a constant.
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If we have some more information we can find C.
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Example

• Find the solution of the initial value problem

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Example

• Find the solution of the initial value problem

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Example

• Find the solution of the initial value problem

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Example

• Find the solution of the initial value problem

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Application

• A car starts from rest and accelerates at a rate
  of -0.6t2 4 m/s2 for 0 lt t lt 12. How long does
  it take for the car to travel 100m?

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Application

• An object is thrown up from a height of 2m at a
  speed of 10 m/s. Find its highest point and when
  it hits the ground.

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(No Transcript)
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Integrals such as are called
indefinite integrals because we can not find a
definite value for the answer.
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Indefinite Integrals

• Review the list of indefinite integrals on p. 307

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Differential Equations General Solution

• Finding the general solution of a differential
  equation means to find the indefinite integral
  (i.e. the antiderivative)

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Find the general solution
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Separation of variables

• If a differential equation has two variables it
  is separable if it is of the form

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Example
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Separation of variables
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Separation of variables
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Separation of variables
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Separation of variables
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Separation of variables
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Initial value problems and differential equations
can be illustrated with a slope field.
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Slope Field Activity

• Given
• Find the slope for your point
• Sketch a tangent segment across your point. Now
  do the same for the rest of the points
• Are you on an equilibrium solution?
• Find your isocline. Is it vertical, horizontal,
  slant, etc.
• Sketch a possible solution curve through your
  point
• Is your point an extremum or point of inflection?
  Is the graph of y increasing/decreasing, CU or
  CD?
• What is the value of d2y/dx2 at your point?

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Slope Field Activity

• Given
• Find the slope for your point
• Sketch a tangent segment across your point. Now
  do the same for the rest of the points
• Are you on an equilibrium solution?
• Find your isocline. Is it vertical, horizontal,
  slant, etc.
• Sketch a possible solution curve through your
  point
• Is your point an extremum or point of inflection?
  Is the graph of y increasing/decreasing, CU or
  CD?
• What is the value of d2y/dx2 at your point?

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Slope Field Activity

• Given
• Find the slope for your point
• Sketch a tangent segment across your point. Now
  do the same for the rest of the points
• Are you on an equilibrium solution?
• Find your isocline. Is it vertical, horizontal,
  slant, etc.
• Sketch a possible solution curve through your
  point
• Is your point an extremum or point of inflection?
  Is the graph of y increasing/decreasing, CU or
  CD?
• What is the value of d2y/dx2 at your point?

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Slope Field Activity

• Given
• Find the slope for your point
• Sketch a tangent segment across your point. Now
  do the same for the rest of the points
• Are you on an equilibrium solution?
• Find your isocline. Is it vertical, horizontal,
  slant, etc.
• Sketch a possible solution curve through your
  point
• Is your point an extremum or point of inflection?
  Is the graph of y increasing/decreasing, CU or
  CD?
• What is the value of d2y/dx2 at your point?

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Slope Field Activity

• Given
• Find the slope for your point
• Sketch a tangent segment across your point. Now
  do the same for the rest of the points
• Are you on an equilibrium solution?
• Find your isocline. Is it vertical, horizontal,
  slant, etc.
• Sketch a possible solution curve through your
  point
• Is your point an extremum or point of inflection?
  Is the graph of y increasing/decreasing, CU or
  CD?
• What is the value of d2y/dx2 at your point?

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Hw p. 312/7-17odd,31-36,39-42