antiderivatives, differential equations, and slope fields.ppt
TRANSCRIPT
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7/27/2019 Antiderivatives, Differential Equations, and Slope Fields.ppt
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AP Calculus AB
Antiderivatives,
Differential Equations,and Slope Fields
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Solution
Review
Consider the equation2xy
2xdy
dx
Find
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Antiderivatives
What is an inverse operation?
Examples include:
Addition and subtraction
Multiplication and division
Exponents and logarithms
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Antiderivatives
Differentiation also has an inverse
antidefferentiation
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Antiderivatives
Consider the function whose derivative is givenby .
What is ?
F 45xxf
xF
xF xf
Solution
We say that is an antiderivative of .
5
F x x
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Antiderivatives
Notice that we say is an antiderivative andnot the antiderivative. Why?
Since is an antiderivative of , we cansay that .
If and , find
and .
xF
xF xf xfxF '
3
5 xxG 25 xxH
xg xh
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Differential Equations
Recall the earlier equation .
This is called a differential equation and could
also be written as .
We can think of solving a differential equation
as being similar to solving any other equation.
dxdy x2
xdxdy 2
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Differential Equations
Trying to find y as a function of x
Can only find indefinite solutions
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Differential Equations
There are two basic steps to follow:
1. Isolate the differential
2. Invert both sidesin other words, find
the antiderivative
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Differential Equations
Since we are only finding indefinitesolutions, we must indicate the ambiguityof the constant.
Normally, this is done through using aletter to represent any constant.Generally, we use C.
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Solution
Differential Equations
Solvedx
dyx2
Cxy
2
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Slope Fields
Consider the following:
HippoCampus
http://www.hippocampus.org/?course=10http://www.hippocampus.org/?course=10 -
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Slope Fields
Aslope fieldshows the general flow of adifferential equations solution.
Often, slope fields are used in lieu ofactually solving differential equations.
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Slope Fields
To construct a slope field, start with adifferential equation. For simplicitys sake welluse Slope Fields
Rather than solving the differential equation,well construct a slope field
Pick points in the coordinate plane
Plug in the x and y values The result is the slope of the tangent line at that
point
xdxdy 2
http://www.ltcconline.net/greenl/courses/106/ApproxOther/slopeFields.htmhttp://www.ltcconline.net/greenl/courses/106/ApproxOther/slopeFields.htm -
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Slope Fields
Notice that since there is no y in our equation,horizontal rows all contain parallel segments.The same would be true for vertical columns if
there were no x.
Construct a slope field for .yxdx
dy