1. Calculus II (MAT 146) Dr. Day Monday November 27, 2017
Integration Applications
Area Between Curves (6.1)
Average Value of a Function (6.5)
Volumes of Solids (6.2, 6.3)
Created by Rotations
Created using Cross Sections
Arc Length of a Curve (8.1)
Probability (8.5)
Methods of Integration
U-substitution (5.5)
Integration by Parts (7.1)
Trig Integrals (7.2)
Trig Substitution (7.3)
Partial-Fraction Decomposition (7.4)
Putting it All Together: Strategies! (7.5)
Improper Integrals (7.8)
Differential Equations
What is a differential equation? (9.1)
Solving Differential Equations
Visual: Slope Fields (9.2)
Numerical: Euler’s Method (9.2)
Analytical: Separation of Variables (9.3)
Applications of Differential Equations
Infinite Sequences & Series (Ch 11)
What is a sequence? A series? (11.1,11.2)
Determining Series Convergence
Divergence Test (11.2)
Integral Test (11.3)
Comparison Tests (11.4)
Alternating Series Test (11.5)
Ratio Test (11.6)
Nth-Root Test (11.6)
Power Series
Interval & Radius of Convergence
New Functions from Old
Taylor Series and Maclaurin Series
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2. Fact or Fiction?
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4. Absolute Convergence and Conditional Convergence
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7. Converge or Diverge?
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8. Power Series
The sum of the series is a function with domain the set of all x values for which the series converges.
The function seems to be a polynomial, except it has an infinite number of terms.
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9. Power Series: Example
If we let cn = 1 for all n, we get a familiar series:
This geometric series has common ratio x and we know the series converges for |x| < 1.
We also know the sum of this series:
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10. Generalized Power Series
This is called:
a power series in (x – a), or
a power series centered at a, or
a power series about a.
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11. Power Series Convergence
For what values of x does each series converge?
Determine the Radius of Convergence and theInterval of Convergence for each power series.
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12. Power Series Convergence
For what values of x does this series converge?
Determine its Radius of Convergence and its Interval of Convergence.
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13. Power Series Convergence
For what values of x does this series converge?
Determine its Radius of Convergence and its Interval of Convergence.
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14. Power Series Convergence
For what values of x does this series converge?
Use the Ratio Test to determine values of x that result in a convergent series.
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15. Power Series Convergence
For what values of x does this series converge?
Use the Ratio Test to determine values of x that result in a convergent series.
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16. Power Series Convergence
For what values of x does this series converge?
Determine its Radius of Convergence and its Interval of Convergence.
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17. Power Series Convergence
For what values of x does this series converge?
Determine its Radius of Convergence and its Interval of Convergence.
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18. Geometric Power Series
If we let cn = 1 for all n, we get a familiar series:
This geometric series has common ratio x and we know the series converges for |x| < 1.
We also know the sum of this series:
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19. Geometric Power Series
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20. Geometric Power Series
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21. Geometric Power Series
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22. Why Study Sequences and Series in Calc II?
Taylor Polynomials applet
Infinite Process Yet Finite Outcome How Can That Be?
Transition to Proof
Re-Expression!
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23. Polynomial Approximators
Our goal is to generate polynomial functions that can be used to approximate other functions near particular values of x.
The polynomial we seek is of the following form:
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27. Polynomial Approximators
Goal: Generate polynomial functions to approximate other functions near particular values of x.
Create a third-degree polynomial approximator for
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28. Create a 3rd-degree polynomial approximator for
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29. Beyond Geometric Series Connections: Taylor Series
How can we describe the cn so a power series can represent OTHER functions? ANY functions?
Now we go way back to the ideas that motivated this chapter’s investigations and connections: Polynomial Approximators!
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30. Taylor Series Demo #1 Taylor Series Demo #2 Taylor Series Demo #3
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31. Taylor Series Example: f(x) = ex, centered around a = 0.
Look at characteristics of the function in question and connect those to the cn.
Example: f(x) = ex, centered around a = 0.
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32. Taylor Series Example: f(x) = ex, centered around a = 0.
And…how far from a = 0 can we stray and still find this re-expression useful?
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33. General Form: Coefficients cn
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34. Examples: Determining the cn
f(x) = cos(x), centered around a = 0.
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35. Examples: Determining the cn
f(x) = sin(x), centered around a = 0.
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36. Examples: Determining the cn
f(x) = ln(1-x), centered around a = 0.
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