1. Calculus II (MAT 146) Dr. Day Wednesday November 29, 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
Wednesday, November 29, 2017

2. 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.
Wednesday, November 29, 2017

3. 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:
Wednesday, November 29, 2017

4. 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.
Wednesday, November 29, 2017

5. 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.
Wednesday, November 29, 2017

6. Power Series Convergence
For what values of x does this series converge?
Determine its Radius of Convergence and its Interval of Convergence.
Wednesday, November 29, 2017

7. Power Series Convergence
For what values of x does this series converge?
Determine its Radius of Convergence and its Interval of Convergence.
Wednesday, November 29, 2017

8. 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.
Wednesday, November 29, 2017

9. 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.
Wednesday, November 29, 2017

10. Power Series Convergence
For what values of x does this series converge?
Determine its Radius of Convergence and its Interval of Convergence.
Wednesday, November 29, 2017

11. Power Series Convergence
For what values of x does this series converge?
Determine its Radius of Convergence and its Interval of Convergence.
Wednesday, November 29, 2017

12. 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:
Wednesday, November 29, 2017

13. Geometric Power Series
Wednesday, November 29, 2017

14. Geometric Power Series
Wednesday, November 29, 2017

15. Geometric Power Series
Wednesday, November 29, 2017

16. 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!
Wednesday, November 29, 2017

17. 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:
Wednesday, November 29, 2017

18. Wednesday, November 29, 2017

19. Wednesday, November 29, 2017

20. Wednesday, November 29, 2017

21. Polynomial Approximators
Goal: Generate polynomial functions to approximate other functions near particular values of x.
Create a third-degree polynomial approximator for
Wednesday, November 29, 2017

22. Create a 3rd-degree polynomial approximator for
Wednesday, November 29, 2017

23. 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!
Wednesday, November 29, 2017

24. Taylor Series Demo #1 Taylor Series Demo #2 Taylor Series Demo #3
Wednesday, November 29, 2017

25. 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.
Wednesday, November 29, 2017

26. 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?
Wednesday, November 29, 2017

27. General Form: Coefficients cn
Wednesday, November 29, 2017

28. Examples: Determining the cn
f(x) = cos(x), centered around a = 0.
Wednesday, November 29, 2017

29. Examples: Determining the cn
f(x) = sin(x), centered around a = 0.
Wednesday, November 29, 2017

30. Examples: Determining the cn
f(x) = ln(1-x), centered around a = 0.
Wednesday, November 29, 2017