1. Calculus II (MAT 146) Dr. Day Thursday, Dec 8, 2016
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)
Semester Exam
Wednesday, Dec 14 (STV 229)
10 am – 12:40 pm
Review Session
Monday, Dec 12, 5:45-6:45 pm (STV 346)
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?
Determining Series Convergence
Divergence Test
Integral Test and p-Series Test
Comparison Tests
Alternating Series Test
Ratio Test
Nth-Root Test
Power Series
Intervals and Radii of Convergence
New Functions from Old
Taylor Series and Maclaurin Series
Thursday, Dec 8, 2016
MAT 146

2. Absolute Convergence and Conditional Convergence
Thursday, Dec 8, 2016
MAT 146

3. Thursday, Dec 8, 2016
MAT 146

4. Power Series Convergence
For what values of x does each series converge?
Determine the Radius of Convergence and the Interval of Convergence for each power series.
Thursday, Dec 8, 2016
MAT 146

5. 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.
Thursday, Dec 8, 2016
MAT 146

6. Power Series Convergence
For what values of x does this series converge?
Determine its Radius of Convergence and its Interval of Convergence.
Thursday, Dec 8, 2016
MAT 146

7. Power Series Convergence
For what values of x does this series converge?
Determine its Radius of Convergence and its Interval of Convergence.
Thursday, Dec 8, 2016
MAT 146

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.
Thursday, Dec 8, 2016
MAT 146

9. Power Series Convergence
For what values of x does this series converge?
Determine its Radius of Convergence and its Interval of Convergence.
Thursday, Dec 8, 2016
MAT 146

10. Power Series Convergence
For what values of x does this series converge?
Determine its Radius of Convergence and its Interval of Convergence.
Thursday, Dec 8, 2016
MAT 146

11. 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:
Thursday, Dec 8, 2016
MAT 146

12. Geometric Power Series
Thursday, Dec 8, 2016
MAT 146

13. Geometric Power Series
Thursday, Dec 8, 2016
MAT 146

14. Geometric Power Series
Thursday, Dec 8, 2016
MAT 146

15. 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!
Thursday, Dec 8, 2016
MAT 146

16. 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:
Thursday, Dec 8, 2016
MAT 146

17. Thursday, Dec 8, 2016
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18. Thursday, Dec 8, 2016
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19. Thursday, Dec 8, 2016
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20. Polynomial Approximators
Goal: Generate polynomial functions to approximate other functions near particular values of x.
Create a third-degree polynomial approximator for
Thursday, Dec 8, 2016
MAT 146

21. Create a 3rd-degree polynomial approximator for
Thursday, Dec 8, 2016
MAT 146