1. Calculus II (MAT 146) Dr. Day Monday November 6, 2017
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
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
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)
Monday, November 6, 2017

2. Sequence Characteristics
Convergence/Divergence: As we look at more and more terms in the sequence, do those terms have a limit?
Increasing/Decreasing: Are the terms of the sequence growing larger, growing smaller, or neither? A sequence that is strictly increasing or strictly decreasing is called a monotonic sequence.
Boundedness: Are there values we can stipulate that describe the upper or lower limits of the sequence?
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3. What is an Infinite Series?
We start with a sequence {an}, n going from 1 to ∞, and define {si} as shown.
The {si} are called partial sums. These partial sums themselves form a sequence.
An infinite series is the summation of an infinite number of terms of the sequence {an}. It is a unique finite value, if it exists.
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4. What is an Infinite Series?
Our goal is to determine whether an infinite series converges or diverges. It must do one or the other.
If the sequence of partial sums {si} has a finite limit as n −−> ∞, we say that the infinite series converges. Otherwise, it diverges.
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5. Notable Series
A geometric series is created from a sequence whose successive terms have a common ratio. When will a geometric series converge?
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6. Notable Series
The harmonic series is the sum of all possible unit fractions.
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7. Notable Series
A telescoping sum can be compressed into just a few terms.
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8. Fact or Fiction?
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9. Our first Series Convergence Test
Our first Series Convergence Test The nth-Term Test also called The Divergence Test
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12. 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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13. 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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17. 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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18. Create a 3rd-degree polynomial approximator for
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30. Converge or Diverge?
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31. Converge or Diverge? (9)
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32. Converge or Diverge? (12)
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34. Absolute Convergence and Conditional Convergence
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