Seating by Group Monday, Nov 7 MAT 146

Calculus II (MAT 146) Dr. Day Monday, November 7, 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) 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 Monday, Nov 7 MAT 146

Population Growth Suppose a population increases by 3% each year and that there are P=100 organisms initially present (at t=0). Write a differential equation to describe this population growth and then solve for P. Monday, Nov 7 MAT 146

Exponential Decay The radioactive isotope Carbon-14 exhibits exponential decay. That is, the rate of change of the amount present (C) with respect to time (t) is proportional to the amount present (C). Carbon-14 has a half-life of 5730 years Write and solve a differential equation to determine the function C(t) to represent the amount, C, of carbon-14 present, with respect to time (t in years), if we know that 20 grams were present initially. Use C(t) to determine the amount present after 250 years. Monday, Nov 7 MAT 146

Applications! Monday, Nov 7 MAT 146

Applications! Monday, Nov 7 MAT 146

Applications! Mixtures A tank contains 2000 L of brine with 30 kg of dissolved salt. A solution enters the tank at a rate of 20 L/min with 0.25 kg of salt per L . The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after t minutes? After 60 minutes? Monday, Nov 7 MAT 146

Applications! Mixtures A tank contains 1000 L of brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after t minutes? How much salt is in the tank after 20 minutes? How many minutes must pass before there are less than 3 kg of salt in the tank? Monday, Nov 7 MAT 146

Application: Ice Growth Details . . . details . . . details! https://plus.maths.org/content/teacher-package-differential-equations Monday, Nov 7 MAT 146

Applications! Spreading a Rumor: Suppose that y represents the number of people that know a rumor at time t and that there are M people in the population. For these parameters, one model for the spread of the rumor is that “the rate at which the rumor is spread is proportional to the product of those who have heard the rumor and those who have not heard it.” Monday, Nov 7 MAT 146

Xeno’s Paradox I’m 20 feet from the door. I walk half the distance to the door. I stop and record the distance I walked. I now repeat this action from my new starting point. Distances Walked: Running Sums of Distances Walked: Total Distance Walked: Monday, Nov 7 MAT 146

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! Monday, Nov 7 MAT 146

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: Monday, Nov 7 MAT 146

Monday, Nov 7 MAT 146

Monday, Nov 7 MAT 146

Monday, Nov 7 MAT 146

Polynomial Approximators Goal: Generate polynomial functions to approximate other functions near particular values of x. Create a third-degree polynomial approximator for Monday, Nov 7 MAT 146

Create a 3rd-degree polynomial approximator for Monday, Nov 7 MAT 146

Some Sequence Calculations If an = 2n−1, list the first three terms of the sequence. The first five terms of a sequence bn are 1, 8, 27, 64, and 125. Create a rule for the sequence, assuming this pattern continues. For the sequence cn = (3n−2)/(n+3) : i) List the first four terms. ii) Are the terms of cn getting larger? Getting smaller? Explain. iii) As n grows large, does cn have a limit? If yes, what is it? If no, why not? Repeat (3) for this sequence: Give an example of L’Hôspital’s Rule in action. Monday, Nov 7 MAT 146

Some Sequence Calculations If an = 2n−1, list the first three terms of the sequence: {1,3,5} The first five terms of a sequence bn are 1, 8, 27, 64, and 125. Create a rule for the sequence, assuming this pattern continues. bn = n3 For the sequence cn = (3n−2)/(n+3) : List the first four terms: 1/4 , 4/5 , 7/6 , 10/7 ii) Are the terms of cn getting larger? Getting smaller? Explain. iii) As n grows large, does cn have a limit? If yes, what is it? If no, why not? Repeat (3) for this sequence: Give an example of L’Hôspital’s Rule in action. Monday, Nov 7 MAT 146

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? Monday, Nov 7 MAT 146

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}. Monday, Nov 7 MAT 146

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. Monday, Nov 7 MAT 146

Notable Series A geometric series is created from a sequence whose successive terms have a common ratio. When will a geometric series converge? Monday, Nov 7 MAT 146

Notable Series The harmonic series is the sum of all possible unit fractions. Monday, Nov 7 MAT 146

Notable Series A telescoping sum can be compressed into just a few terms. Monday, Nov 7 MAT 146

Fact or Fiction? Monday, Nov 7 MAT 146