Calculus II (MAT 146) Dr. Day Wednesday, Oct 11, 2017 Probability (Sec 8.5) Differential Equations (Chapter 9) Friday: Test #2, STV 308 Help Session: Thursday, 7:15 p – 8:15 pm STV 347A Wednesday, October 11, 2017 MAT 146

A Discrete Probability Distribution: Rolling a Die Wednesday, October 11, 2017 MAT 146

Continuous Random Variables Wednesday, October 11, 2017 MAT 146

Continuous Random Variables Wednesday, October 11, 2017 MAT 146

Continuous Random Variables Wednesday, October 11, 2017 MAT 146

Probability Density Functions (PDFs) Wednesday, October 11, 2017 MAT 146

Probability Representations Wednesday, October 11, 2017 MAT 146

Probability Representations Wednesday, October 11, 2017 MAT 146

Wednesday, October 11, 2017 MAT 146

Wednesday, October 11, 2017 MAT 146

Wednesday, October 11, 2017 MAT 146

Wednesday, October 11, 2017 MAT 146

Wednesday, October 11, 2017 MAT 146

(A) What Does it Mean for Something to be a Solution to an Equation? (i) Is x = −1/5 a solution to the equation 4x + 7 = 2x – 3 ? (ii) State all solutions to the equation x2 – x = 12. (iii) Is x = 4 a solution to the equation 3ex = 12 ? (B) What Information About a Function is Revealed Through Its First Derivative? g’(x) = –8 –6x2. Gordon looked at the derivative of g(x) and shouted, “The function g is always decreasing!” How did he know? Wednesday, October 11, 2017 MAT 146

What is a Differential Equation?   A differential equation is an equation that contains one or more derivatives. Here’s a differential equation you have already solved: y’ = 2x What is the solution of this differential equation? Wednesday, October 11, 2017 MAT 146

What is a Solution to a Differential Equation?   A general solution to a differential equation is a family of functions that satisfies a given differential equation. A particular solution to a differential equation (also called the solution to an initial-value problem) is a particular function that satisfies both a given differential equation and some specified ordered pair for the function. Wednesday, October 11, 2017 MAT 146

DE Questions (1) For the differential equation y’ + 2y = 2ex, Leonard claims that the following function is a solution. Describe and illustrate at least two different ways we can verify or refute Leonard’s claim. (2) Repeat problem (1) for the following differential equation and the proposed solution. Wednesday, October 11, 2017 MAT 146

DE Questions (3) Population growth for a certain organism is modeled by this differential equation, with P measured in millions, t in years: (a) For what values of the population P will the population be growing? (b) For what values of the population P will the population be diminishing? (a) What, if any, are the equilibrium values of the population? Wednesday, October 11, 2017 MAT 146

DE Questions (4) Solve this initial-value problem: Wednesday, October 11, 2017 MAT 146