Calculus II (MAT 146) Dr. Day Wednesday, February 28, 2018 Methods of Integration: Improper Integrals (7.8) Return to Applications Using a Definite Integral as a Summing Tool Arc Length (8.1) Probability (8.5) Wednesday, February 28, 2018 MAT 146

Arc Length: The length of a curve between two points on the curve Wednesday, February 28, 2018 MAT 146

Arc Length: How Can We Use the Sum of an Infinite Number of Lengths? Wednesday, February 28, 2018 MAT 146

Arc Length: The length of a typical segment Wednesday, February 28, 2018 MAT 146

Probability Write your own brief definition for probability. Include a practical example to illustrate a probability calculation consistent with the definition you just wrote. Wednesday, February 28, 2018 MAT 146

A Discrete Probability Distribution: Rolling a Die Wednesday, February 28, 2018 MAT 146

Continuous Random Variables Wednesday, February 28, 2018 MAT 146

Continuous Random Variables Wednesday, February 28, 2018 MAT 146

Continuous Random Variables Wednesday, February 28, 2018 MAT 146

Probability Density Functions (PDFs) Wednesday, February 28, 2018 MAT 146

Probability Representations Wednesday, February 28, 2018 MAT 146

Probability Representations Wednesday, February 28, 2018 MAT 146

Wednesday, February 28, 2018 MAT 146

Wednesday, February 28, 2018 MAT 146

Wednesday, February 28, 2018 MAT 146

Wednesday, February 28, 2018 MAT 146

Wednesday, February 28, 2018 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, February 28, 2018 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, February 28, 2018 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, February 28, 2018 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, February 28, 2018 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, February 28, 2018 MAT 146

DE Questions (4) Solve this initial-value problem: Wednesday, February 28, 2018 MAT 146