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Calculus II (MAT 146) Dr. Day Wednes day March 5, 2014

Calculus II (MAT 146) Dr. Day Wednes day March 5, 2014. Test Prep: Questions for Test #2? Introduction Differential Equations (Chapter 9). Trig Integrals: Which is Easiest? Toughest?. Continuous Random Variables. Probability Density Functions (PDFs). Probability Representations.

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Calculus II (MAT 146) Dr. Day Wednes day March 5, 2014

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  1. Calculus II (MAT 146)Dr. Day Wednesday March 5, 2014 • Test Prep: Questions for Test #2? • Introduction Differential Equations (Chapter 9) MAT 146

  2. Trig Integrals:Which is Easiest? Toughest? MAT 146

  3. MAT 146

  4. ContinuousRandom Variables MAT 146

  5. Probability Density Functions (PDFs) MAT 146

  6. Probability Representations MAT 146

  7. Probability Representations MAT 146

  8. MAT 146

  9. MAT 146

  10. Assignments WebAssign • Methods of Integration Review (Ch 7) due tonight Test #2: Thursday and Friday Help Session Tonight, 6-7 pm, STV 210 MAT 146

  11. (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? (i) What does y’ reveal about the function y? Example: y’ = 6x2 –3x. (ii) Gordon looked at the derivative of g(x) and shouted, “That function is always decreasing!” How did he know? MAT 146

  12. 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? MAT 146

  13. What is a Solution to a Differential Equation? • A general solution is a family of functions that satisfies a given differential equation. • A particular solution (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. MAT 146

  14. 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. MAT 146

  15. DE Questions (3) Solve this initial-value problem: MAT 146

  16. DE Questions (4) 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? MAT 146

  17. DE Warm-Ups • For the differential equation here, what are the constant solutions? • For the following differential equation, determine whether any of the functions that follow are solutions. MAT 146

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