1. 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
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2. Arc Length: The length of a curve between two points on the curve
Wednesday, February 28, 2018
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3. Arc Length: How Can We Use the Sum of an Infinite Number of Lengths?
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4. Arc Length: The length of a typical segment
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5. 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
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6. A Discrete Probability Distribution: Rolling a Die
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7. Continuous Random Variables
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8. Continuous Random Variables
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9. Continuous Random Variables
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10. Probability Density Functions (PDFs)
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11. Probability Representations
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12. Probability Representations
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18. (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

19. 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

20. 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

21. 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

22. 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

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