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Applied Calculus (MAT 121) Dr. Day Thursday Jan 26, 2012

Applied Calculus (MAT 121) Dr. Day Thursday Jan 26, 2012. Highlights and Questions from Last Class: Linear Relationships (1.4) Two-Point Slopes to One-Point Slopes: Extending the Concept of Rate of Change (2.6) Assignments. Linear Functions (1.4). Characteristics of linear functions

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Applied Calculus (MAT 121) Dr. Day Thursday Jan 26, 2012

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  1. Applied Calculus (MAT 121)Dr. Day Thursday Jan 26, 2012 • Highlights and Questions from Last Class: Linear Relationships (1.4) • Two-Point Slopes to One-Point Slopes: Extending the Concept of Rate of Change (2.6) • Assignments MAT 121

  2. Linear Functions (1.4) • Characteristics of linear functions • Representations for linear relationships • Writing symbolic representations for linear relationships • Applications MAT 121

  3. Two-Point Slope to One-Point Slope (2.6) • Determining Slope for a Function: Average Rate of Change • Applying the Idea of a Limit to the Slope • What Do We Get? Instantaneous Rate of Change • Practice the Algebra! MAT 121

  4. Two-Point Slope to One-Point Slope (2.6) Determining Slope for a Function (Average Rate of Change) MAT 121

  5. Two-Point Slope to One-Point Slope (2.6) Applying the Idea of a Limit to the Slope • The LIMIT QUESTION: What happens to the outputs as the inputs get closer and closer to some value? Example: f(x) = x2: As the input values get closer and closer to 5, do the corresponding output values seem to approach some value? MAT 121

  6. Two-Point Slope to One-Point Slope (2.6) The LIMIT QUESTION: What happens to the outputs as the inputs get closer and closer to some value? • Example: f(x) = x2: As the input values get closer and closer to 5, do the corresponding output values seem to approach some value? The table and graph seem to indicate that: As x approaches 5, x2 approaches 25. x5 leads to x225 MAT 121

  7. Two-Point Slope to One-Point Slope (2.6) Now ask this same limit question for the slope calculation we made: As the distance between the two x values decreases, do the calculated slopes approach some value? We’ll use the variable h to describe the distance between the two x values, so our slope calculation is: And we now look at what happens as h—the distance between the x values—gets closer and closer to 0. As h approaches 0, the slope approaches 2. h0 leads to slope2 MAT 121

  8. Two-Point Slope to One-Point Slope (2.6) • What Do We Get? Instantaneous Rate of Change • And if we start this process at any old point in the domain, represented by x, for any old function f, we get a more general result: • This is called the derivative of f, symbolized by f’(x), read “f prime atx.” MAT 121

  9. Two-Point Slope to One-Point Slope (2.6) Practice the Algebra! Show how to determine f’(x) for f(x) = x2. • Show the definition of derivative. • Use the given function rule in that derivative definition. • Simplify! Look for opportunity to reduce h/h. • Label and restate your response. MAT 121

  10. WebLinks: Developing the Concept of the Derivative • http://www.sosmath.com/calculus/diff/der00/der00.htmlhttp://www.youtube.com/watch?v=vzDYOHETFlohttp://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDerivative.aspxhttp://www.math.hmc.edu/calculus/tutorials/limit_definition/http://calculusapplets.com/derivfunc.htmlhttp://www.univie.ac.at/future.media/moe/galerie/diff1/diff1.htmlhttp://www.calculus-help.com/tutorialshttp://www.slu.edu/classes/maymk/GeoGebra/SecantToTangent.html?FunctionField=f%28x%29+%3D+x^2 MAT 121

  11. Assignments WebAssign • 1.4 due tonight • Quiz #1 due Sunday night (WA tasks through section 1.4) • 2.6 due Monday night Test #1: Wednesday, Feb 1 MAT 121

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