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Calculus I (MAT 145) Dr. Day Friday Feb 8, 2013

Calculus I (MAT 145) Dr. Day Friday Feb 8, 2013. Continuity of a Function (2.5) Slope at a Point: The Derivative (2.7) Assignments. Approachment. What patterns of approachment in outputs are apparent when the inputs approach some value?

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Calculus I (MAT 145) Dr. Day Friday Feb 8, 2013

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  1. Calculus I (MAT 145)Dr. Day Friday Feb 8, 2013 • Continuity of a Function (2.5) • Slope at a Point: The Derivative (2.7) • Assignments MAT 145

  2. Approachment • What patterns of approachment in outputs are apparent when the inputs approach some value? • Reminder! We’re not concerned with behavior AT the specific point, only the behavior that’s apparent as we approach it! • Numerical, Graphical, and Symbolic Perspectives • One-Sided and Two-Sided Limits MAT 145

  3. MAT 145

  4. What Makes a Function Continuous? (PIP pp 18-19) Informal Perspective Formal Definition One-Sided Continuity Continuity on an Interval Connections: Limits and Continuity MAT 145

  5. MAT 145

  6. MAT 145

  7. The Derivative at a Point (2.7) • Return to the task we posed at the start of the chapter: For y=f(x)=√(4x), what is the slope of the tangent line to the graph at the point P=(1,2)? • Use limits and our 2-point slope definition to create a 1-point slope calculator. This is called the derivative of the function at the point P. Tells us: • Slope of the line tangent to f at P • Slope of the curve y =f(x) at P • The instantaneous rate of change of f at P. • Try this with two other functions, point specified. MAT 145

  8. The Derivative at a Point (2.7) • Focus on Graphical Interpretation (PIP p. 21) • Contextual situations involving instantaneous rate of change (PIP p. 21) • More graphical interpretations (PIP p. 22) MAT 145

  9. MAT 145

  10. MAT 145 Exercise 2.7.11, p. 151

  11. MAT 145 Fig. 2.8.1, p. 154

  12. The Derivative as a Function (2.8) • Revising our 1-point slope (derivative) definition • Will a derivative always exist? Why not? • Derivative notation. • Beyond the first derivative. MAT 145

  13. From velocity: What can we learn about position? MAT 145

  14. Assignments • WebAssign • 2.6 due tonight • 2.5 due Monday night • 2.7 due Wednesday night • 2.8 due Thursday night Take-Home Assignment #2: Out today, due on Monday Test #2: Friday, February 15 MAT 145

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