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Assignment 4

Assignment 4. Section 3.1 The Derivative and Tangent Line Problem. The Basic Question is…. How do you find the equation of a line that is tangent to a function y=f(x) at an arbitrary point P? To find the equation of a line you need: a point and a slope.

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Assignment 4

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  1. Assignment 4 Section 3.1 The Derivative and Tangent Line Problem

  2. The Basic Question is… • How do you find the equation of a line that is tangent to a function y=f(x) at an arbitrary point P? • To find the equation of a line you need: a point and a slope

  3. How do you find the slope when the line is a tangent line?

  4. First, we approximate with the secant line.

  5. How do we make the approximation better? • Choose h smaller… • And smaller… • And smaller… • And smaller… • How close to zero can it get? • Infinitely

  6. Definition of slope of the tangent line If f(x) is defined on an open interval (a,b) then the slope of the tangent line to the graph of y=f(x) at an arbitrary point (x,f(x)) is given by:

  7. Example: • #6—Find the slope of the tangent line to the graph of the function at the given point. • (-2, -2)

  8. The limit that is the slope of the tangent line is actually much more.. • Definition of the Derivative of a Function The derivative of f at x is given by Provided the limit exists. For all x for which the limit exists, is a function of x.

  9. Notations for derivative

  10. Find the derivative by the limit process. #20 #24

  11. Find an equation of the tangent line to th graph of f at the given point. • #26 • ( - 3, 4)

  12. #34 Find an equation of the line that is tangent to the graph of f and parallel to the given line.

  13. Sketch the graph of f’ #46

  14. What destroys the derivative at a point? • Cusps • Corners • Vertical tangents

  15. And…Points of Discontinuity Fact: If a function is differentiable at x=c, then f is continuous at x=c

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