Lesson3.1 The Derivative And The Tangent Line
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Lesson3.1 The Derivative And The Tangent Line






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Lesson3.1 The Derivative And The Tangent Line Lesson3.1 The Derivative And The Tangent Line Presentation Transcript

  • The Derivative and the Tangent Line Problem Lesson 3.1
  • Definition of Tan-gent
  • Tangent Definition
    • From geometry
      • a line in the plane of a circle
      • intersects in exactly one point
    • We wish to enlarge on the idea to include tangency to any function, f(x)
    View slide
  • Slope of Line Tangent to a Curve
    • Approximated by secants
      • two points of intersection
    • Let second point get closer and closer to desired point of tangency
    • View spreadsheet simulation • • View slide
  • Animated Tangent
  • Slope of Line Tangent to a Curve
    • Recall the concept of a limit from previous chapter
    • Use the limit in this context
    • •
  • Definition of a Tangent
    • Let Δ x shrink from the left
  • Definition of a Tangent
    • Let Δ x shrink from the right
  • The Slope Is a Limit
    • Consider f(x) = x 3 Find the tangent at x 0 = 2
    • Now finish …
  • Animated Secant Line
  • Calculator Capabilities
    • Able to draw tangent line
    • Steps
    • Specify function on Y= screen
    • F5-math, A-tangent
    • Specify an x (where to place tangent line)
    • Note results
  • Difference Function
    • Creating a difference function on your calculator
      • store the desired function in f(x) x^3 -> f(x)
      • Then specify the difference function (f(x + dx) – f(x))/dx -> difq(x,dx)
      • Call the function difq(2, .001)
    • Use some small value for dx
    • Result is close to actual slope
  • Definition of Derivative
    • The derivative is the formula which gives the slope of the tangent line at any point x for f(x)
    • Note: the limit must exist
      • no hole
      • no jump
      • no pole
      • no sharp corner
    A derivative is a limit !
  • Finding the Derivative
    • We will (for now) manipulate the difference quotient algebraically
    • View end result of the limit
    • Note possible use of calculator limit ((f(x + dx) – f(x)) /dx, dx, 0)
  • Related Line – the Normal
    • The line perpendicular to the function at a point
      • called the “normal”
    • Find the slope of the function
    • Normal will have slope of negative reciprocal to tangent
    • Use y = m(x – h) + k
  • Using the Derivative
    • Consider that you are given the graph of the derivative …
    • What might the slope of the original function look like?
    • Consider …
      • what do x-intercepts show?
      • what do max and mins show?
      • f `(x) <0 or f `(x) > 0 means what?
    To actually find f(x), we need a specific point it contains f `(x)
  • Derivative Notation
    • For the function y = f(x)
    • Derivative may be expressed as …
  • Assignment
    • Lesson 3.1
    • Page 123
    • Exercises: 1 – 41, 63 – 65 odd