Lesson3.1 The Derivative And The Tangent Line
Upcoming SlideShare
Loading in...5
×
 

Lesson3.1 The Derivative And The Tangent Line

on

  • 1,183 views

 

Statistics

Views

Total Views
1,183
Views on SlideShare
1,183
Embed Views
0

Actions

Likes
0
Downloads
7
Comments
0

0 Embeds 0

No embeds

Accessibility

Categories

Upload Details

Uploaded via as Microsoft PowerPoint

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

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