the derivative and the tangent line problem
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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). •. •.

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tangent definition
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)
slope of line tangent to a curve

Slope of Line Tangent to a Curve
  • Approximated by secants
    • two points of intersection
  • Let second point get closerand closer to desiredpoint of tangency

View spreadsheet simulation

GeogebraDemo

slope of line tangent to a curve6

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
Definition ofa Tangent
  • Let Δx shrinkfrom the left
definition of a tangent8
Definition ofa Tangent
  • Let Δx shrinkfrom the right
the slope is a limit
The Slope Is a Limit
  • Consider f(x) = x3 Find the tangent at x0= 2
  • Now finish …
calculator capabilities
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
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 functiondifq(2, .001)
  • Use some small value for dx
  • Result is close to actual slope
definition of derivative
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
Finding the Derivative
  • We will (for now) manipulate the difference quotient algebraically
  • View end result of the limit
  • Note possible use of calculatorlimit ((f(x + dx) – f(x)) /dx, dx, 0)
related line the normal
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
Using the Derivative
  • Consider that you are given the graph of the derivative …
  • What might theslope of the originalfunction look like?
  • Consider …
    • what do x-intercepts show?
    • what do max and mins show?
    • f `(x) <0 or f `(x) > 0 means what?

f \'(x)

To actually find f(x), we need a specific point it contains

derivative notation
Derivative Notation
  • For the function y = f(x)
  • Derivative may be expressed as …
assignment
Assignment
  • Lesson 3.1
  • Page 123
  • Exercises: 1 – 41 EOO, 63, 65
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