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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The derivative and the tangent line problem l.jpg

The Derivative and theTangent Line Problem

Lesson 3.1



Tangent definition l.jpg
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 l.jpg

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 l.jpg

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 l.jpg
Definition ofa Tangent

  • Let Δx shrinkfrom the left


Definition of a tangent8 l.jpg
Definition ofa Tangent

  • Let Δx shrinkfrom the right


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The Slope Is a Limit

  • Consider f(x) = x3 Find the tangent at x0= 2

  • Now finish …



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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 l.jpg
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


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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 l.jpg
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)


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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 l.jpg
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


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Derivative Notation

  • For the function y = f(x)

  • Derivative may be expressed as …


Assignment l.jpg
Assignment

  • Lesson 3.1

  • Page 123

  • Exercises: 1 – 41 EOO, 63, 65


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