The Derivative and the Tangent Line Problem

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# the derivative and the tangent line problem - PowerPoint PPT Presentation

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 theTangent Line Problem

Lesson 3.1

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
• Approximated by secants
• two points of intersection
• Let second point get closerand closer to desiredpoint of tangency

Slope of Line Tangent to a Curve
• Recall the concept of a limit from previous chapter
• Use the limit in this context

Definition ofa Tangent
• Let Δx shrinkfrom the left
Definition ofa Tangent
• Let Δx shrinkfrom the right
The Slope Is a Limit
• Consider f(x) = x3 Find the tangent at x0= 2
• Now finish …
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 functiondifq(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 calculatorlimit ((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 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
• For the function y = f(x)
• Derivative may be expressed as …
Assignment
• Lesson 3.1
• Page 123
• Exercises: 1 – 41 EOO, 63, 65