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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

• 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

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

• Now finish …

• 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

• 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

• 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 !

• 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)

• 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

• 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

• For the function y = f(x)

• Derivative may be expressed as …

• Lesson 3.1

• Page 123

• Exercises: 1 – 41 EOO, 63, 65