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## PowerPoint Slideshow about 'the derivative and the tangent line problem' - Thomas

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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

•

View spreadsheet simulation

GeogebraDemo

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

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