1. The Derivative and the Tangent Line Problem
12.3

2. 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)

3. Animated Tangent

4. Slope of Line Tangent to a Curve
Approximated by secants
two points of intersection
Let second point get closer and closer to desired point of tangency • • •

5. Animated Secant Line

6. Slope of Line Tangent to a Curve
How do you find the slope of a line?
What does this have to do with the difference quotient? • •

7. The Slope Is a Limit Consider f(x) = x3 Find the tangent at x= 2
Now finish …

9. 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

10. 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 function difq(2, .001)
Use some small value for dx
Result is close to actual slope

11. 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 !

12. Finding the Derivative
We will (for now) manipulate the difference quotient algebraically
View end result of the limit
Note possible use of calculator limit ((f(x + dx) – f(x)) /dx, dx, 0)

13. 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

14. To actually find f(x), we need a specific point it contains
Using the Derivative
Consider that you are given the graph of the derivative …
What might the slope of the original function 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

15. Derivative Notation For the function y = f(x)
Derivative may be expressed as …

16. Assignment
Lesson 3.1
Page 123
Exercises: 1 – 41 EOO, 63, 65

17. Slope of Line Tangent to a Curve
Approximated by secants
two points of intersection
Let second point get closer and closer to desired point of tangency • • •
View spreadsheet simulation
Geogebra Demo

18. Definition of a Tangent
Let Δx shrink from the left

19. Definition of a Tangent
Let Δx shrink from the right