1. Aim: What do slope, tangent and the derivative have to do with each other?
Do Now: What is the equation of the line tangent to the circle at point (7, 8)?

2. A tangent to a circle is a line in the plane of
Tangents & Secants
A tangent to a circle is a line in the plane of
the circle that intersects the circle in exactly
one point.
A secant of a circle is a line that intersects the
circle in two points. B C

3. Tan  1 y radius = 1 center at (0,0) (x,y) cos , sin  -1 sin  1 -1

4. slope is falling: m is (-)
Tangents to a Graph
(x3, y3)
slope is level: m = 0
(x2, y2)
slope is falling: m is (-)
(x4, y4)
(x1, y1)
slope is steep!
Unlike a tangent to a circle, tangent lines of curves can intersect the graph at more than one point.

5. Finding the Slope (tangent) of a Graph at a Point2
(1, 1) 1
This is an approximation. How can we be sure this line is really tangent to f(x) at (1, 1)?

6. Slope and the Limit Process
A more precise method for finding the slope of the tangent through (x, f(x)) employs use of the secant line.
(x + h,
f(x + h))
f(x + h) – f(x)
f(x + h) – f(x) is the change in y
x, f(x) h
h is the change in x
If h is change in x, what are coordinates of second point?
What is expression for change in y?
Difference Quotient!
How can the approximation get better?
This is a very rough approximation of the slope of the tangent at the point (x, f(x)).

7. Slope and the Limit Process
h is the change in x
f(x + h) – f(x) is the change in y
(x + h,
f(x + h))
f(x + h) – f(x)
x, f(x)
As (x + h, f(x + h)) moves down the curve and gets closer to (x, f(x)), the slope of the secant more approximates the slope of the tangent at (x, f(x). h

8. Slope and the Limit Process
h is the change in x
f(x + h) – f(x) is the change in y
(x + h,
f(x + h))
What is happening to h, the change in x?
x, f(x)
f(x + h) – f(x) h
It’s approaching 0,
or its limit at x as h approaches 0.

9. Slope and the Limit Process
As h  0, the slope of the secant, which approximates the slope of the tangent at (x, f(x)) more closely as (x + h, f(x + h)) moved down the curve. At reaching its limit, the slope of the secant equaled the slope of the tangent at (x, f(x)).

10. Definition of slope of a Graph
The slope m of the graph of f at the point (x, f(x)) , is equal to the slope of its tangent line at (x, f(x)), and is given by
provided this limit exists.
difference quotient

11. Find the slope of the graph f(x) = x2 at the point (-2, 4).
Model Problem
Find the slope of the graph f(x) = x2 at the point (-2, 4).
set up difference quotient
Use f(x) = x2
Expand
Simplify
Factor and divide out
Simplify
Evaluate the limit

12. Slope at Specific Point vs. Formula
What is the difference between the following two versions of the difference quotient?
(1) Produces a formula for finding the slope of any point on the function.
(2) Finds the slope of the graph for the specific coordinate (c, f(c)).

13. Definition of the Derivative
The function found by evaluating the limit of the difference quotient is called the derivative of f at x. It is denoted by f ’(x), which is read “f prime of x”.
The derivative of f at x is
provided this limit exists.
The derivative f’(x) is a formula for the slope of the tangent line to the graph of f at the point (x,f(x)).

14. Find the derivative of f(x) = 3x2 – 2x.
Finding a Derivative
Find the derivative of f(x) = 3x2 – 2x.
factor out h

15. Aim: What is the connection between differentiability and continuity?
Do Now:
Find the equation of the line tangent to

16. Differentiability and Continuity
What is the relationship, if any, between differentiability and continuity?
f(x) is a continuous function
(x, f(x))
f(x) – f(c)
x – c x c
(c, f(c))
Is there a limit as x approaches c?
YES
alternative form of derivative

17. Differentiability and Continuity
Is this step function differentiable at x = 1?
By definition the derivative is a limit. If there is no limit at x = c, then the function is not differentiable at x = c.
Does this step function, the greatest integer function, have a limit at 1?
NO: f(x) approaches a different number from the right side of 1 than it does from the left side.

18. Differentiability and Continuity
If f is differentiable at x = c, then f is continuous at x = c. NO
Is the Converse true?
If f is continuous at x = c, then f is differentiable at x = c.

19. Graphs with Sharp Turns – Differentiable?
alternative form of derivative
f(x) = |x – 2|
m = 1
m = -1
Is this function continuous at 2?
YES
Is this function differentiable at 2?
One-sided limits are not equal, f is therefore not differentiable at 2. There is no tangent line at (2, 0)

20. Graph with a Vertical Tangent Line
f(x) = x1/3
Is f continuous at 0?
YES
Does a limit exist at 0? NO
f is not differentiable at 0; slope of vertical line is undefined.

21. Differentiability Implies Continuity
corner
vertical tangent a b c d
f is not continuous at a therefore not differentiable
f is continuous at b & c, but not differentiable
f is continuous at d and differentiable

22. Summary 1. If a function is differentiable at x = c,
then it is continuous at x = c.
Thus, differentiability implies continuity.
2. It is possible for a function to be continuous at x = c
and not be differentiable at x = c.
Thus continuity does not imply differentiability.