1. Calculus 2.1 Introduction to Differentiation
Mrs. Kessler

2. Rate of Change
To begin our study of rates and changes we must realize that the rate of change is the change in y divided by the change in x.
We are assuming two points
With a straight line, this has been the slope, m.
with
However, with a curve, this is a little more complicated.

3. Examples of tangents of curves
Since the curve is increasing and decreasing at different rates, we are looking for the instantaneous rate of change at a particular point.
This means we need the slope of the tangent line at the point P.

4. Approximating the slope of the tangent line with the secant line
x + x

5. What happens as x gets smaller and sMALLER?

6. Definition of the Slope of a Graph

7. Definition of Derivative

8. Example 1 Find the slope of f(x) = 7x - 4 using the limit process
But you already knew that. 

9. Ex 2. Find f ′(x) if using the limit process
Notice that the slope, or derivative, has a variable.

10. Ex. 3 Find the equation of the line tangent to at (1, 2)
From before
Let’s look graphically at this:

11. Ex. 3 Find the equation of the line tangent to at (1, 2)
From before
At x =1
m = ¼ point = (1, 2) y - y1= m (x – x1)

12. Find f′(x) if using the limit process

13. Find f′(x) if using the limit process

14. Find equation of the line tangent to and parallel to y = 2x -7

15. That’s what we are looking for! Find its equation.
Find equation of the line tangent to and parallel to y = 2x -7
Original function
That’s what we are looking for! Find its equation.
Original function plus line

16. Find equation of the line tangent to and parallel to y = 2x -7

17. is the slope of the tangent line at any point x.
Continued:
is the slope of the tangent line at any point x.
We want a tangent line parallel to y = 2x – 7
We can see m = 2 in y = 2x – 7
and from the calculus, m = .
Now set the slopes equal and solve for x.
Now sub back into the original function, and find y.
y = 1/4

18. Now find the equation y – y1 =m ( x – x1)
At the point (17/16, ¼), the tangent line is parallel to
Now find the equation
y – y1 =m ( x – x1)

19. Day
Questions?????????

20. y = x2. The two tangent lines intersect at (1, -3).
Find their equations.
(x1, y1)
(1, -3)
Let the red dot be the point (x1, ,y1). By the delta process the derivative is 2x.
At the point (x1 ,y1), the derivative which is the slope = 2x1.
Now use the slope formula setting the two =.

21. The tangent points are ( 3, 9) and (-1, 1) If x =3, f(x) = y = 9
y = x2. The two tangent lines intersect at (1, -3).
Find their equations.
(x1, y1)
Cross multiply and set equal to 0. Note: y1=(x1)2
(1, -3)
The tangent points are
( 3, 9) and (-1, 1)
If x =3, f(x) = y = 9
If x = -1, f(x) = y = 1

22. Now find the equation of the line.
y = x2. The two tangent lines intersect at (1, -3).
Find their equations.
The tangent points are
( 3, 9) and (-1, 1)
Now find the equation of the line.
y - y1= m ( x – x1)
(x1, y1)
(1, -3)
For first equation use ( 3, 9) and (1, -3)
For second equation use ( -1, 1) and (1, -3)
y = 6x – and y = - 2x -1 Graph and check.

23. Weird Tangents - no derivative

24. Alternative form of the derivative
provided the limit exists.
exist and are equal.

25. Alternative form of the derivative
at x = 4
Since the limit from left is not equal to the limit from the right, there is no limit, which means there is no derivative at x = 4 even though the function is continuous there.

26. Continued

28. Using the TI 83/84 to draw the tangent line
Enter the function into the y -editor.
Graph the function:
y = x sin(x)
Suppose I want the tangent at x = π/3.
Press 2nd Draw 5
Type π/3
Note: Decimal place was set a 3 fixed places.