1. Section 2.1 – The Derivative and the Tangent Line Problem

2. A line that passes through two points on a curve.
Secant Line
A line that passes through two points on a curve.

3. The blue line intersects the pink curve twice. Yet it is a tangent.
Tangent Line
Most people believe that a tangent line only intersects a curve once. For instance, the first time most students see a tangent line is with a circle: Although this is true for circles, it is not true for every curve:
Every blue line intersects the pink curve only once. Yet none are tangents.
The blue line intersects the pink curve twice. Yet it is a tangent.

4. Local Linearity
If a function has a tangent line at a point, it is at least locally linear.
Tangent Line Exists.

5. Local Linearity
If a function has a tangent line at a point, it is at least locally linear.
Tangent Line Does Not Exist.
The function is NOT smooth at this point.

6. A Tangent Line Exists at Every Point.
Local Linearity
If a function has a tangent line at a point, it is at least locally linear.
A Tangent Line Exists at Every Point.
The function is Always smooth.

7. Tangent Line
As two points of a secant line are brought together, a tangent line is formed. The slope of which is the instantaneous rate of change

8. Tangent Line
As two points of a secant line are brought together, a tangent line is formed. The slope of which is the instantaneous rate of change

9. Tangent Line
As two points of a secant line are brought together, a tangent line is formed. The slope of which is the instantaneous rate of change

10. Tangent Line
As two points of a secant line are brought together, a tangent line is formed. The slope of which is the instantaneous rate of change

11. Tangent Line
As two points of a secant line are brought together, a tangent line is formed. The slope of which is the instantaneous rate of change

12. Tangent Line
As two points of a secant line are brought together, a tangent line is formed. The slope of which is the instantaneous rate of change

13. Slope of a Tangent Line
In order to find a formula for the slope of a tangent line, first look at the slope of a secant line that contains (x1,y1) and (x2,y2):
(x2,y2) Δx
(x1,y1)
In order to find the slope of the tangent line, the change in x needs to be as small as possible.

14. Instantaneous Rate of Change
If f is defined on an open interval containing c, and if the limit:
exists, then the line passing through ( c, f(c) ) with slope m is the tangent line to the graph of f at the point ( c, f(c) ).
f(x)
The blue line is a Tangent Line with Slope m that also contains the point ( c, f(c) ) m
(c, f(c) )

15. Slope can only be positive, negative, zero, or undefined.
Classifying Slope
Determine the best way to describe the slope of the tangent line at each point. A.
Since the curve is decreasing, the slope will also be decreasing. Thus, the slope is Negative. B.
The vertex is where the curve goes from increasing to decreasing. Thus, the slope must be Zero. D A C.
Since the curve is increasing, the slope will also be increasing. Thus the slope is Positive. C D.
Since the curve has a sharp turn, the tangent line will be vertical. Thus the slope is Undefined. B
Slope can only be positive, negative, zero, or undefined.

16. c is the x-coordinate of the point on the curve
Example 1
Find the instantaneous rate of change to at (3,-6).
Substitute into the function
c is the x-coordinate of the point on the curve
Simplify in order to cancel the denominator
Direct substitution

17. c is the x-coordinate of the point on the curve
Example 2
Find the equation of the tangent line to at (2,10).
Substitute into the function
Simplify in order to cancel the denominator
c is the x-coordinate of the point on the curve
Just the slope. Now use the point-slope formula to find the equation
Direct substitution

18. A Function to Describe Slope
In the preceding notes, we considered the slope of a tangent line of a function f at a number c. Now, we change our point of view and let the number c vary by replacing it with x.
A variable.
A constant.
A function whose output is the slope of a tangent line at any x.
The slope of a tangent line at the point x = c.
The slope of a tangent line or the instantaneous rate of change, will be referred to as a Derivative of a function at a value of x.
The slope function or an instantaneous rate of change function, will be referred to as the Derivative of a Function.

19. The Derivative of a Function
The limit used to define the slope of a tangent line is also used to define one of the two fundamental operations of calculus:
The derivative of f at x is given by
Provided the limit exists. For all x for which this limit exists, f ' is a function of x.
READ: “f prime of x.”
Other Notations for a Derivative:

20. Vocab
Differentiation: The process of finding the derivative of a function. Differentiable: The derivative exists. Able to be differentiated. Possess a derivative. Etc.

21. Example 1
Derive a formula for the slope of the tangent line to the graph of
Multiply by a common denominator
Simplify in order to cancel the denominator
Substitute into the function
Find the Derivative
A formula to find the slope of any tangent line at x.
Direct substitution

22. Substitute into the function
Example 2
Differentiate
Simplify in order to cancel the denominator
Substitute into the function
Make the problem easier by factoring out common constants
Direct substitution

23. Example 3
Find the tangent line equation(s) for such that the tangent line has a slope of 12.
Find when the derivative equals 12
Find the derivative first since the derivative finds the slope for an x value
Find the output of the function for every input
Use the point-slope formula to find the equations

24. Alternate Definition of a Derivative
The slope of a secant line between x and x0 is:
The derivative of a function provides us with a measure of the instantaneous rate of change. Thus, we get the derivative at x0 or f'(x0) if we take the limit as the denominator goes to 0:
Known as The Difference Quotient at a particular point

25. Substitute into the function
Example
Find f '(2) if :
Since the derivative needs to be evaluated at a point, the alternate definition can be used.
Simplify in order to cancel the denominator
Factor
Substitute into the function
Direct substitution

26. This is a difficult limit to evaluate.
AP Type Question
Evaluate the limit:
Notice that it is just applying the definition of a derivative at a point
This is a difficult limit to evaluate.
The limit is equal to the derivative of sine at Pi/2. We have not calculated this derivative YET, so we are currently not able to answer this question.

27. How Do the Function and Derivative Function compare?
f is not differentiable at x = -½
Domain:
Domain:

28. Differentiability Justification 1
In order to prove that a function is differentiable at x = c, you must show the following:
In other words, the derivative from the left side MUST EQUAL the derivative from the right side.
Common Example
of a way for a
derivative to fail:
Other common examples: Corners, Cusps
Not differentiable at x = -4

29. Differentiability Justification 2
In order to prove that a function is differentiable at x = c, you must show the following:
In other words, the function must be continuous.
Common Example
of a way for a
derivative to fail:
Other common examples: Gaps, Jumps, Asymptotes
Not differentiable at x = 0

30. Differentiability Justification 3
In order to prove that a function is differentiable at x = c, you must show the following:
In other words, the tangent line can not be a vertical line.
An Example of a Vertical
tangent where the
derivative to Fails to
exist:
Not differentiable at x = 0

31. Differentiability Implies Continuity
If f is differentiable at x = c, then f is continuous at x = c.
The contrapositive of this statement is true: If f is NOT continuous at x = c, then f is NOT differentiable at x = c. The converse of this statement is NOT always true: If f is continuous at x = c, then f is differentiable at x = c. The inverse of this statement is NOT always true: If f is NOT differentiable at x = c, then f is NOT continuous at x = c.

32. Example 1
Determine whether the following derivatives exist for the graph of the function.

33. Example 2 Show that does exist if Prove that it is Continuous
The limit exists
f(1) exists
Since the function is continuous at x=1.
Prove the Right Hand Derivative is the same as the Left Hand Derivative (and non-infinite)
The one-sided derivatives are equal and non-infinite.
Thus the derivative exists, at x=1.

34. First rewrite the absolute value function as a piecewise function
Example 3
Show that does not exist if
First rewrite the absolute value function as a piecewise function
Find the Left Hand Derivative
Find the Right Hand Derivative
Since the one-sided limits are not equal, the derivative does not exist

35. Function v Derivative
Compare and contrast the function and its derivative.
Positive Slopes
x-intercept -5
Vertex
Negative Slopes
Decreasing
Increasing -5
FUNCTION
DERIVATIVE

36. Function v Derivative
Compare and contrast the function and its derivative.
Local Max
Decreasing
Positive Derivatives
Positive Derivatives
Increasing
Increasing
x-intercept
x-intercept -5 5 -5 5
Negative Derivatives
Local Min
FUNCTION
DERIVATIVE

37. Example 1
Accurately graph the derivative of the function graphed below at left.
The Derivative does not exist at a corner.
Make sure the x-value does not have a derivative
The slope from -∞ to -7 is -2
The slope from 2 to ∞ is 1
The slope from -7 to 2 is 0

38. Example 2
Sketch a graph of the derivative of the function graphed below at left.
Negative
Negative
Increasing
Increasing
Increasing
Decreasing
Decreasing
Positive
Positive
Positive
1. Find the x values where the slope of the tangent line is zero (max, mins, twists)
2. Determine whether the function is increasing or decreasing on each interval

39. Example 3
Sketch a graph of the derivative of the function graphed below at left.
Positive
Negative
Decreasing
Increasing
Increasing
Positive
1. Find the x values where the slope of the tangent line is zero (max, mins, twists)
2. Determine whether the function is increasing or decreasing on each interval