1. Chapter 3 Introduction to the Derivative Sections 3. 5, 3. 6, 4
Chapter 3 Introduction to the Derivative Sections 3.5, 3.6, 4.1 and 4.2

2. Introduction to the Derivative
Average Rate of Change
The Derivative

3. Average Rate of Change The change of f (x) over the interval [a,b] is
The average rate of change of f (x) over the interval [a , b] is
Difference Quotient

4. Average Rate of Change
Is equal to the slope of the secant line through the points (a , f (a)) and (b , f (b)) on the graph of f (x)
secant line has slope mS

5. Average Rate of Change as h0
The average rate of change of f over the interval [a , b] can be written in two different ways:

6. Average Rate of Change as h0
We now look at the behavior of the average rate of change of f (x) as b a  h gets smaller and smaller, that is, we will let h tend to 0 (h0) and look for a geometric interpretation of the result.
For this, we consider an example of values of
and their corresponding secant lines.

7. Average Rate of Change as h0
Secant line tends to become the Tangent line

8. Average Rate of Change as h0
Secant line tends to become the Tangent line

9. Average Rate of Change as h0
Secant line tends to become the Tangent line

10. Average Rate of Change as h0
Secant line tends to become the Tangent line

11. Average Rate of Change as h0
Secant line tends to become the Tangent line

12. Average Rate of Change as h0
Secant line tends to become the Tangent line

13. Average Rate of Change as h0
Secant line tends to become the Tangent line
Zoom in

14. Average Rate of Change as h0
Secant line tends to become the Tangent line

15. Average Rate of Change as h0
Secant line tends to become the Tangent line

16. Average Rate of Change as h0
We observe that as h approaches zero,
The secant line through P and Q approaches the tangent line to the point P on the graph of f.
and consequently, the slope mS of the secant line approaches the slope mT of the tangent line to the point P on the graph of f.

17. Instantaneous Rate of Change of f at x = a
That is, the limiting value, as h gets increasingly smaller, of the difference quotient
is the slope mT of the tangent line to the graph of the function f (x) at x = a.
(slope of secant line)

18. Instantaneous Rate of Change of f at x = a
This is usually written as
tends to, or
approaches
That is, mS approaches mT as h tends to 0.

19. Instantaneous Rate of Change of f at x = a
More briefly using symbols by writing
and read
as “the limit as h approaches 0 of ”

20. Instantaneous Rate of Change of f at x = a
that is, the limit symbol indicates that the value of
can be made arbitrarily close to mT by taking h to be a sufficiently small number.

21. Instantaneous Rate of Change of f at x = a
Definition: The instantaneous rate of change of f (x) at x = a is defined to be the slope of the tangent line to the graph of the function f (x) at x = a. That is,
Remark: The slope mT gives a precise indication of how fast the graph of f (x) is increasing or decreasing at x = a.

22. A Concrete Example
Consider the function f (x)  – x2/4 + 9/4, whose graph is given below, at the points a  – 3 and a  – 1
At which of these two points is the function increasing faster?
Intuition says at x  – 3 because we notice the graph is steeper at the point x  – 3. Why?

23. A Concrete Example
Consider the function f (x)  – x2/4 + 9/4, whose graph is given below, at the points a  – 3 and a  – 1
To make this more obvious we zoom in
Because our brain makes no distinction between the graph of f and the tangent line to the graph at the point in question.

24. A Concrete Example
Consider the function f (x)  – x2/4 + 9/4, whose graph is given below, at the points a  – 3 and a  – 1
We see how the tangent line basically coincides with the graph of f near the point of contact. What is the slope of each line?

25. A Concrete Example
Consider the function f (x)  – x2/4 + 9/4, whose graph is given below, at the points a  – 3 and a  – 1
That is, at the points of contact, the function is increasing as fast as its corresponding tangent line.

26. A Concrete Example
Consider the function f (x)  – x2/4 + 9/4, whose graph is given below, at the points a  – 3 and a  – 1
We now verify the claims by explicitly computing for this function, the limit,

27. A Concrete Example
For the function f (x)  – x2/4 + 9/4 at any point a we have

28. A Concrete Example
Thus, for the function f (x)  – x2/4 + 9/4 at any point a we have
Therefore, as h tends to 0, mS approaches – a/2  mT .

29. A Concrete Example
For the function f (x)  – x2/4 + 9/4 at any point a we have
Thus,

30. A Concrete Example
Let us try some other values of a for the same function.

31. Remark
The example clearly indicates that the slope mT of the tangent line is itself a function of the point a we choose.
We need a better notation reflecting the fact that this object is a function of a

32. The Derivative
The instantaneous rate of change mT at a is also called the derivative of f at x = a and it is denoted by f '(a). That is,
f '(a) is read “f prime of a”
mT = f '(a) = “slope of the tangent line to f (x) at a”
= “instantaneous rate of change of f at a”

33. The Derivative
The instantaneous rate of change mT at a is also called the derivative of f at x = a and it is denoted by f '(a). That is,
In our previous example, the derivative of the function f (x)  – (x2 – 9)/4 at any point a is given by

34. Rates of Change
Average rate of change of f over the interval [a, a+h] is
Slope of the secant line through the points (a , f (a)) and (a , f (a+h))
Instantaneous rate of change of f at x=a is
Slope of the tangent line at the point (a , f (a))

35. Tangent Line and Secant Line
tangent line at a

36. Using the point-slope form of the line
Equation of Tangent Line at x = a
Using the point-slope form of the line
tangent line at a

37. Terminology
Finding the derivative of the function f is called differentiating f.
If f '(a) exists, then we say that f is differentiable at x = a.
For some functions f , f '(a) may not exist. In this case we say that the function f is not differentiable at x = a.

38. Quick Approximation of the Derivative
Recall that f '(a) is the limiting value of the expression
as we make h increasingly smaller. Therefore, we can approximate the numerical value of the derivative using small values of h. h = often works. In this case,

39. Quick Approximation - Example
Demand: The demand for an old brand of TV is given by
where p is the price per TV set, in dollars, and q is the number of TV sets that can be sold at price p . Find q (190) and estimate q '(190) . Interpret your answers.

40. Solution

41. Geometric Interpretation
At a  190, q(p) decreases as fast as the tangent line does, that is, at the rate
mT  q '(190)  2.5 TVs/$
What does this means?
It means that at the price of p  $190, the demand will decrease by 2.5 TV sets per dollar we increase the price.
a  190
If we now set the price at $200, how many TV sets do we expect to sell?

42. Geometric Interpretation
At a  190, the equation of the tangent line is
y  2.5( p-190 )+500
Thus, when we set the price at p  $200, the line shows that we expect to sell y  475 TVs
The actual number we expect to sell according to the demand function is
q (200)   476 TVs
which is very close to the prediction given by the tangent line.
p  200

43. Now Recall The Example
The slope mT  f '(a) of the tangent line depends on the point a we choose on the graph of f .

44. Chapter 4 Techniques of Differentiation with Applications Sections 4
Chapter 4 Techniques of Differentiation with Applications Sections 4.1 and 4.2

45. Techniques of Differentiation
Derivatives of Powers, Sums and Constant asMultiples
Marginal Analysis

46. The Derivative as a Function
If f is a function, its derivative function is the function whose value at x is the derivative of f at x. That is,
Notice that all we have done is substituted x for a in the definition of f '(a). This way, when we are done with the algebra, the answer will be given in terms of x.

47. The Derivative as a Function
Example: Given the function

48. Geometric Verification
At any point on the graph of f , the tangent line agrees with the graph of which already is a straight line of slope 1

49. The Derivative as a Function
Example: Given the function

50. Geometric Verificationx
y=x2 -3 -2 -1 1 2 3 x
y'=2x -3 -2 -1 1 2 3

51. The Derivative as a Function
Example: Given the function

52. The Power Rule
Example:
Example:
Example:

53. Verification when f (x)  x1/2
Example: Given the function

54. Leibniz’ d Notation For Derivatives
Average rate of change
Instantaneous rate of change
means the same as

55. Differential Notation: Differentiation
means “the derivative with respect to x”.
The derivative of y  f (x) with respect to x is written
or as
Example:

56. Differential Notation: Differentiation
Example:
To find f '(2), the derivative of y  f (x) at x  2, we first need to find the function f '(x).
In Leibniz’ notation f '(2) is denoted by the symbol

57. Differentiation Rules
Example:
Example:

58. Differentiation Rules
Example:

59. Differentiation Rules
Example:

60. More Examples Example: Find the derivative of
First write f (x) as constant times a power

61. Example: Find the derivative of
First write f (x) as

62. Functions Not Differentiable at a Point
a) Vertical tangent x = a
b) Cusp x = a
c) f (a) = Undefined

63. Example: The Absolute Value of x.

64. Example: The cube root of x.

65. Introduction to the Derivative
Average Rate of Change
The Derivative
Derivatives of Powers, Sums and Constant asMultiples
Marginal Analysis

66. C(x)= “variable costs” + “fixed costs”
Cost Functions
A cost function specifies the total cost C as a function of the number of items x produced. Thus, C(x) is the cost of x items. The cost functions is made up of two parts:
C(x)= “variable costs” + “fixed costs”
The marginal cost function is the derivative C′(x) of C(x). It measures the instantaneous rate of change of cost with respect to x.

67. By definition the Marginal Cost Function is C′(x).
The units of marginal cost are units of cost (say, $) per item.
Interpretation: The marginal cost function C′(x) approximates the change in actual cost of producing an additional unit when x items are already being produced.

68. Interpretation of Marginal Cost
The Marginal Cost Function is C′(x). Thus, for small h we have the quick approximation given by
If we let h=1 (one more item being produced ), then

69. Interpretation of Marginal Cost
That is, for h=1, we have
Notice that C(x+1) - C(x) is the actual (exact) cost of producing one additional item when the production level is x.
In practice, h=1 is considered to be small in relation to production level x, which in general is a large number.

70. Average Cost Functions
Given a cost function C(x) , the average cost function given by:

71. Example
The total monthly cost function C, in dollars, of producing x computer games is given by:
Find the marginal cost function and use it to estimate the cost of producing the 1,001st game.
Find the actual cost of producing the 1,001st game.

72. Solution The marginal cost function is given by:
An estimate for the cost of producing the 1,001st game is
That is, it costs an additional $18 to produce one more game when the production level is at 1,000 games a month

73. Solution
The the actual cost of producing the 1,001st game

74. Example Continued
The total monthly cost function C, in dollars, of producing x computer games is given by:
Find the average cost per game if 1,000 games are manufactured.
Find the average cost function.

75. Solution
The average cost function:

76. More Marginal Functions
The Marginal Revenue Function measures the rate of change of the revenue function. It approximates the revenue from the sale of an additional unit.
The Marginal Profit Function measures the rate of change of the profit function. It approximates the profit from the sale of an additional unit.

77. More Marginal Functions
Given a revenue function, R(q), The Marginal Revenue Function is:
Given a profit function, P(q), The Marginal Profit Function is:

78. Marginal Functions - Examples
The monthly demand for T-shirts is given by
where p denotes the wholesale unit price in dollars and q denotes the quantity demanded. The monthly cost for these T-shirts is $8 per shirt.
Find the revenue and profit functions.
Find the marginal revenue and marginal profit functions.

79. Solution Revenue = qp Profit = revenue – cost
Find the revenue and profit functions.
Revenue = qp
Profit = revenue – cost

80. Solution Marginal profit
Find the marginal revenue and marginal profit functions.
Marginal revenue
Marginal profit