Chapter 3 Limits and the Derivative Section 4 The Derivative (Part 1)

2 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 3.4 The Derivative ■ Part One ■ The student will be able to: ■ Calculate slope of the secant line. ■ Calculate average rate of change. ■ Calculate slope of the tangent line. ■ Calculate instantaneous rate of change.

3 Introduction  In Calculus, we study how a change in one variable affects another variable.  In studying this, we will make use of the limit concepts we learned in the previous lessons of this chapter. Barnett/Ziegler/Byleen Business Calculus 12e

4 Slopes  Slope of a secant  Slope of a tangent Barnett/Ziegler/Byleen Business Calculus 12e

5 Example 1 Revenue Analysis  The graph below shows the revenue (in dollars) from the sale of x widgets.  When 100 widgets are sold, the revenue is $1800.  If we increase production by an additional 300 widgets, the revenue increases to $4800. Barnett/Ziegler/Byleen Business Calculus 12e

6 Example 1 (continued) Barnett/Ziegler/Byleen Business Calculus 12e

7 Rate of Change  This is an example of the “rate of change” concept.  The average rate of change is the ratio of the change in y over the change in x.  You know this as the “slope” between two points. Barnett/Ziegler/Byleen Business Calculus 12e

8 The Rate of Change For y = f (x), the average rate of change from x = a to x = a + h is The above expression is also called a difference quotient. It can be interpreted as the slope of a secant. See the picture on the next slide for illustration.

9 Barnett/Ziegler/Byleen Business Calculus 12e Graphical Interpretation Average rate of change = slope of the secant line

10 What if…  Suppose the 2 nd point (a+h, f(a+h)) gets closer and closer to the first point (a, f(a)). What happens to the value of h? Barnett/Ziegler/Byleen Business Calculus 12e Answer: h approaches zero

11 Barnett/Ziegler/Byleen Business Calculus 12e The Instantaneous Rate of Change If we find the slope of the secant line as h approaches zero, that’s the same as the limit shown below. Now, instead of the average rate of change, this limit gives us the instantaneous rate of change of f(x) at x = a. And instead of the slope of a secant, it’s the slope of a tangent.

12 Barnett/Ziegler/Byleen Business Calculus 12e Visual Interpretation Slope of tangent at x = a is the instantaneous rate of change. Tangent line at x=a

13 Barnett/Ziegler/Byleen Business Calculus 12e Instantaneous Rate of Change

14 Example 3A Barnett/Ziegler/Byleen Business Calculus 12e

15 Example 3B Barnett/Ziegler/Byleen Business Calculus 12e Continued on next slide…

16 Example 3B - continued Barnett/Ziegler/Byleen Business Calculus 12e

17 Example 3C Barnett/Ziegler/Byleen Business Calculus 12e

18 Application - Velocity  A watermelon that is dropped from the Eiffel Tower will fall a distance of y feet in x seconds.  Find the average velocity from 2 to 5 seconds. Answer: Barnett/Ziegler/Byleen Business Calculus 12e

19 Velocity (continued)  Find the instantaneous velocity at x = 2 seconds. Barnett/Ziegler/Byleen Business Calculus 12e

20 Summary  Slope of a secant  Average rate of change  Average velocity  Slope of a tangent  Instantaneous rate of change  Instantaneous velocity Barnett/Ziegler/Byleen Business Calculus 12e

21 Homework #3-4A Pg 175 (1-4, 27-30) Barnett/Ziegler/Byleen Business Calculus 12e

Chapter 3 Limits and the Derivative Section 4 The Derivative (Part 2)

23 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 3.4 The Derivative ■ Part Two ■ The student will be able to: ■ Calculate the derivative. ■ Identify the nonexistence of a derivative.

24 Introduction  In Part 1, we learned that the limit of a difference quotient, can be interpreted as: instantaneous rate of change at x=a slope of the tangent line at x=a instantaneous velocity at x=a  In this part of the lesson, we will take a closer look at this limit where we replace a with x. Barnett/Ziegler/Byleen Business Calculus 12e

25 Barnett/Ziegler/Byleen Business Calculus 12e The Derivative For y = f (x), we define the derivative of f at x, denoted f ( x), to be

26 Barnett/Ziegler/Byleen Business Calculus 12e Same Meaning as Before If f is a function, then f has the following interpretations: ■ For each x in the domain of f, f (x) is the slope of the line tangent to the graph of f at the point (x, f (x)). ■ For each x in the domain of f, f (x) is the instantaneous rate of change of y = f (x) with respect to x. ■ If f (x) is the position of a moving object at time x, then v = f (x) is the instantaneous velocity of the object with respect to time.

27 Barnett/Ziegler/Byleen Business Calculus 12e Finding the Derivative To find f (x), we use a four-step process: Step 1. Find f (x + h) Step 2. Find f (x + h) – f (x) Step 3. Find Step 4. Find f (x) = *Feel free to go directly to Step 4 when you’ve got the process down!

28 Barnett/Ziegler/Byleen Business Calculus 12e Example 1

29 Example 1 (continued) Step 4: Find f (x) = Barnett/Ziegler/Byleen Business Calculus 12e For x=a, where a is in the domain of f(x), f (a) is the slope of the line tangent to f(x) at x=a. Find the slope of the line tangent to the graph of f (x) at x = 0, x = 2, and x = 3. f (0) = -3 f (2) = 1 f (3) = 3

30 Barnett/Ziegler/Byleen Business Calculus 12e Example 2

31 Example 2 (continued) Step 4: Find f (x) = Barnett/Ziegler/Byleen Business Calculus 12e Find the slope of the line tangent to the graph of f (x) at x = -2, x = 0, and x = 1. f (-2) = 14 f (0) = 2 f (1) = -4

32 Barnett/Ziegler/Byleen Business Calculus 12e Example 3

33 Example 3 (continued) Step 4: Find f (x) = Barnett/Ziegler/Byleen Business Calculus 12e

34 Barnett/Ziegler/Byleen Business Calculus 12e Nonexistence of the Derivative The existence of a derivative at x = a depends on the existence of the limit If the limit does not exist, we say that the function is nondifferentiable at x = a, or f (a) does not exist.

35 Barnett/Ziegler/Byleen Business Calculus 12e Nonexistence of the Derivative (continued) Some of the reasons why the derivative of a function may not exist at x = a are ■ The graph of f has a hole or break at x = a, or ■ The graph of f has a sharp corner at x = a, or ■ The graph of f has a vertical tangent at x = a.

36 Examples of Nonexistent Derivatives Barnett/Ziegler/Byleen Business Calculus 12e In each graph, f is nondifferentiable at x=a.

37 Application – Profit Barnett/Ziegler/Byleen Business Calculus 12e

38 Application – Profit (continued) Barnett/Ziegler/Byleen Business Calculus 12e The avg change in profit when production increases from 800 to 850 car seats is $3.75 per seat.

39 Application – Profit (continued) Barnett/Ziegler/Byleen Business Calculus 12e

40 Application – Profit (continued) Barnett/Ziegler/Byleen Business Calculus 12e

41 Application – Profit (continued) Barnett/Ziegler/Byleen Business Calculus 12e At a production level of 800 car seats, the profit is $15,000 and it is increasing at a rate of $5 per car seat.

42 Homework Barnett/Ziegler/Byleen Business Calculus 12e #3-4B Pg 176 (7, 21, 25, 31-41, 61, 63)