Chapter 5 Graphing and Optimization Section 2 Second Derivative and Graphs (Part I)

2 Barnett/Ziegler/Byleen Business Calculus 12e Objectives for Section 5.2 Second Derivatives and Graphs ■ The student will be able to: ■ Part I: Use the 2 nd derivative to determine the concavity and inflection points of functions ■ Part II: Analyze graphs and do curve sketching.

3 Graph Comparison  How are the graphs below similar?  How are they different? Barnett/Ziegler/Byleen Business Calculus 12e  They are both increasing over the interval (0,  )  The left graph is concave up, while the right one is concave down.

4 Barnett/Ziegler/Byleen Business Calculus 12e Mathematical Definition of Concavity A graph is concave up on the interval (a,b) if any secant connecting two points on the graph in that interval lies above the graph. It is concave down on (a,b) if all secants lie below the graph. up down

5 Barnett/Ziegler/Byleen Business Calculus 12e Concavity Tests The concavity of a graph can be determined by using the 2 nd derivative. For y = f (x), the second derivative of f, provided it exists, is the derivative of the first derivative: f is concave up if f  (x) > 0 f is concave down if f  (x) < 0

6 Determining Concavity Barnett/Ziegler/Byleen Business Calculus 12e

7 Example 1 Find the intervals where the graph of f is concave up or concave down. f (x) = x x x f is concave down: (- , -8) f is concave up: (-8,  )

Barnett/Ziegler/Byleen Business Calculus 12e f is concave down: (- , -8) f is concave up: (-8,  )

9 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 Find the intervals where the graph of f is concave up or concave down. f (x) = e x f is concave up: (- ,  )

Barnett/Ziegler/Byleen Business Calculus 12e f is concave up: (- ,  )

11 Barnett/Ziegler/Byleen Business Calculus 12e Example 3 Find the intervals where the graph of f is concave up or concave down. f (x) = ln x f is concave down: (0,  ) Note: Domain of ln x is (0,  )

Barnett/Ziegler/Byleen Business Calculus 12e f is concave down: (0,  )

13 Barnett/Ziegler/Byleen Business Calculus 12e Inflection Points An inflection point is a point on the graph where the concavity changes from upward to downward or downward to upward.

14 Barnett/Ziegler/Byleen Business Calculus 12e Inflection Points

15 Barnett/Ziegler/Byleen Business Calculus 12e Example 4 Find the intervals where the graph of f is concave up or concave down & find the inflection points. f (x) = x 3 - 9x x f is concave down: (- , 3) f is concave up: (3,  ) Inflection point at (3, 8)

Barnett/Ziegler/Byleen Business Calculus 12e f is concave down: (- , 3) f is concave up: (3,  ) Inflection point at (3, 8)

17 Barnett/Ziegler/Byleen Business Calculus 12e Example 5 Find the intervals where the graph of f is concave up or concave down & find the inflection points. f is concave up: (- , -.5) f is concave down: (-.5,  ) f(-.5)=undefined No inflection point.

Barnett/Ziegler/Byleen Business Calculus 12e f is concave up: (- , -.5) f is concave down: (-.5,  ) No inflection point because f(-.5)=undefined.

19 Homework Barnett/Ziegler/Byleen Business Calculus 12e

Chapter 5 Graphing and Optimization Section 2 Second Derivative and Graphs (Part II)

21 Barnett/Ziegler/Byleen Business Calculus 12e Objectives for Section 5.2 Second Derivatives and Graphs ■ The student will be able to: ■ Part I: Use the 2 nd derivative to determine the concavity and inflection points of functions ■ Part II: Analyze graphs and do curve sketching.

22 Barnett/Ziegler/Byleen Business Calculus 12e Curve Sketching Graphing calculators and computers produce the graph of a function by plotting many points. Although quite accurate, important points on a plot may be difficult to identify. Using information gained from the function and its derivative, we can sketch by hand a very good representation of the graph of f (x). This process is called curve sketching and is summarized on the following slides.

23 Barnett/Ziegler/Byleen Business Calculus 12e Curve Sketching Process

24 Barnett/Ziegler/Byleen Business Calculus 12e  Step 3. Analyze f  (x). Construct a sign chart for f  (x) to determine the intervals where the graph of f is concave up and concave down and find the location of inflection points.  Step 4. Sketch the graph of f using the information from steps 1-3. Plot additional points as needed to complete the sketch. Curve Sketching Process

25 Example 1 Barnett/Ziegler/Byleen Business Calculus 12e x y

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

27 Example 2 (continued) Barnett/Ziegler/Byleen Business Calculus 12e

28 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 (continued)

29 Barnett/Ziegler/Byleen Business Calculus 12e x y Example 2 (continued)

30 Homework Barnett/Ziegler/Byleen Business Calculus 12e Do NOT use your calculator to graph!!!