MyMathLab Homework: 11. Section 12.1 & 12.2 Derivatives and the Graph of a Function

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1 DERIVATIVES AND FUNCTION GRAPHS Text References: Section " 2.1 & 12.2 MyMathLab Homeork: 11. Section 12.1 & 12.2 Derivatives and the Graph of a Function By inspection of the graph of the function, e have learned to identify: the intervals of B-values here a function is increasing and here it is decreasing, the coordinates of turning points, and the maximum or minimum C-value of a function. We ill no learn ho to calculate these graph properties of the function by using the derivative of the function. DERIVATIVES AND THE DIRECTION OF A GRAPH Recall that: (i) the tangent line at the point ith Bœ+ appears to coincide ith the graph at this point; (ii) 0 Ð+Ñequals the slope of the tangent line at B œ + ; (iii) a line ith positive slope is rising and a line ith negative slope is falling. Therefore, e conclude the folloing. 1. If 0 ÐBÑ! on an interval of B-values, then 0ÐBÑ is increasing on the interval. This means the function graph is rising on the interval. 2. If 0 ÐBÑ! on an interval of B-values, then 0ÐBÑis decreasing on the interval. This means the function graph is decreasing on the interval. Since e ill only be considering polynomial functions, e ill also say that: 3. If 0 Ð+Ñ! then 0ÐBÑis increasing at Bœ+. 4. If 0 Ð+Ñ! then 0ÐBÑis decreasing at B œ +.

2 Text, page 616 You-Try-It Let 0ÐBÑ œ % )B B #. A. Find 0 ÐBÑ. B. Find 0 Ð &Ñ. Does this result imply that 0ÐBÑ is increasing or decreasing at B œ &? C. Find 0 Ð!Ñ. Does this result imply that 0ÐBÑ is increasing or decreasing at B œ!?

3 TURNING POINTS ARE LOCAL EXTREMA If the turning point ÐBß CÑ œ Ð+ß,Ñ is a peak point on the graph of the function, then e say Cœ,œ0Ð+Ñis a local maximum, or that the function has a local maximum at Bœ+. See the figure belo. If the turning point ÐBß CÑ œ Ð+ß,Ñ is a valley point on the graph of the function, then e say Cœ,œ0Ð+Ñis a local minimum, or that the function has a local minimum at Bœ+. See the figure belo. We say Cœ,œ0Ð+Ñis a local extremum if it is either a local maximum or a local minimum.

4 You-Try-It Exercises 12.1, page 626

5 DERIVATIVES AND TURNING POINTS Let's closely examine the derivative values of the function near a local maximum and near a local minimum. Derivative values near the local maximum: (i) The function has a local maximum at Bœ", and 0 Ð"Ñœ!. (ii) Nearby, on the left side of Bœ", the function is increasing so 0 ÐBÑ!. (iii) Nearby, on the right side of Bœ", the function is decreasing so 0 ÐBÑ!. Therefore, the derivative 0 ÐBÑ changes sign from positive to negative at the local maximum. Derivative values near the local minimum. (i) The function has a local minimum at Bœ$, and 0 Ð$Ñœ!. (ii) Nearby, on the left side of Bœ$, the function is decreasing so 0 ÐBÑ!. (iii) Nearby, on the right side of Bœ$, the function is increasing so 0 ÐBÑ!. Therefore, the derivative 0 ÐBÑ changes sign from negative to positive at the local minimum. These properties about local extremum are true in general.

6 USE 0 ÐBÑ TO FIND THE POSSIBLE TURNING POINTS AND CLASSIFY EACH AS A LOCAL MAXIMUM OR A LOCAL MINIMUM OR NEITHER 1. Solve 0 ÐBÑ œ!. The solutions are also called critical values of 0. If B œ + is a solution, then ÐBß CÑ œ Ð+ß 0Ð+ÑÑ is a possible turning point. 2. At each possible turning point, e calculate the rate of change 0 ÐBÑat a nearby point on the left side and at a nearby point on the right side. 3. If the rate of change 0 ÐBÑchanges sign from negative to positive, then the function 0ÐBÑ changes from decreasing to increasing, and you can conclude that the point is actually a turning point and that it is a local minimum. If the rate of change 0 ÐBÑ changes sign from positive to negative, then the function 0ÐBÑ changes from increasing to decreasing, and you can conclude that the point is actually a turning point and that it is a local maximum. If the rate of change 0 ÐBÑ does not change sign, then the function 0ÐBÑ does not change direction and the point is not a turning point. See the graphs belo. Text, page 622

7 TWO TYPES OF CRITICAL VALUES If 0 Ð-Ñ œ!, then B œ - is a critical value of the function, if it is in the domain of the function. If 0 Ð-Ñ œ DNE, then B œ - is a critical value of the function, if it is in the domain of the function. The left and right properties described above apply to both types of critical values.

8 SIGN CHART FOR 0 ÐBÑ You-Try-It Exercises 12.1, page 549

9 You-Try-It MyMathLat #11 Derivatives and Graphs Handout #6: Use Derivatives to Describe the Graph of f(x)

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12 SECOND DERIVATIVES AND CONCAVITY OF A GRAPH The second derivative of a function 0ÐBÑ is the derivative of 0 ÐBÑ and it is ritten as 0 ÐBÑ, C and #.C.B# Therefore the second derivative is also a rate of change function. For example, 0ÐBÑ œ B # 0 ÐBÑ œ #B 0 ÐBÑ œ Ð#BÑ œ # You-Try-It Exercises 12.2, page 644

13 Second derivatives are used to describe the concavity of a function graph. A function graph may be concave up or concave don on an interval of B-values. See the figure belo. An inflection point is a point on the graph of the function here the concavity changes from upard to donard or from donard to upard. In the figure belo, both functions have an inflection point at Bœ-.

14 SECOND DERIVATIVES AND CONCAVITY Let's consider the properties of the derivatives of a function and the concavity of the function graph. We are only considering polynomial functions. 1. Graph is concave up. 1. Graph is concave don. 2. From left to right, the tangent 2. From left to right, the tangent line slopes 0 ÐBÑ increase. line slopes 0 ÐBÑ decrease. 3. From left to right, the 3. From left to right, the rates 0 ÐBÑ increase. rates 0 ÐBÑ decrease. 4. Therefore, 0 ÐBÑ!. 4. Therefore, 0 ÐBÑ!.

15 SECOND DERIVATIVES AND INFLECTION POINTS If 0ÐBÑ has an inflection point at B œ -, then 1. 0 Ð-Ñ œ! 2. The second derivative changes sign from positive to negative at Bœ-and so the graph changes from concave up to concave don. Or, the second derivative changes sign from negative to positive at Bœ-and so the graph changes from concave don to concave up at Bœ-. USE 0 ÐBÑ TO FIND THE POSSIBLE INFLECTION POINTS 1. Solve 0 ÐBÑ œ!. If B œ + is a solution and C œ 0Ð+Ñ, then Ð+ß 0Ð+ÑÑ is a possible inflection point. 2. At each possible inflection point, e calculate the rate of change nearby point on the left side and at a nearby point on the right side. 3. If the rate of change 0 ÐBÑ changes sign from negative to positive, or changes sign from positive to negative, then you can conclude that the point is actually an inflection point. 0 ÐBÑ If the rate of change 0 ÐBÑ does not change sign, then the function 0ÐBÑ does not change concavity and the point is not an inflection point. at a

16 You-Try-It Handout #6: Use Derivatives to Describe the Graph of 0ÐBÑ

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18 INFLECTION POINT: POINT OF DIMINISHING RETURNS An inflection point is a point of greatest slope slope 0 ÐBÑ, compared to other nearby points. 0 ÐBÑ or a point of least If a function is increasing, this means a function is increasing at the fastest rate at the inflection point. That is, before the inflection point the function is increasing at an increasing rate, and after the inflection point the function is increasing at a decreasing rate. The inflection point is called the point of diminishing returns. Here is a graph of RÐBÑ. Check that there is an inflection point at Bœ'. # R ÐBÑ œ B Ð* BÑ and R ÐBÑ œ $BÐ' BÑ. Up to the advertising level of $ ' thousand, sales are increasing at an increasing rate, and afterards, sales are increasing at a decreasing rate. Therefore, the sales level Bœ' thousand dollars is a point of diminishing returns.

19 SALES EXAMPLE In the folloing table, a company's monthly sales WÐBÑ are given for each month B of a year. Let's analyze the company's sales at to different levels. first 6 months second 6 months " " " " " " " " " " " B (months) WÐBÑ (# sold) " # $ % & ' ( ) * "! "" "# & ' "" #" $' &' )" "!" ""' "#' "$" "$# " & "! "& #! #& #! "& "! & " sales groth increasing sales groth decreasing A graph of the function WÐBÑ is given here. Properties of Sales Properties of WÐBÑ 1. The company's sales are increasing 1. The function WÐBÑ is increasing throughout the year. on the interval Ò"ß "#Ó. WÐBÑis positive on the interval Ò"Þ "#Ó. 2. The first ' months of the year, 2. The rates W ÐBÑare increasing the monthly sales groth is increasing. on the interval Ò"ß 'Ó. The function graph is concave up on the interval Ò"ß 'Ó. W ÐBÑis positive on the interval Ò"ß 'Ó. 3. The second ' months of the year, 3. The rates W ÐBÑare decreasing the monthly sales groth is decreasing. on the interval Ò'ß "#Ó. The function is concave don on the interval Ò'ß "#Ó. W ÐBÑis negative on the interval Ò"ß 'Ó.

20 You-Try-It Handout #6: Use Derivatives to Describe the Graph of 0ÐBÑ Here is a graph of MÐ>Ñ.