3-1 First Derivative and Graphs

3. Graphing and Optimization

Increasing and Decreasing Functions

/ / Graph of
+ / increasing / rising
- / decreasing / falling

Given the function ,

What values of x correspond to horizontal tangents?

Where is increasing? Decreasing?

Graph ; add horizontal tangents.

Critical Values

Definition. The values of x in the domain of f where or where does not exist are called the critical values of f.

Find the critical values of , and intervals where it is increasing/decreasing.

Local Extrema

Use the graph to find intervals where f is increasing/decreasing, local max/min.

Theorem

If f is continuous on the interval (a, b), c is a number in (a, b), and f (c) is a local extremum, then either f ¢(c) = 0 or f ¢(c) does not exist. That is, c is a critical point. (A critical value is not necessarily a local extremum.)

First Derivative Test

Let c be a critical value of f [f (c) is defined, and either f ¢(c) = 0 or f ¢(c) is not defined.] Construct a sign chart for f ¢(x) close to and on either side of c.

For ,

Find the critical values of f.

Find local max/min.

Graph.

Theorem 3.

If ,

is an nth-degree polynomial, then f has at most n x-intercepts and at most (n – 1) local extrema.

Over the past few decades, The US has exported more ag products than it imported. Graph approximates rate of change of balance of trade.

Graph of total revenue from sale of bookcases

Describe; graph .

3-2 Second Derivative and Graphs

Using Concavity as a Graphing Tool

Consider and for .

The term concave upward (or simply concave up) is used to describe a portion of a graph that opens upward. Concave downward is used to describe a portion of a graph that opens downward.

Concavity

The graph of a function f is concave upward on the interval (a,b) if f ¢(x) is increasing on (a,b), and is concave downward on the interval (a,b) if f ¢(x) is decreasing on (a,b).

For y = f (x), the second derivative of f, provided it exists, is , or or .

For , find .

Discuss concavity upward/downward for

Inflection Points

An inflection point is a point on the graph where the concavity changes from upward to downward or downward to upward.

Theorem 1

If is continuous on and has an inflection point at x = c, then either f ¢¢(c) = 0 or f ¢¢(c) does not exist.

Find the inflection points for

Analyzing graphs

Curve Sketching

Graphing Strategy

§ Step 1. Analyze .
Find the domain and the intercepts. The x intercepts are the solutions to, and the y intercept is .

§ Step 2. Analyze.
Find the partition points critical values of. Construct a sign chart for, determine the intervals where f is increasing and decreasing, and find local maxima and minima.

§ Step 3. Analyze .
Find the partition numbers of . Construct a sign chart for , determine the intervals where the graph of f is concave upward and concave downward, and find inflection points.

§ Step 4. Sketch the graph of f.
Locate intercepts, local maxima and minima, and inflection points. Sketch in what you know from steps 1-3. Plot additional points as needed and complete the sketch.

Follow the graphing strategy and analyze the function .

Step 1. Analyze .

Step 2. Analyze.

Step 3. Analyze .

Step 4. Sketch the graph of f.

Follow the graphing strategy and analyze the function .

Step 1. Analyze .

Step 2. Analyze.

Step 3. Analyze .

Step 4. Sketch the graph of f.

Point of Diminishing Returns

If a company decides to increase spending on advertising, they would expect sales to increase. At first, sales will increase at an increasing rate and then increase at a decreasing rate. The value of x where the rate of change of sales changes from increasing to decreasing is called the point of diminishing returns. This is also the point where the rate of change has a maximum value. Money spent after this point may increase sales, but at a lower rate.

A discount appliance store is selling 200 large-screen TV sets monthly. If the store invests $x thousand in advertising, the ad company estimates that sales will increase to

When is the rate of change of sales increasing and when is it decreasing? What is the point of diminishing returns and the maximum rate of change of sales? Graph N and N¢.

(6-2 Second-Order Partial Derivatives)

Let

Find and .

Find

3- 4 Curve Sketching Techniques

When we summarized the graphing strategy in a previous section, we omitted one very important topic: asymptotes.

Modifying the Graphing Strategy

Step 1. Analyze .

• Find the domain of f.

• Find the intercepts.

• Find asymptotes

Step 2. Analyze.

• Find the partition numbers and critical values of f ¢(x).

• Construct a sign chart for f ¢(x).

• Determine the intervals where f is increasing and decreasing

• Find local maxima and minima

Step 3. Analyze .

• Find the partition numbers of f ¢¢(x).

• Construct a sign chart for f ¢¢(x).

• Determine the intervals where the graph of f is concave upward and concave downward.

• Find inflection points.

Step 4. Sketch the graph of f.

• Draw asymptotes and locate intercepts, local max and min, and inflection points.

• Plot additional points as needed and complete the sketch

Using the Graphing Strategy

Analyze and graph

Step 1. Analyze .

• Find the domain of f.

• Find the intercepts.

• Find asymptotes

Step 2. Analyze.

• Find the partition numbers and critical values of f ¢(x).

• Construct a sign chart for f ¢(x).

• Determine the intervals where f is increasing and decreasing

• Find local maxima and minima

Step 3. Analyze .

• Find the partition numbers of f ¢¢(x).

• Construct a sign chart for f ¢¢(x).

• Determine the intervals where the graph of f is concave upward and concave downward.

• Find inflection points.

Step 4. Sketch the graph of f.

• Draw asymptotes and locate intercepts, local max and min, and inflection points.

• Plot additional points as needed and complete the sketch

Using the Graphing Strategy

Analyze and graph

Step 1. Analyze .

• Find the domain of g.

• Find the intercepts.

• Find asymptotes

Step 2. Analyze.

• Find the partition numbers and critical values of g ¢(x).

• Construct a sign chart for g ¢(x).

• Determine the intervals where g is increasing and decreasing

• Find local maxima and minima

Step 3. Analyze .

• Find the partition numbers of g ¢¢(x).

• Construct a sign chart for g ¢¢(x).

• Determine the intervals where the graph of g is concave upward and concave downward.

• Find inflection points.

Step 4. Sketch the graph of g.

• Draw asymptotes and locate intercepts, local max and min, and inflection points.

• Plot additional points as needed and complete the sketch

Using the Graphing Strategy

Analyze and graph

Step 1. Analyze .

• Find the domain of f.

• Find the intercepts.

• Find asymptotes

Step 2. Analyze.

• Find the partition numbers and critical values of f ¢(x).

• Construct a sign chart for f ¢(x).

• Determine the intervals where f is increasing and decreasing

• Find local maxima and minima

Step 3. Analyze .

• Find the partition numbers of f ¢¢(x).

• Construct a sign chart for f ¢¢(x).

• Determine the intervals where the graph of f is concave upward and concave downward.

• Find inflection points.

Step 4. Sketch the graph of f.

• Draw asymptotes and locate intercepts, local max and min, and inflection points.

• Plot additional points as needed and complete the sketch

Using the Graphing Strategy

Analyze and graph

Step 1. Analyze .

• Find the domain of f.

• Find the intercepts.

• Find asymptotes

Step 2. Analyze.

• Find the partition numbers and critical values of f ¢(x).

• Construct a sign chart for f ¢(x).

• Determine the intervals where f is increasing and decreasing

• Find local maxima and minima

Step 3. Analyze .

• Find the partition numbers of f ¢¢(x).

• Construct a sign chart for f ¢¢(x).

• Determine the intervals where the graph of f is concave upward and concave downward.

• Find inflection points.

Step 4. Sketch the graph of f.

• Draw asymptotes and locate intercepts, local max and min, and inflection points.

• Plot additional points as needed and complete the sketch

Modeling average cost

Let 

Step 1. Analyze .

• Find the domain of .

• Find the intercepts.

• Find asymptotes

Step 2. Analyze.

• Find the partition numbers and critical values of .

• Construct a sign chart for .

• Determine the intervals where is increasing and decreasing

• Find local maxima and minima

Step 3. Analyze .

• Find the partition numbers of .

• Construct a sign chart for .

• Determine the intervals where the graph of is concave up and concave down.

• Find inflection points.

Step 4. Sketch the graph of f.

• Draw asymptotes and locate intercepts, local max and min, and inflection points.

• Plot additional points as needed and complete the sketch

3-5 Absolute Maxima and Minima

is an absolute maximum of f if for all x in the domain of f.

is an absolute minimum of f if for all x in the domain of f.

Theorem 1. Extreme Value Theorem

A function f that is continuous on a closed interval has both an absolute maximum value and an absolute minimum value on that interval.

Theorem 2. Absolute extrema (if they exist) must always occur at critical values of the derivative, or at end points.

Finding Absolute Extrema on a closed interval

Step 1. Check to make sure f is continuous over .

Step 2. Find the critical values in the interval .

Step 3. Evaluate f at the end points a and b and at the critical values found in step 2.

Step 4. The absolute maximum on [a, b] is the largest of the values found in step 3.

Step 5. The absolute minimum on [a, b] is the smallest of the values found in step 3.

Find the absolute minimum and maximum values of on .

on .

Second Derivative and Extrema

Suppose and

/ / graph of f is /
0 / + / concave up / local minimum
0 / - / concave down / local maximum
0 / 0 / ? / test does not apply

Find local max/min for

Theorem 3

Second Derivative Test for Absolute Extremum

Let f be continuous on an interval I with only one critical value c in I.

If and , then is the absolute minimum of f on I.

If and , then is the absolute maximum of f on I.

Find the absolute minimum of on the open interval

3- 6 Optimization

A homeowner has $320 to spend on fencing for his garden. 3 sides of the fence will be with wire at a cost of $2/foot. The fourth side will be wood at a cost of $6/foot. Find the dimensions and the area of the largest garden that can be enclosed for $320.

Strategy for Solving Optimization Problems

1. Introduce variables, look for relationships among these variables, and construct a math model of the form:

Maximize (minimize) f (x) on the interval I.

2. Find the critical values of f (x).

3. Find the maximum (minimum) value of f (x) on the interval I (last section).

4. Use the solution to the mathematical model to answer all the questions asked in the problem.

The homeowner judges that 800 ft² is too small, and decides to increase area to 1,250 ft². What is the minimum cost of this fence? What are the dimensions of this fence?

An office supply company sells x permanent markers per year at $p per marker. The price-demand equation for these markers is . What should the company charge for the markers to maximize revenue?

The total annual cost of manufacturing x permanent markers is . What is the company’s maximum profit, what should the company charge per marker, and how many should be produced?

The government decides to tax the company $2/marker produced. Taking this into account, how many markers should the company manufacture each week to maximize its weekly profit? What is the maximum weekly profit? How much should the company charge for the markers to maximize profit?

When a management training company prices its seminar on management techniques at $400 per person, 1,000 people will attend the seminar. The company estimates that for each $5 reduction in price, an additional 20 people will attend the seminar. How much should the company charge in order to maximize its revenue? What is the maximum revenue?

After additional analysis, the company decides its attendance estimate was too high. The new estimate is that only 10 additional will attend for each $5 reduction. How much should the company charge now in order to maximize its revenue? What is the new maximum revenue?

Inventory Control

A multimedia company anticipates that there will be a demand for 20,000 copies of a DVD during the next year. It costs the company $0.50 to store a DVD for a year. Each setup for DVDs costs $200. How many DVDs should the company make in each run to minimize its total storage and setup costs?