3.1 Graphing Using the First Derivative

Transcription

1 3.1 Graphing Using the First Derivative Hartfield MATH 2040 Unit 3 Page 1 Recall that a derivative indicates the instantaneous rate of change at an x-value (or other input value) which is related to the slope of a line tangent to that x-value. Definition: A relative maximum point is a point where the y-value of the point is greater than the y-value of any point nearby. Definition: A relative minimum point is a point where the y-value of the point is less than the y-value of any point nearby. f is increasing f is decreasing An interest area for us in calculus is the points where f stops increasing or stops decreasing. Relative Extreme Points Collectively we call relative maximum points and relative minimum points together relative extreme points.

2 Relative Extreme Values and Critical Numbers Hartfield MATH 2040 Unit 3 Page 2 Note the emphasis on the y-value in the definition of relative extreme points. A related set of definitions focus exclusively on the function values: Definition: f has a relative maximum value at c if f(c) > f(x) for all x in an open interval containing c. Definition: f has a relative minimum value at c if f(c) < f(x) for all x in an open interval containing c. Using this pair of definitions, we would say that the relative extreme point occurs at (c, f(c)) if a relative extreme value occurs at x = c. From the previous images of relative extreme points, it is important to note the behavior of the derivative at an extreme value. This also brings us to an important definition. Definition: A critical number of a function f is an x-value in the domain of f at which either the derivative of f is zero or undefined.

3 Hartfield MATH 2040 Unit 3 Page 3 Finding the critical numbers of a function leads us to finding the relative extreme points of the function. Find the relative extreme points (more specifically, the x-values of the relative extreme points) allows us to divide up the graph into intervals of increasing or decreasing behavior. We can use this information to sketch graphs of functions by hand. Graphing Functions We will be using sign diagrams to assist in sketching graphs. The process for using a sign diagram goes as follows. We will use the function f(x) = x 3 6x 2 96x + 20 as an example. Step 1: Find critical numbers of the function by finding f and setting it equal to 0.

4 Hartfield MATH 2040 Unit 3 Page 4 Step 2: Create a sign diagram by diving up a number line using the critical number. In the intervals between critical numbers, use a test number with f to determine whether f is increasing or decreasing. Step 3: Calculate the coordinates of the relative extreme points using the critical numbers and then sketch the graph.

5 First Derivative Test & Relative Extreme Values Hartfield MATH 2040 Unit 3 Page 5 We will return to sketching graphs in a moment but at this point we should highlight a related topic, The First Derivative Test. The First Derivative Test: If f has a critical number c, then at x = c the function has a 1. relative maximum if f > 0 just before c and f < 0 just after c. 2. relative minimum if f < 0 just before c and f > 0 just after c. The First Derivative Test allows us take the results of our sign diagram and use them to determine what type of relative extreme value exists at a critical number (if one exists at all). Critical numbers may not indicate relative extreme points however.

6 Hartfield MATH 2040 Unit 3 Page 6 Ex. 2: Sketch a graph of the given function by finding its critical numbers and creating an appropriate sign diagram. Identify the open intervals where the function is increasing and where it is decreasing. Then use the First Derivative Test to identify its relative extreme points. f(x) = x 4 4x x 2 10

7 Hartfield MATH 2040 Unit 3 Page 7 Ex. 3: Sketch a graph of the given function by finding its critical numbers and creating an appropriate sign diagram. Identify the open intervals where the function is increasing and where it is decreasing. Then use the First Derivative Test to identify its relative extreme points. f(x) = x 2 (x 5) 3

8 Hartfield MATH 2040 Unit 3 Page 8 Ex. 4: Sketch a graph of the given function by finding its critical numbers and creating an appropriate sign diagram. Identify the open intervals where the function is increasing and where it is decreasing. Then use the First Derivative Test to identify its relative extreme points. f(x) = 2 3x 2x 6

9 3.2 Graphing Using 1 st and 2 nd Derivatives Hartfield MATH 2040 Unit 3 Page 9 Applying the first derivative, we can find where a function is increasing or decreasing. By adding in the second derivative, we can have a more full visualization of the way a function is changing. Compare the following graphs below: No concavity Concave Up Concave Down Definition: A curve (or a portion of a curve) is said to be concave up if the curve curls upward. Definition: A curve (or a portion of a curve) is said to be concave down if the curve curls downward. Definition: An inflection point is a point on a curve where concavity changes. Ex.: Identify the parts of the function which are concave up and concave down and find any inflection points.

10 Calculus-Based Definition of Concavity and Inflection Points Hartfield MATH 2040 Unit 3 Page 10 Definition: Over an open interval, if f > 0, then f is concave up. Over an open interval, if f < 0, then f is concave down. An inflection point occurs at (c, f(c)) if concavity changes on either side of x = c. f (c) will be zero or undefined. Ex. 1: Find the intervals over which f is concave up or concave down and then identify any inflection points When it comes to sketching the graph, use a f( x) x 4x 48x 24 sign-chart with the second derivative for concavity (similar to how you can use it with the first derivative to determine increasing and decreasing intervals).

11 Hartfield MATH 2040 Unit 3 Page 11 Ex. 2: Find the intervals over which f is concave up or concave down and then identify any inflection points. f( x) x x 5 4

12 Second Derivative Test & Relative Extreme Values Hartfield MATH 2040 Unit 3 Page 12 While a second derivative is directly used to find concavity, it can also be used to gives you information about extreme values. The Second Derivative Test: If f is twice-differentiable and has a critical number at x = c, then 1. f (c) > 0 means that f has a relative minimum at x = c. 2. f (c) < 0 means that f has a relative maximum at x = c. Important Observations about the Second Derivative Test: 1. Remember that critical numbers are found using the first derivative. Setting the second derivative equal to zero is related to finding inflection points. 2. The Second Derivative Test can only be applied in cases where the second derivative exists at x = c. Also, if the second derivative is equal to zero, the test fails and you must return to the First Derivative Test to determine whether a relative maximum or relative minimum exists at the critical number.

13 Hartfield MATH 2040 Unit 3 Page 13 Ex. 1: Sketch a graph of the given function by finding its critical numbers and creating an appropriate sign diagram for the first derivative. Identify the open intervals where the function is increasing and where it is decreasing. Then find where the second derivative is zero or undefined and create an appropriate sign diagram for the second derivative. Identify the open intervals where the function is concave up or concave down and any inflection points. Further use either the First Derivative Test or the Second Derivative Test to find the relative extreme points. Finally sketch a graph of the function by hand. f(x) =2x 3 9x 2 24x

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15 Hartfield MATH 2040 Unit 3 Page 15 Ex. 2: Sketch a graph of the given function by finding its critical numbers and creating an appropriate sign diagram for the first derivative. Identify the open intervals where the function is increasing and where it is decreasing. Then find where the second derivative is zero or undefined and create an appropriate sign diagram for the second derivative. Identify the open intervals where the function is concave up or concave down and any inflection points. Further use either the First Derivative Test or the Second Derivative Test to find the relative extreme points. Finally sketch a graph of the function by hand. f(x) = x 4 + 8x x 2 + 8

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17 Hartfield MATH 2040 Unit 3 Page 17 Ex. 3: Sketch a graph of the given function by finding its critical numbers and creating an appropriate sign diagram for the first derivative. Identify the open intervals where the function is increasing and where it is decreasing. Then find where the second derivative is zero or undefined and create an appropriate sign diagram for the second derivative. Identify the open intervals where the function is concave up or concave down and any inflection points. Further use either the First Derivative Test or the Second Derivative Test to find the relative extreme points. Finally sketch a graph of the function by hand. f(x) = 1 3 x3 (4 x)

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19 3.3/4/5 Optimization, part 2 Hartfield MATH 2040 Unit 3 Page 19 An application of finding extreme values is optimization. Frequently we may be interested in finding where a function is minimized or maximized and so long as the domain of a function is satisfied, the extreme value will always be found either at a critical number of at the boundaries of the domain. (Note that not all functions have explicitly defined domains.) How many jerseys should the manufacturer produce is a run so that the average cost per jersey is minimized? What will be the average cost per jersey at that point? (Source: Finite Mathematics & Calculus Applied to the Real World (1996) p. 846, example 1) Ex. 1: A high-end sports apparel manufacturer produces hockey jerseys for sale in college bookstores. Its cost function, in 2 dollars, is C( x) x 0.2x where x is the number of jersey manufactured.

20 Hartfield MATH 2040 Unit 3 Page 20 We will focus on applications related to business and economics in this section. Several concepts either to be reviewed or introduced: A: How many bicycles should be produced to maximize profit? B: At what price is profit maximized? C: What would be the maximum profit? 1. Profit = Revenue Cost 2. Revenue = (Unit Price) (Quantity) 3. A price function p(x) gives the price that consumers will buy x units of a product. 4. Maximum profit occurs when marginal revenue and marginal cost are equal. Ex. 2: City Cycles Inc. finds that it costs $70 to manufacture each bicycle with fixed costs of $100 per day. The price function for sales is p(x) = x, where p(x) is the price in dollars when exactly x bicycles are sold. (Source: 4 th edition p. 206, #34)

21 Hartfield MATH 2040 Unit 3 Page 21 Ex. 3: An automobile dealer can sell four cars per day at a price of $ She estimates that for each $200 price reduction, she can sell two more cars per day. Each car costs her $10000 and she has fixed costs of $100 per day. (Source: 4 th edition p. 216, #2) A: At what price is her profit maximized? B: How many cars would she sell at this price? C: What would be the maximum profit?

22 Hartfield MATH 2040 Unit 3 Page 22 Ex. 4: A peach grower finds that if he plants 40 trees per acre, each tree will yield 60 bushels of peaches. He also estimates that for each additional tree that he plants per acre, the yield of each tree will decrease by 2 bushels. (Source: 4 th edition p. 216, #6) A: How many trees should he plant to maximize his harvest? B: What would be the maximum yield per acre?

23 Hartfield MATH 2040 Unit 3 Page 23 Ex. 5: A wine warehouse expects to sell bottles of wine in a year. Each bottle costs $9 with a fixed charge of $200 per order. Storing a bottle for a year costs $3. Answer the following questions with the goal of minimizing inventory costs. (Source: 4 th edition p. 225, #4) A: How many bottles should be ordered at a time? B: How many orders should the warehouse place in a year?

24 Hartfield MATH 2040 Unit 3 Page 24 Ex. 6: A compact disc manufacturer estimates the yearly demand for a CD to be 10,000. It costs $400 to set the machinery for the CD plus $3 for each CD produced. It costs $2 to store a CD for a year. Answer the following questions with the goal of minimizing costs. (Source: 4 th edition p. 225, #10) A: How many CDs should be burned at a time? B: How many production runs will be needed in one year?