Extreme Values: Graphically Return to Table of Contents Slide 5 / - - PDF document

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AP Calculus

Analyzing Functions Using Derivatives

2015-11-04 www.njctl.org

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Mean Value Theorem Connecting Graphs of f, f', and f''

click on the topic to go to that section

1st Derivative Test Concavity & 2nd Derivative Test Extreme Values - Graphically Curve Sketching Optimization Rolle's Theorem Newton's Method

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Extreme Values: Graphically

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Recall from the previous unit... we analyzed graphs and discovered the locations of horizontal tangent lines. a b c Looking at locations a, b, and c, while they all share the trait that they have horizontal tangents, what is different about each point?

Horizontal Tangents Slide 5 / 163

a Looking specifically at point a, we know at the peak, the slope is zero. What do you notice about the slope on either side of a?

Slopes Surrounding Point a Slide 6 / 163

c Similarly, we have a change in slopes at point c, however the slope is changing from negative to positive at this point.

Slopes Surrounding Point c Slide 7 / 163

b Now, consider point b. We know the slope is zero at b; however, the function's slope does not change signs at this point.

Slopes Surrounding Point b Slide 8 / 163 Local (Relative) Extrema

Local Maximum: a high point on any interval relative to points around it. At this point, the slope changes from positive to negative, and the function changes from increasing to decreasing. Local Minimum: a low point on any interval relative to points around

• it. At this point the slope changes from negative to positive, and the

function changes from decreasing to increasing. NOTE: Local max/mins CANNOT occur at endpoints!

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Absolute (Global) Extrema

Absolute Maximum: occurs at c if f(c)>f(x) for all x in domain Absolute Minimum: occurs at c if f(c)<f(x) for all x in domain NOTE: Absolute max/mins can occur at endpoints!

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Identify/Label each of the following with Local or Absolute Maximum or

• Minimum. What do you notice about what is occuring at the star?

Extrema Slide 11 / 163 Slide 12 / 163

Critical Value

When asked to find local extrema, only critical values must be

• considered. If asked to find absolute extrema, critical values as well

as endpoints are considered. A critical value (or critical point) is a point on the interior of the domain

• f a function at which the slope is zero or undefined.

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An extrema can only occur at critical values or endpoints (absolute); however, the presence of a critical value does not guarantee an extrema at that value. What does this mean?

Extrema & Endpoints Slide 14 / 163

1 Using the given graph, which of the following are

• ccurring at point b?

A B C D E F local maximum local minimum absolute minimum absolute maximum slope is zero slope is undefined

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2 Using the given graph, which of the following are

• ccurring at point d?

A B C D E F local maximum local minimum absolute minimum absolute maximum slope is zero slope is undefined

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3 Using the given graph, which of the following are

• ccurring at point c?

A B C D E F local maximum local minimum absolute minimum absolute maximum slope is zero slope is undefined

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4 Using the given graph, which of the following are

• ccurring at point a?

A B C D E F local maximum local minimum absolute minimum absolute maximum slope is zero slope is undefined

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5 On which interval(s) is the function increasing? A B C (a,b) (b,c) (c,d)

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6 Using the given graph, which of the following are critical values? A B C D E F G H

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7 Using the given graph, which of the following are

• ccurring at ?

A B C D E F local maximum local minimum absolute minimum absolute maximum slope is zero slope is undefined

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8 On which interval(s) is the function decreasing? A B C D E F G

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12 On which interval(s) is the function increasing?

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13 If a function has a critical value at x=3, then there must be a local or absolute extrema at that value. True False

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1st Derivative Test

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We have discovered how to graphically interpret local and absolute extrema, and now we will extend our understanding to calculate extrema algebraically.

Calculating Extrema Algebraically Slide 30 / 163

A local maxima/minima occurs when the slope of the function changes from positive to negative, or negative to positive at a critical value. Another way of thinking is that the original function changes from increasing to decreasing, or decreasing to increasing. For a function to contain an absolute extrema, it must be the highest

• r lowest extrema on the interval, including endpoints.

Recall: Slide 31 / 163

Algebraically, we can find critical values and test points on either side to determine the change in slope, if there is one. This is known as the 1st Derivative Test. Helpful Steps:

• 1. Find the derivative of the given function.
• 2. Find x-values where the derivative equals zero or is
• undefined. These are the critical values (locations of possible

extrema).

• 3. Test x-values on either side of the critical values, substitute

into the derivative and observe sign change for maximum or minimum.

• 4. Substitute x-values into original function to get

corresponding y-values for the extrema.

The 1st Derivative Test Slide 32 / 163

First let's practice finding critical values. Remember, a critical value is the potential location for an extreme value. Find the location of any critical value(s):

Critical Values Slide 33 / 163

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For the function, identify the intervals of increasing/decreasing:

Example Slide 35 / 163

For the previous function, find any local extrema.

Example, Continued Slide 36 / 163

For the following function, identify the intervals of increasing/decreasing.

Example Slide 37 / 163

For the previous function, find any local extrema:

Example, Continued Slide 38 / 163

Find any local extrema for

Example Slide 39 / 163

Find any local extrema for

Example Slide 40 / 163 Grading on the AP Exam

Drawing a sign chart is a great way to help visualize what is happening; however, a sign chart alone is not enough of an explanation on the AP Exam. Be sure to defend your answer, describing the sign change of the derivative. For example: The function has a local maximum at (-3,7) because f '(x) changes from positive to negative at x=-3.

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Now that we feel comfortable solving for local extrema, we can go further and determine where absolute extrema occur. Remember, for absolute extrema we must compare all extrema within the given interval, as well as the endpoints of that interval.

Absolute Extrema Slide 42 / 163

Extreme Value Theorem

Given that is continuous on the interval then must attain an absolute maximum and minimum on that interval.

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Which of the following graph(s) meet the criteria for the Extreme Value Theorem? Circle your answer, and for those graph(s) that do not, explain why.

Extreme Value Theorem Slide 44 / 163

Let's take a look back at an example we previously worked with: Example: Find the absolute extrema for on We found this function had a local minimum at and a local maximum at Now, to find the absolute extrema we need to compare these with the endpoints.

Absolute Extrema Slide 45 / 163

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20 If a function has a local minimum at then True False

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Concavity & 2nd Derivative Test

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As we saw in the previous section, derivatives can tell us a lot of information about the original function. Next, we will discover what information the 2nd derivative provides about the original function.

Relationship Between Derivatives & Functions Slide 56 / 163

As we know, in mathematics, not all functions are linear. There are infinite curves created by functions and they take on unique shapes. Let's cover some new vocabulary to describe these curves.

Concavity Slide 57 / 163

Recall that the sign of the 1st derivative told us whether the function was increasing or decreasing. The sign of the second derivative will tell us where the function is concave up or concave down. Why is this helpful?

Concavity Slide 58 / 163

Some functions remain concave up or down for all x in their domain, but many others can change concavity at multiple points. For example: Label each section with the correct concavity.

Concavity Slide 59 / 163

A point x=c is an inflection point (or point of inflection) if a function is continuous at that point and the function changes concavity at that point. Note: Each star represents an inflection point for the function above.

Inflection Point Slide 60 / 163

Use 2 colors to shade the different regions (either concave up or concave down) or shade one with pencil and leave the other unshaded. Then identify any points of inflection.

Concavity & Points of Inflection Slide 61 / 163

23 Which accurately describes the shape at the point ? A B C D E F none of the above concave up; decreasing concave down; decreasing point of inflection concave up; increasing concave down; increasing

a b c d e f g h i j k m n p q r s

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24 Which accurately describes the shape at the point ? A B C D E F none of the above concave up; decreasing concave down; decreasing point of inflection concave up; increasing concave down; increasing

a b c d e f g h i j k m n p q r s

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25 Which accurately describes the shape at the point ? A B C D E F none of the above concave up; decreasing concave down; decreasing point of inflection concave up; increasing concave down; increasing

a b c d e f g h i j k m n p q r s

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28 Which accurately describes the shape at the point ? A B C D E F none of the above concave up; decreasing concave down; decreasing point of inflection concave up; increasing concave down; increasing

a b c d e f g h i j k m n p q r s

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29 Which accurately describes the shape at the point ? A B C D E F none of the above concave up; decreasing concave down; decreasing point of inflection concave up; increasing concave down; increasing

a b c d e f g h i j k m n p q r s

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30 Which accurately describes the shape at the point ? A B C D E F none of the above concave up; decreasing concave down; decreasing point of inflection concave up; increasing concave down; increasing

a b c d e f g h i j k m n p q r s

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31 Which accurately describes the shape at the point ? A B C D E F none of the above concave up; decreasing concave down; decreasing point of inflection concave up; increasing concave down; increasing

a b c d e f g h i j k m n p q r s

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32 Which accurately describes the shape at the point ? A B C D E F none of the above concave up; decreasing concave down; decreasing point of inflection concave up; increasing concave down; increasing

a b c d e f g h i j k m n p q r s

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33 Which accurately describes the shape at the point ? A B C D E F none of the above concave up; decreasing concave down; decreasing point of inflection concave up; increasing concave down; increasing

a b c d e f g h i j k m n p q r s

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Since we are able to graphically interpret concavity, now we will discover how to algebraically determine concavity and points of inflection.

Algebraically Determining Concavity Slide 73 / 163

The first step in finding intervals of concavity algebraically is to find critical values for the 2nd derivative. These critical values will occur when the 2nd derivative equals zero or is undefined.

Critical Values Slide 74 / 163

The sign of the 2nd derivative determines the concavity of the original function. If for on an interval, , then is concave up

• n that interval.

If for on an interval, , then is concave down

• n that interval.

If has a critical value at and changes signs at , then is a point of inflection.

Sign of 2nd Derivative Slide 75 / 163

Find the critical values for the 2nd derivative using the following function:

Example Slide 76 / 163

Find the intervals of concavity as well as any points of inflection:

Example, Continued Slide 77 / 163

Find the intervals of concavity as well as any points of inflection:

Example Slide 78 / 163

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Slide 82 / 163 Slide 83 / 163 The 2nd Derivative Test

This test allows us to identify local extrema for functions by combining information about the 1st and 2nd Derivative. Given that is a critical value of , and is continuous at . Then, if then is a local maximum and if then is a local minimum.

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Note: The 2nd Derivative Test does not replace the 1

st Derivative Test.

Both are useful tools in finding extrema of functions. The 2nd Derivative Test simply provides an alternative method for classifying the extrema. That being said, there are situations where the 2nd Derivative Test would fail to supply any information about extrema: · does not exist · but does not exist · and In any of these scenarios, the 1st Derivative Test would need to be used.

2nd Derivative Test Slide 85 / 163

Using the 2nd Derivative Test, determine if the function below has any relative extrema.

Example Slide 86 / 163

39 The 2nd Derivative Test is used to find absolute extrema of functions. True False

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Connecting Graphs

• f f, f', and f''

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In mathematics, we are often accustomed to being given a picture of a graph and being asked questions about that specific function which is graphed (much like we were practicing in the first section). However, as we venture further in to Calculus, we may be given the graph of the 1st or 2nd derivative and asked questions about the

• riginal function. We must become familiar with the information that

each graph provides us in order to accurately answer the questions.

Analyzing Graphs Slide 93 / 163

So, let's review what we discovered in the first few sections: is increasing when ____________________ is decreasing when ____________________ has a local max when ____________________ has a local min when _____________________ is concave up when ______________________ is concave down when ____________________ has a point of inflection when __________________

Review Slide 94 / 163

The #1 piece of advice when proceeding with questions involving graphs from now on: Ask yourself: Which graph am I looking at? · the original function? · the first derivative? · the second derivative?

Important Advice Slide 95 / 163

At right is the graph of the derivative

• f a function, f, whose domain is the

set of all real numbers and is continuous everywhere. Using the graph, determine the following:

• 1. Intervals on which f is increasing: ________________
• 2. Intervals on which f is decreasing: _______________
• 3. Relative extrema: ___________________
• 4. Intervals where f is concave up: _________________
• 5. Intervals where f is concave down: _______________
• 6. Point(s) of inflection: __________________

Example Slide 96 / 163

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Curve Sketching

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Interpreting information from the first or second derivative can allow us to come up with a fairly accurate sketch of the original function without knowing the equation itself. Think of yourself as a criminal sketch artist - although you may have never seen the actual person, based on eye witness description and detail, you can create a fairly accurate picture.

Curve Sketching Slide 106 / 163

Below is the graph of the derivative of a function f, whose domain is the set of all real numbers and which is continuous everywhere. Use the provided information to create a sketch of the original function, if the

• nly zeros of f(x) are x=-4 and x=0.

Example Slide 107 / 163 Slide 108 / 163

Slide 109 / 163 Curve Sketching Practice Questions Slide 110 / 163

52 Below is the graph of the derivative of a function f, whose domain is the set of all real numbers and which is continuous everywhere. Use the provided information to create a sketch of the original function, if the only zeros

• f f(x) are x = -3.5, x = 0, x = 6, and x = 9.5.

When you are finished, type "done" with your SMART Responder.

Students type their answers here

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53 Below is the graph of the derivative of a function f, whose domain is the set of all real numbers and which is continuous everywhere. Use the provided information to create a sketch of the original function, if the only zeros

• f f(x) are x = -7.5, x = -1.5, and x = 3.

When you are finished, type "done" with your SMART Responder.

Students type their answers here

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54 Below is the graph of the derivative of a function f, whose domain is the set of all real numbers and which is continuous everywhere. Use the provided information to create a sketch of the original function, if the only zeros of f(x) are x = -3, x = 1, and x = 4. When you are finished, type "done" with your SMART Responder.

Students type their answers here

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57 A B C D If the graph of is below, which of the graphs at right is the graph of ?

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58 A B C D If the graph of is below, which of the graphs at right is the graph of ?

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59 A B C D If the graph of is below, which of the graphs at right is the graph of ?

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Rolle's Theorem

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Interesting fact about French mathematician Michel Rolle: He originally criticized the subject of calculus, thinking it did not use sound reasoning and resulted in errors. Later

• n, he published Rolle's theorem and thus approved the

usefulness of calculus. He also contributed to the work on Gaussian elimination.

Michel Rolle (1652-1719) Slide 120 / 163

On the graph below, connect the dots using any function that is continuous and differentiable. · Identify any points on your graph with a horizontal tangent. · Is it possible to connect the dots without creating a horizontal tangent, while still maintaining continuity and differentiability?

Activity Slide 121 / 163 Rolle's Theorem

Suppose is a function that satisfies all of the following: · is continuous on the closed interval [a,b] · is differentiable on the open interval (a,b) · Then there exists a number c, such that a<c<b, and

Slide 122 / 163 A Visual Understanding of Rolle's Theorem: Slide 123 / 163

Find the value, c, that satisfies Rolle's Theorem for the function:

Example Slide 124 / 163 Slide 125 / 163 Slide 126 / 163

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63 Find the value(s), c, that satisfies Rolle's Theorem for the following function on the given interval:

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64 Find the value(s), c, that satisfies Rolle's Theorem for the following function on the given interval:

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Mean Value Theorem

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Consider going on a road trip with your friends. You are driving along and at noon you decide to enter a toll road. You pick up a toll entrance ticket and notice you are at mile marker 1. You also notice the posted speed of 55mph and see no police. At 2:00pm you exit the toll road at mile marker 141. You exit and hand the attendant your toll ticket. After a moment, the attendant says, "That will be $5.00 for the toll and $125.00 for the speeding violation." You sit in the car a few moments thinking; you didn't notice any police cars or speed cameras. Were you speeding on the toll road? And if so, how did the attendant know?

Road Trip! Slide 131 / 163 Mean Value Theorem

Suppose is a function that satisfies the following: · is continuous on the closed interval [a,b] · is differentiable on the open interval (a,b) Then there exists a number c, such that a<c<b, and

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A Visual Understanding of the Mean Value Theorem Slide 133 / 163 Discussion:

How are Rolle's Theorem and Mean Value Theorem related? How are they different?

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Find the value, c, which satisfies the Mean Value Theorem for the function:

Example Slide 135 / 163

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65 Find the value(s), c, which satisfy the Mean Value Theorem for the function:

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68 Find the value(s), c, which satisfy the Mean Value Theorem for the function:

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69 Find the value(s), c, which satisfy the Mean Value Theorem for the function:

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70 Find the value(s), c, which satisfy the Mean Value Theorem for the function:

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Newton's Method

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Newton's Method is a tool for approximating zeros (solutions) of

• functions. It has very useful applications, as this is a process we

complete often in mathematics. The basic idea of Newton's Method is to start with an approximation for a zero and use the tangent line to narrow in on the actual

• solution. This process will become more clear as we work through

some examples.

Newton's Method Slide 144 / 163

We are wanting to find the solution to a function f(x). We start by choosing an estimate, Notice: The solution to the tangent line at produces a closer approximation to the actual solution of f(x).

A Visual Representation of Newton's Method Slide 145 / 163

If we continue this process, of using the tangent line, we can gain closer and closer approximations to the actual solution. Now, we need to define a way to calculate each of these subsequent x values.

Newton's Method Slide 146 / 163 Newton's Method

If is an approximate solution of and if then the next approximation is given by:

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· How do we decide on our first "guess" for the root? –This can be done a variety of ways: · the "guess" is provided for you in the question · graphically or knowledge of the function, itself · Intermediate Value Theorem · How many times should we apply Newton's Method? –This depends on the accuracy you seek for the solution to the original function. Typically, the question will inform you

• f how many digits are needed.

· Note: If the question asks for 6 decimal places, this does not mean that you simply get an answer with at least 6 decimal places. You must get 2 consecutive answers to agree for at least 6 decimal places.

2 Frequently Asked Questions: Slide 148 / 163 Slide 149 / 163

Calculators are extremely useful during this section, due to the fact that the numbers/results contain so many digits, and need to be substituted time and time again. Next, we will cover how to effectively use your calculator on Newton's Method questions.

Newton's Method With a Calculator Slide 150 / 163

Newton's Method Calculator Instructions

• 1. Use your initial guess and substitute value into algorithm:
• 2. Using the ANS button, enter the same algorithm.
• 3. Keep pushing ENTER until you get the desired number of digits to

match/agree.

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73

Calculator OK

Use Newton's Method to approximate the root to accurate to 6 decimal places. Choose your own initial estimate.

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Optimization

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Often in businesses and real world situations, there is a need to maximize profit, minimize cost, minimize travel time, etc. We call these optimization problems. Now that we have a deeper understanding of calculus, we can apply it to these types of problems using the given constraints.

Optimization in the Real World Slide 157 / 163 Helpful Steps in Solving Optimization Problems

• 1. Assign variables to unknown quantities (drawing pictures is helpful).
• 2. Define the constraints for your variables, if there are any.
• 3. Decide on the equation that needs to be maximized or minimized.

Write the equation in terms of only one variable.

• 4. Take the derivative of the equation from step 3 and set equal to

zero, calculating max/min.

• 5. Check your answer against the constraints to make sure it is

feasible.

• 6. Answer question fully.

Note: Make sure to test endpoints if the interval is closed.

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What is the minimum distance between and the point ?

Example Slide 159 / 163

A farmer is creating a rectangular pen for animals and has 160 feet of

• fence. The side of a barn will be used for one side of the fence. What

length and width would produce the largest area for the pen? What would the area be?

Example Slide 160 / 163

75 A can of tomatoes is being constructed from aluminum. The volume of the can is 180 cubic inches. What should the height of the can be in order to minimize the amount

• f aluminum used?

Calculator OK

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76 An open top box is created by cutting out squares of side length x from each corner, and bending up the sides. The cardboard is 18 by 24 inches. What is the maximum possible volume of the box?

Calculator OK

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