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maximum popcorn challenge
Maximum Popcorn Challenge

You wanted to make an open-topped box out of a rectangular sheet of paper 8.5 in. by 11 in. The student must cut congruent squares out of each corner of the sheet and then bend the edges of the sheet upward to form the sides of the box. For what dimensions does the box have the greatest possible volume?

x varies from box to box

Draw a Picture

11 – 2x

Eliminate Variable(s) with other Conditions

x

Use Calculus to Solve the Problem

x

8.5 – 2x

8.5

x

x

The slope of a tangent is 0 at a max

11

What needs to be Optimized?

Quad. Form.

You can’t cut an 4.9x4.9 in. square out of an 8.5x11 in. paper

Volume needs to be maximized:

1.585 in x 7.829 in x 5.329 in

a beginning to optimization problems
A Beginning to Optimization Problems

One of the principal goals of calculus is to investigate the behavior of various functions. There exists a large class of problems that involve finding a maximum or minimum value of a function, if one exists. These problems are referred to as optimization problems and require an introduction to terminology and techniques.

Example of an optimization problem:

A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?

extrema of a function
Extrema of a Function

f(c)

f(c) is an absolute maximum of f on I if f(c) ≥ f(x) for all x in I.

f(c) is an absolute minimum of f on I if f(c) ≤ f(x) for all x in I.

c

I

I

c

f(c)

Let f be a function defined on an interval I that contains the number c. Then:

These values are also referred to as maximum/minimum, extreme values, or absolute extrema.

example 1
Example 1

The highest point occurs at x=b

f(x)

x

a

c

d

e

f

b

The lowest point occurs at x=d

Absolute Maximum:

f(b)

Absolute Minimum:

f(d)

The graph of a function f is shown below. Locate the extreme values of f defined on the closed interval [a,b].

example 2
Example 2

The function may have a limit at the highest point BUT there is no absolute maximum value

The function may have a limit at the lowest point BUT there is no absolute minimum value

f(x)

0.5

.9

.9

.1

.1

.99

.99

.01

.01

x

.999

.999

.001

.001

1

.9999

.9999

.0001

.0001

Absolute Maximum:

None

Absolute Minimum:

None

The graph of a function f is shown below. Locate the extreme values of f defined on the open interval (0,1).

example 3
Example 3

The highest point occurs at x=1 & -1

f(x)

There is an issue because this function is not continuous on the closed interval [-1,1]

1

There is no lowest point because a discontinuity exists at the border

x

-0.5

0.5

Absolute Maximum:

2

Absolute Minimum:

None

The graph of a function f is shown below. Locate the extreme values of f defined on the closed interval [-1,1].

white board challenge
White Board Challenge

Sketch a graph of the function with the following characteristics:

It is defined on the open interval (-7,-1).

It is not differentiable at x=-4

It has a maximum of 5 and a minimum of -4.

the extreme value theorem
The Extreme Value Theorem

Key Word.

Absolute Maximum

f(x)

This function is continuous and defined on the intervals.

x

a

c

d

e

f

b

Absolute Minimum

Absolute Maximum:

f(b)

Absolute Minimum:

f(d)

A function f has an absolute maximum and an absolute minimum on any closed, bounded interval [a,b] where it is continuous.

example 11
Example 1

g(x)

f(x)

1

2

x

x

1

1

2

2

Even though the function has no minimum, it does not contradict the EVT because it is not defined on a closed interval.

Even though the function has no maximum, it does not contradict the EVT because it is no continuous on [0,2].

In each case, explain why the given function does not contradict the Extreme Value Theorem:

white board challenge1
White Board Challenge

The function below describes the position a particle is moving in a horizontal straight line.

Find the average velocity between t = 2 and 4.

relative extrema of a function

Plural = Relative maxima/minima

Relative Extrema of a Function

f(x)

Typically relative extrema of continuous functions occur at “peaks” and “valleys.”

x

f(c) is a relative maximum at x=c

f(d) is a relative minimum at x=d

f(e) is a relative maximum at x=e

a

c

d

e

f

b

Endpoints are not relative extrema.

f(f) is a relative minimum at x=f

A function f has a relative maximum (or local maximum) at cif f(c) ≥ f(x) when x is nearc. [This means that f(c) ≥ f(x) for all x in some open interval containing c.]

A function f has a relative minimum (or local minimum) at cif f(c) ≤ f(x) when x is nearc. [This means that f(c) ≤ f(x) for all x in some open interval containing c.]

relative extrema and derivatives
Relative Extrema and Derivatives

Since relative extrema exist at “peaks” and “valleys,” this suggests that they occur when:

The derivative is zero (horizontal tangent)

The derivative does not exist (no tangent)

critical numbers and critical points
Critical Numbers and Critical Points

Suppose f is defined at c and either f '(c)=0 or f '(c) does not exist. Then the number c is called a critical number of f, and the point (c, f(c)) on the graph of f is called a critical point.

2 is a critical number and (2,3) is a critical point

-3 is a critical number and (-3,7) is a critical point

example 12
Example 1

Find the critical numbers for .

Domain of Function:

All Real Numbers

Take the Derivative

Solve the Derivative for 0

Both values are in the domain.

When is the derivative undefined?

Now find when the derivative is 0 and/or undefined for x values in the domain.

The derivative is defined for all real numbers.

example 21
Example 2

Find the critical numbers for .

Domain of Function:

All Real Numbers except 2

Take the Derivative

Solve the Derivative for 0

Both values are in the domain.

When is the derivative undefined?

The derivative is not defined for x = 2.

Now find when the derivative is 0 and/or undefined for x values in the domain.

BUT x = 2 is not in the domain of the function.

white board challenge2
White Board Challenge

Consider the function below:

Find the equation of the tangent line to the function at the vertex.

example 31
Example 3

Find the critical numbers for .

Domain of Function:

All Real Numbers greater than or equal to 0

Take the Derivative

Solve the Derivative for 0

and 0

2 is in the domain.

When is the derivative undefined?

The derivative is not defined for 0 or negative numbers.

Now find when the derivative is 0 and/or undefined for x values in the domain.

Since 0 is in the domain, it is also a critical point.

example 4
Example 4

Find the critical points for .

Domain of Function:

All Real Numbers

Take the Derivative

Solve the Derivative for 0

Both values are in the domain.

When is the derivative undefined?

The derivative is defined for all real numbers.

Find the y-value(s)

Now find when the derivative is 0 and/or undefined for x values in the domain.

example 5
Example 5

Find the critical numbers for .

Domain of Function:

All Real Numbers

Take the Derivative

Solve the Derivative for 0

The derivative never equals 0.

When is the derivative undefined?

The derivative is undefined for x=-1.

Now find when the derivative is 0 and/or undefined for x values in the domain.

Since -1 is in the domain

critical number theorem
Critical Number Theorem

If a continuous function has a relative extremum at c, then c must be a critical number of f.

NOTE: The converse is not necessarily true. In other words, if c is a critical number of a continuous function f, c is NOT always a relative extremum.

important note
Important Note

Not every critical point is a relative extrema.

Take the Derivative

Solve the Derivative for 0

Find the y-value(s)

is NOT a relative extrema

white board challenge3
White Board Challenge

Find the derivative of the function below:

how do we find absolute extrema
How do we Find Absolute Extrema?

Absolute Maximum

f(x)

On a closed interval, extrema exist at endpoints or at relative extrema.

x

a

c

d

e

f

b

Absolute Minimum

Absolute Maximum:

f(b)

Absolute Minimum:

f(d)

Suppose we are looking for the absolute extrema of a continuous function f on the closed, bounded interval [a,b]. Since the EVT says they must exist, how can we narrow the list of candidates for points where extrema exist?

procedure for finding absolute extrema on an closed interval
Procedure for Finding Absolute Extrema on an Closed Interval

To find the absolute maximum and minimum values of a continuous function f on a closed interval [a,b]:

  • Find the values of f at the critical numbers of f in (a,b).
  • Find the values of f at the endpoints of the interval.
  • The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.
summary of procedure
Summary of Procedure

Find the absolute maximum and minimum of the function graphed below.

Find the values of f at critical numbers

The value of the function at the critical number 2 is:

-3

smallest

Find the values of f at the endpoints

The value of the function at the enpoint 0 is:

1

The value of the function at the enpoint 3 is:

-2

largest

Find the largest and smallest values from the above work

1 is the maximum and -3 is the minimum

example 13
Example 1

Find the absolute extrema of the function defined by the equation on the closed interval [-1,2].

Find the values of f at critical numbers

Domain of f:

All Reals

Find the values of f at the endpoints

smallest

largest

Not a critical point since it’s an enpoint

Answer the Question

The maximum occurs at x=2 and is 11; the minimum occurs at x=-1 and 1 and is 2

smallest

example 22
Example 2

Find the absolute extrema of the function defined by the equation on the closed interval [0,2π].

Find the values of f at critical numbers

Domain of f:

All Reals

Find the values of f at the endpoints

Answer the Question

smallest

The maximum occurs at x=5π/3and is 6.97; the minimum occurs at x= π/3 and is -0.68

largest

example 32
Example 3

Find the absolute extrema of the function defined by the equation on the closed interval [-1,2].

Find the values of f at critical numbers

Domain of f:

All Reals

Find the values of f at the endpoints

largest

x=0 is a critical number too since it makes the derivative undefined.

Answer the Question

The maximum occurs at x=-1and is 7; the minimum occurs at x=0and is 0

smallest