etch-a-sketch (Polynomial Functions)

Due to there not being as many lessons to learn in the last stretch of the semester this blog will cover a topic visited in a previous post. In this post I will demonstrate how to successfully sketch a polynomial graph. As stated in my other post, in order to be able to sketch graphs certain information about the function has to be given. This information consists of what kind of function it is, a table of coordinate points, limits, and sign charts. The information given may or may not contain everything listed. For this post the function will be polynomial. Just to refresh the memory about things to look out for when dealing with polynomials I will give the list of their characteristics again (repetition is key)

Given information helps by:

1. When given what kind of function: you can know what characteristics the graph should contain and which characteristics you can omit

2. coordinate points: helps with were the graph will actually pass through

3. limits: shows were you will have vertical or horizontal asymptotes as well as holes.

4. first derivative: shows were the graph is increasing or decreasing when read from left to right. also used to help determine if a sharp bend is causing the nd at a certain point. sharp bends usually change direction. From observing this sign chart we can get our local extrema

• +,- is a local max
• -,+ is a local min
• I provided visual examples of how these look in a previous blog over local extrema, refer there for examples

5. second derivative: shows were the graph will be concave up or down. also used to help determine if there is a vertical tangent causing the nd at a certain point. vertical tangents change concavity but not direction. From observing this sign chart we can see were inflection points are present

• when there is a change in the sign (+,-) then we know there is one at that point (refer back to blog that covered local extrema for visual examples)

Characteristics of Polynomial Functions:

1. Always have domain of all real numbers

2. Always smooth no sharp bends

3. Always continuous everywhere

Also to refresh your memory is the list of other things that a function graphed could contain:

Asymptotes can be left out because polynomial functions do not have these

We can also leave out the three reasons for not determined (ND) because:

1. discontinuities: polynomials are continuous everywhere

2. sharp bends: polynomial functions are always smooth

3. vertical tangent lines: polynomial and rational functions do not consist of these

Knowing these traits about polynomials is useful so you do not put anything on the graph that shouldnt be there, especially when it states that its a polynomial function. So lets start.

Given info:

f is a polynomial function

Given this info and information on how to sketch the graphs from this post along with the other post we can now sketch our graph. After sketching we want to label any asymptotes local extrema, and inflection points.

Optimizing what you have to the extrema

I think that I may have said what I am about to say enough times to drive some one crazy. Anyhow, as we all know math is a particular subject that does indeed continue to build off of previously learned material. With that being said I will show how optimization and absolute extrema can be found in one problem using knowledge learned in finding each individual one. Since I am a microbiology major, what better question to use than finding the maximum population of bacteria in a culture during a given time period. This post will not contain the specific rules, guidelines etc. as they can be found in the previously posted blogs pertaining to optimization and absolute extrema.

Assume that the number of bacteria R(t) (in millions) present in a certain culture at time t (in hours) is given by

a. at what time between  0-10 hours will the population be maximized

b. find the maximum population

In order to attain the time the population will be maximized we must first find the derivative of the given function

Derivative:

Remember we will want to have it in its simplest form so we can find the zeros

after the derivative is in simplest form we set it equal to zero

since 3=0 is not a true statement we can ignore that part of the function, so:

and

We  know that there cant be a negative time so this gives us an interval that we can use when finding our absolute extrema [0,10]

The population will be maximized at hours

To find what the max population will be we plug 6 in for t in our original function.

The max population will be 2932 million bacteria     OPTIMIZATION HAS BEEN FOUND

Now we can find the absolute extrema, since the question is only interested from 0-10 hours and we know there will be a max population at a certain time we can infer that there will be an absolute max. We also know that when bacteria reach a maximum population then it becomes stationary and then drops due to nutrients running out.

Since we have an interval [a,b] we know that we will have both an absolute max and min.

It is also obvious that we have a max due to the rule that if for every x in the domain and is obvious for min due to its similar rule of if for every x in the domain.

Reflection

One thing that I would like to point out to anyone and everyone is how few people we had in our class. I enjoyed that very much because I felt like I could get the one-on-one attention, even in class, whenever I had a question. For most of my educational career I had not been fond of math because I did not allow myself to learn concepts due to telling myself it was too hard. I always built this barrier and tried to avoid math at all costs. However, this calculus class that I immediately took advantage of when I heard how it was structured, changed my whole perception of math. This was not your average calculus  class, it differed from most in the fact that we were required to make blogs of previously covered lessons.

Having to make blogs over material that we had covered over a certain time frame, was very beneficial to me; I feel as if that could be the only case. The reason I say that is because to achieve a solid grade at each blog check adequate information, tables, and graphs had to be present. Being a student that aims for high grades I always tried to incorporate these things into my posts; doing so instilled the information that much more into my head as to not forget it. When making graphs I would utilize “paint” most often to synthesize my own graphs. This ultimately ensured that I had a more than sufficient handle on the material. To make a correct graph, sign chart or whatever it may have been I had to have a good foundation of the material. Having to make blogs was not the only factor in me realizing my newly found interest in mathematics.

Aside from blogging, having a good teacher in mathematics made the learning less stressful and made me more willing to try to understand what was going on. In other courses of math that I have taken I felt as if the teachers did not explain things as clearly as they could. It does not impress me that they try to explain it as hard as possible…know what I mean? Luckily, my professor for this 2321 class always explained and broke things down in the simplest way. Not to mention, great notes were also always presented in class. I feel like that is necessary for when one goes home to do practice problems; like I have always said practice makes perfect, especially in math! You can tell when your professor gives excellent notes when you go to class but you are not all there and are just going through the motions writing the notes, then go home and can teach yourself what was going on.

Naturally, as a student I have to complain about how tedious making each blog post was. This was the only characteristic of blogging that I did not favor. This slight fall back of making blogs did not compare to the benefits that I gained. After getting the hang of how everything functioned I noticed an increase in my exam grades, and that was no coincidence. I think that this project does not only help the person making the blogs but other students as well. When ever I make a blog I try and put more than enough information in the most simple terms so virtually anyone could follow.

Overall, I strongly believe that people who struggle with math should be introduced to math classes structured like this one. Making blogs of  the calculus material was substantially helpful in retaining the knowledge. Posting blogs encouraged one to use their own voice and language which I feel was a big factor in its effectiveness.

The Raschals

In preparation for finals this blog will consist of the things that I missed on previous exams so I can avoid making the same mistake. After all, math is a subject that demands repetition to have a good grasp on how to do certain things; so this post emphasizes just that. 

Exam 1:

Question 5

Suppose n is a constant and f  is a piecewise function defined as follows:

 

What number must n be equal to in order to make f continuous?

1st: you want to factor out

2nd: set the denominator equal to 0 and solve for x 

 

3rd: after you find the value of x you plug it in to the remaining numerator

So, n must be equal to 8 in order to make f continuous.

Question 13

Find for the function

1. put it in form that is easier to work with

2. bring down exponent and subtract 1

3. lastly put it back in its original form

Question 14

Find the derivative of

1. identify what kind of function it is so we can know what rule to use

Chain function

2. take derivative of outside first

3. take derivative of inside

4. lastly multiply the two together

Question 15

Find the derivative of

1. Identify what kind of function it is so we can know what rule to use

Chain Function

2. put it in a form easy to work with

 

3. bring down the exponent and subtract 1

4. take derivative of inside

5. Multiply the two together

6. lastly Put it back in original form

Question 17

Find the derivative of

Question 18

Find derivative by first simplifying the function using logarithm rules

Exam 2:

Question 18

For this problem we were given a graph of the original function and were told to choose the correct graph of the derivative.

To find this a couple things need to be known

• if a function is increasing then the derivative is positive (above the x axis when graphed)
• if a function is decreasing then the derivative is negative (below the x axis when graphed)

In order to solve this one should look at the intervals that the original function is increasing or decreasing (from left to right)

Identify were there are Zeros present (when the derivative is graphed these points will be on the x axis)

Just from this little list of things the right graph can be identified.

Exam 3:

Find all local extrema for the function

1. find derivative

2. set derivative equal to zero

3. create sign chart with critical values

4. plug x values in to derivative to get the signs for sign chart

5. identify local extrema

1.

2.

 

Local max at x= -3, local min at x= 7 

Optimizing What You Have

If there has been one lesson throughout my educational career that I can say is essential to know for everyday life, it would be optimization. Optimization is utilized when trying to figure out how to get things done using the minimal amount of what you have or maximizing, getting the most out of what you have. Whether you notice it or not you are partaking in optimization too on a daily basis.

For example say your parents have too much stuff crowding the house and they want to organize it and put it in a designated spot, they decide to build a storage room. They need it to be a certain size and want to save as much money as possible. In order to figure out the necessary information for building this they will use what we learned in calculus as optimization.

GUIDELINES AND COST OF MATERIALS:

• Storage has to have a volume of 125 cubic feet
• concrete for base costs 3$ per sq ft
• material for roof costs 4$ per sq ft
• material for sides costs $3.50 per sq ft

With the given information we must find the dimensions of the most economical storage room.

The first thing to do when trying to solve an optimization problem is find the right function to use. In this case we will utilize a cost function.

• it is also beneficial to identify and state the constraints off to the side of your work and give variables to things that will be used. Putting variables in is needed to show how each contributes to the problem at hand in an equation format.

Cost Function:

c=(# of sides)(price)(area)

                                                           

Constraints:

Once you know your constraints you need to form an equation for reaching this value, since its a storage room in the shape of the box with a square base we know two of the variables will be the same in the formula for volume of a box.

IMPORTANT:

As you might have noticed our cost function contains two variables (x and h) when multiplying for the area. This poses a problem so we need to get everything in terms of x this is done by using the equation formed, due to the constraints, and isolating h.

        Instead of working with a hard fraction you can bring the denominator up, remembering to change the sign of the exponent. Dont forget to put it back in its original form later.

Now that we have everything in terms of x we can plug in values and start to solve

Now you can plug in h.

Once you have gotten your cost function to its simplest form, the next step in achieving the desirable dimensions is to find the derivative of the cost function.

                   DERIVATIVE

In order to find the value of x you set the derivative equal to zero and solve for x 

since x was the same length we have the length of two sides, to find the length of the missing side h, we simply plug our x value into the equation that we found h to equal.

Now we have all the lengths of each side necessary for building a storage room that has a volume of 125 cubic ft in the most economical fashion.

The dimensions are 5ft x 5ft x 5ft

Ancient times, Sketching Graphs (Rational Functions)

Before taking this calculus class I would have never imagined being able to come up with a graph without my graphing calculator, however it became apparent that it is possible. With a few characteristics to remember and previous knowledge from old posts I will show how to sketch a graph when given some information. First off I will give a breakdown of different things to look out for and remember, based off of the information provided. So far the information that has been provided to us in coming up with a sketch has been: The type of function it is, a table of coordinate points, limits (if there are any), and the first and second derivative sign charts. Reviewing how to interpret the different sign charts would be beneficial if you have forgotten what they do.

I have constructed a list of characteristics that can be present in a graph, knowing these will help you successfully sketch the graph:

THINGS TO KNOW:

Polynomial Functions:

1. Always have domain of all real numbers

2. Always smooth no sharp bends

3. Always continuous everywhere

Rational Functions:

1. Polynomial in numerator and polynomial in denominator

2. Continuous everywhere they are defined

3. Can have vertical asymptotes

4. Can have only 1 horizontal asymptote

5. Can have holes

6. Always smooth no vertical tangent lines

Vertical Asymptote:

1. Obvious when x is approaching a number c and is equal to infinity Ex.

Horizontal Asymptote:

1. Obvious when x is approaching infinity and is equal to some number Ex.

2. These only matter at the ends of the second derivative sign charts

Holes:

1. Present when x is approaching a number c and is equal to some number l Ex.

Reasons For Not Determined (ND):

1. Discontinuity

2. Sharp bend (can be spotted when direction changes from first derivative sign chart)

3. Vertical tangent line (can usually be identified when concavity changes from second derivative sign chart)

Using this list of things to know and previous topics covered, you should know be able to sketch a graph. lets try it.

Given Information:

f is a rational function, with a domain of all real numbers except -6

The way I go about arriving at the correct sketch is using the list of things I know to out rule certain characteristics that I THINK the graph could consist of then plot my points and use what I figured out. For example, for this problem it gives that its a rational function; therefore it can have only one horizontal asymptote, vertical asymptotes and/or holes, its always smooth and doesnt have vertical tangent lines.

Being able to know these characteristics shows you what is not on the graph and lowers the chance of sketching the graph wrong.

Things that CAN NOT be the case:

• For ND

1. Vertical tangent (rational functions do not have these)

2. sharp bends (rational functions are always smooth)

-With that being said the reason for being ND is due to discontinuity which can be due to a couple reasons. However previously in this blog for things to know it was stated how to recognize when a hole is present. A piece of the information given matches up with one of the things to know and reveals that there is a hole present.

• You now know the reason for nd is due to a hole at the coordinate stated (-6,2).

Using the list it also reveals that there is a horizontal asymptote at y=3  because x is approaching infinity and equals some number l 

Remember: horizontal asymptotes only matter at the ends of the second derivative sign chart

Sign charts:

First derivative sign chart: tells where the graph is increasing or decreasing. And local extrema.

You also use this to graph your zeros and other critical values. Since a rational function is always smooth you can go ahead and draw in the curve at these points.

Second derivative sign chart: tells where the graph is concave up or down. And inflection points.

Everything that I have just stated is sufficient to sketch the correct graph first plot coordinates then apply what you have figured out. the graph should look like this:

Extrema that are Absolutely there

This particular post deals with absolute extrema. Absolute extrema are similar to that of local extrema however different guidelines are used when determining absolute extrema. For example, an absolute maximum is present when x equals some number c,  and that number is greater than or equal to any other number c at some x coordinate for every x in the domain. The same thing can be said for an absolute minimum but the value c is less than or equal to. When finding absolute extrema we can see that a domain is necessary.

These two things can be more easily explained by:

Absolute Maximum: if for every x in the domain.

Absolute Minimum: if for every x in the domain.

Graphical representation Examples:

 

 

The first thing that should be done in finding the absolute extrema is identifying what can even be present(absolute max or absolute min). Something used in making this identification a little easier is the Extreme Value Theorem.

Extreme Value Theorem states: If f is continuous on the closed interval [a, b] then it is guaranteed that it will have an absolute max and min on [a, b]

Reminder: being continuous means you can draw the graph with out lifting your pencil.

Note: [ ] indicate that the interval is closed (greater than or equal to, less than or equal to) where as ( ) indicates that the interval is opened (greater than, less than). this will be useful when trying to answer questions.

Now that you have a better understanding of absolute extrema I will list then demonstrate the steps necessary for finding the extrema for: on [0, 2]

!st: take derivative of the function given

2nd: find the critical values by setting derivative equal to zero

3rd: Check critical values to make sure they are within the boundaries (these will be used)

4th: Plug values of x into the original function given (critical values, if in boundaries, and values given from boundaries)

5th: Make a chart of the x values and their corresponding y values to make it easier to see your absolute extrema. (this is not necessary but it reduces room for error)

Lets begin: Find absolute extrema for on [0,2]

Note: we know polynomial functions are continuous everywhere and we are given a closed interval, both of which meet expectations for the extreme value theorem. According to the Extreme Value Theorem, we will have an absolute max and min present in this problem.

1st: Derivative-

2nd: Critical Values-

Remembering basic rules we can factor out a 6 and solve for zeros to give us:

and

and

3rd: Check Critical Values- -2 is not in the interval so you will not be using it in finding the absolute max and min.

4th: Plug In Values- 1, 0, 2 each into

5th: Make A Chart- which ever number is lowest for y values in the given x domain then that is where the absolute min is. which ever is highest in the given domain then that is where an absolute max is present.

 

Answer: Absolute max of 8 at x=2 and absolute min of -3 at x=1 and this function graphed it is clear that these are the absolute extrema.

 

For this graph when the interval is opened, (0,2) there is no absolute extrema because there are other values that can be higher or lower for this particular graph. According to what I said in the beginning of this blog it all matches up.

 

 

 

Derivatives applied to real world scenarios

You might have been thinking that derivatives have no relevance to everyday life. However, derivatives are important in figuring out the rate at which things are occurring. This could be crucial for our environment as well as the animals and things that make up the environment. To give an example of this I have chosen a word problem from the text book for calculus for life science 1 to demonstrate, this and many more problems similar can be found here:

Greenwell, Raymond, Nathan Ritchey, and Margaret Lial. Calculus For The Life Science . 1st ed. United States of America : Person Education , 2003. Ex 214. Print.

51. The net U.S greenhouse gas emissions from human activities for the years 1990-1997 can be approximated by:

.                                                            

Where t is the year since 1900 & E is measured in millions of metric tons of carbon equivalent. assuming that the emissions continue to follow this formula, answer the following:

a) Calculate the net U.S greenhouse gas emissions for the year 2002 (we do this by finding t how many years since 1900 and then plugging  it in to the original function).

where t = 2002 – 1900

t= 102

 million metric ton/ 102 years

This function graphed is represented below. With years being on the x axis and the amount of greenhouse gases emitted on the y-axis.

b) Calculate E'(102) & interpret your answer (this is found by taking the derivative of the original function and plugging in 102 for t).

million metric tons/year

The derivative of the function is telling us the rate in which the green house gases are changing per year. By the year 2002 the rate of change will be increasing by 37.3 million metric tons. A graphical representation looks like;

Here using the derivative gives us an idea of how much the green house gases are increasing by each year. This can be useful in determining ways to minimize the production of green house gases. It also allows us to see just how much damage we are causing each year and may introduce ways to accommodate for it.

Concavity and Sign Charts

Concavity is another quality of a function that we can get from a sign chart, the sign chart from the second derivative. Applying all the information given in the last blog in addition with info from this blog you will see how they are used together. Concavity is found from the sign chart of the second derivative. the second derivative is the derivative of the derivative.

Notation of 2nd Derivative:

• F”(x)
• 
• 

Things to know:

• If the second derivative is positive then the function you were originally given is concave up when graphed.
• If the second derivative is negative then the function you were originally given is concave down when graphed.

Even though a function can be concave up or down it can still be either increasing or decreasing. So there can be any combination to a function of Increasing/decreasing and concave up/down.

The picture above found at http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx gives a good visual of what is being said.

A term not talked about in the previous blog is inflection point. An Inflection point is a point where the function changes concavity, that is from up to down or vice versa. This information on concavity is retrieved from the sign chart of the second derivative. Now to combine everything from this post and last post we can figure out concavity using sign charts.

1.

                          

Now we determine the second derivative and its sign chart:

  

Once we have made our sign charts we can now determine our concavity and inflection points for the original function Concave up is denoted on the sign chart for second derivative when there is a positive sign (+) and down when minus sign (-), so:

Concave up: and

Concave down: (0,2)

Inflection points: 0 and 2

Inflection points are obvious because its where the sign changes. Based on everything we found it should match up with the graph below, just as it does.

Becoming Familiar with Key Terms and Ideas (sign charts & functions)

Unlike many of my other blogs that utilize sequence of equations to relay the help, this blog will be used to familiarize you with the terms and concepts associated with sign charts and information about functions. In order to know how to do things one must know what is being talked about as well as what is being asked of them. So again, this is a lesson on sign charts and qualities of a function in general terms.

SIGN CHART: A sign chart relays information of a function, such as where the function is positive, negative, has zeros or discontinuities.

Note: Whenever the word “where” is used to explain what a sign chart is used for, it is always talking about the values on the x-axis that make a function positive when the values on the y-axis are also positive.

Something good to remember is that a function can only change sign at its zeros and discontinuities. You will see something similar to the graph below if the function has these qualities.

Partition Points: points on a function where it is zero, and points where it is discontinuous.

For the sign chart (which is a number line) we label only partition points and indicate why they are a partition point. Based off of the above chart I will identify the zeros and discontinuities of the functions. Zeros of the functions are simply the value of x when y is zero, and discontinuities are where you have to pick your pencil up to continue the graph, so:

ZEROS: 8

DISCONTINUITIES: -1.5, & 2

Labeling partition points:

• 0: used for when it is a zero
• dc: used for when there is discontinuity
• nd: used when it is not defined (this is a special kind of discontinuity)
• Also don’t forget to label which function you are dealing with

For the function that is graphed a labeled sign chart looks like:

For this graph we can see that there are only zeros present so putting nd, or dc is not necessary. however if there were points of nd, or dc then we would simply place those labels above its respective value just as we did for the “0” above -2 and 2.

THINGS TO REMEMBER THAT ARE EQUIVALENT AND WILL BE USED AGAIN HERE:

• The derivative
• The limit of the difference quotient
• slope of tangent line
• instantaneous rate of change
• slope of the function

Based off the following list some conclusions can be made:

Increasing:

First off, if a function is increasing then it is going up as you move from left to right (like reading a book). If this is the case then its slope is positive which also means the derivative is positive.

Decreasing:

If a function is going down as you move from left to right then it is decreasing. a decreasing function has a negative slope which also means its derivative has to be negative.

IMPORTANT TO KNOW: An x-value, c, in the domain of the function f is a critical value if either:

•   OR
• 

LOCAL MAX: Highest point in its area on graph (neighborhood)

LOCAL MIN: Lowest point in its area on graph (neighborhood)

NOTE: Local extrema can be found from sign chart by looking at where the derivative changes signs.

• If derivative changes from (+) to (-) there is local max
• If derivative changes from (-) to (+) there is local min

Given this info and a sign chart of the derivative we can identify where the function is increasing or decreasing as well as local max and local min.

Example:

Looking back at the rules about how local extrema can be found we see that -5 and 4 are local max and 1 is a local min. The function is increasing from  and  and is decreasing from  and