by Nicole Dowling | Nov 2, 2015 | Math
Derivatives and Curviness
How to use Derivatives when Describing “
Curvy-ness” of Functions
Most students when they first learn about derivatives in Calculus are not exposed to the many uses of this important topic. However, students have heard the term “derivative” disguised as words like SLOPE and RATE OF CHANGE! So how do we use derivatives when describing how curvy a function is? Well, since derivative is just a fancy word for slope, how about finding the slope at select points to see if we change from a positive value to a negative value, or vice-versa, or if we even change signs at all!
For example, I was tutoring a student who had to find the maximum and minimum of various functions on some closed interval. Well we know that if we have a closed window we are looking into, then at some point the curve we are analyzing will have a highest peak and lowest trough somewhere, even if those points are the endpoints of the specified closed interval. Think about a linear function: the derivative of any linear function will be constant. What does that mean when describing the “curvy-ness” of linear functions? It means the slopes at each point in some closed interval is fixed, i.e. if the derivative is negative, then the slope is negative, then the maximum will be the leftmost end-point and the minimum will the rightmost endpoint. *Note the max is the rightmost endpoint and the min is the leftmost endpoint when the derivative is positive* Now what if we have an exponential or logarithmic function. Well, the same argument applies: if the derivative of either function is positive, then the rightmost point will be higher than the leftmost point; if the derivative of either function is negative, then the opposite is true. Why is this true? Think if someone pulled a linear function with some slope (other than 0) straight in either the x direction or the y direction; now we have a bend in our curve, yet we will still have a positive or negative slope AT EVERY POINT!
Now let’s talk about some more interesting functions. What about wiggly and curvy functions??? A wiggly function, like sin(x) or cos(x), will oscillate between some y-axis interval. Meaning we have a maximum y-value and a minimum y-value between some closed x-axis interval. The question is WHERE? We can no longer assume the max/min will be at the endpoints. Remember a term called “critical values”? I can almost hear the moans and groans bringing this term up again, but do not fret. Basically, we need to know when the signs of our slope changes. The method? You guessed it! Take the derivative, and this time set that derivative equal to zero. Why? We want to find out when our function has zero slope. Think about that moment at the very peak of your favorite roller coaster: there is a very brief moment when the machine no longer has to trek your car up a very steep slope, and you are motionless as you look upon the horizon straight in front of you. Or when your car starts its trudge down the other side of this steep slope, down the pits of gravity when all weight is thrusted perpendicular on top of your shoulders into the seat of your car. This is the very most bottom of the hill, at least for that section of the roller coaster, i.e. some interval of this curvy coaster! So the same argument applies for polynomials of some high degree, where the function wiggles many times in some interval. Just find the places where we have zero slope and calculate whether we have positive or negative slopes BETWEEN our critical values and voila! We know where our peaks and troughs are, and therefore we know the “curvy-ness” of any function.
All this can be analyzed WITHOUT graphing these functions at all! Pretty cool right?
by Nicole Dowling | Aug 17, 2015 | Math
Mathematics IS Truth
When I first came to college, I went through an emotional culture shock. I had no idea what kind of box we were all in throughout high school, but once I took the next step toward supporting myself in the adult world, I see now that it’s a scary thing breaking away from the habit of high school. Ordered. Structured. Dates were set for you; your whole schedule seemed all planned out. Of course, personal success depended on each and every student outside of the classroom, but you were surrounded by support: teachers, other students, the atmosphere of it all.
College has the same feel, don’t get me wrong. But you are not in school 8 hours a day, 5 days a week anymore, from someone else’s design anyway. You are setting your own schedule, creating your own support groups and may even have time in between classes to do with how ever you please. Those who are diligent will continue that “all day at school” mentality in order to keep up with your own studies. Success can look like many things, sometimes it can just be encouragement from a friend to continue pushing through the hard studies. This is supposed to be a team effort.
The acceptance of change in one’s own life can seem like being on the tallest hill and down below you oversee the obstacles that lay ahead, as dangerous as they may appear. Sometimes you are on that hill and all you see when you look down is fog. Unknowing to all that is ahead of you. But you know in your heart it’s time to keep going. For something. Chest out. Head high. One foot at a time, till you’re on your path of finding your own truth. Spoiler alert: Mathematics IS Truth.
Mathematics uses proof and logic in order to state the truths about our universe. The use of a proof is simply a description of why something is true and uses all possibilities to obtain these proofs.
But we have a slight advantage: there was a curious spark, an omnipotent insight that pops in when truth is all around. It feels good. And if you pay attention, the world seems to flow in harmony.
So, if you ever find yourself on top of that tallest hill, delve into mathematics. There’s a branch for everyone’s interests and you might find that creative brainwave needed to take your first step. Every student, thus every human, has an influence within the world in which we live. Influenced by movement. Not necessarily which choices to take next, but the act of choosing as they come. This is a fast existence. Darwin should have called his studies: Survival of the Adaptive.
by Nicole Dowling | Jun 25, 2015 | Math
Sacred Geometry, Mandalas, and Mathematics…
How many different shapes do you see in this image? There are circles and triangles and hexagons; oh my! A little closer look might reveal more intricate shapes…
I encourage everyone to draw this image. It may be messy at first, but you’ll start to see how these shapes are all pieced together. And you might discover more shapes during your artistic voyage!
Drawing Mandalas have been a big part of the Hindu and Buddhist history. The word “Mandala” is a Sanskrit term for “circle” and can be thought of as a schematic visual representation o
f the Universe. These images have been used in meditation: symbolizing the journey around (or in) the space one wanted to discover in greater detail. That space can be the around the Earth, in the Universe, on even in one’s own Mind. By visualizing a path to traverse around the Mandala, one is thought to have a clearer mind on how to move through the Universe as a whole.
Mathematics is about communicating the patterns that one observes about the Universe with great precision; thus the more practice one has in creating these detailed diagrams, the more likely one notices the details within our beautiful, yet complex, Universe. Mental discipline is a useful tool when it comes to mathematical thinking… Some people draw these mandalas with immense accuracy in the sand; then, once done, smooth out the sand as if no image was there in the first place. The emphasis here is in the process. The act of drawing is what’s important, thus when the image is done it’s immediately erased. The lesson is to show us how to let go of things we work so hard on; to be okay with washing away the hard work we have done. To practice being adaptable! So draw on young explorers and produce your perspective with critical specificity!
Thank you to those who took my class: FractalRock!
by Nicole Dowling | Apr 18, 2015 | Math
Finding Your Way…
Calculus 1. Check. Calculus 2. Check. Calculus 3. Check. Mathematical modeling. Check. Linear algebra. Check. Modern algebra. Check. Differential equations. Check. Probability & statistics. Check & check. …I have only three “checks” left before I receive a piece of paper saying I’m a somebody in the political world.
At least that’s the mentality I had for awhile before I realized why the heck I’m putting myself through a mathematics degree…
We all go through that BiG QuEsTiOn phase in our lives; that moment when you realize the thing you’ve been forcing to work for so many years is actually not the right path for you. And that’s okay. But it’s taken me about a year to accept my change. This is a personal story.
My freshman year in college my mind was so..well..fresh, so ready to soak up All The Maths! My interests were all over the place: fractal dimension, fluid & air dynamics, physics, graph theory, topology, sacred geometry, knot theory. I’ve been fascinated with how this world fits together and, more impressively, the constant dance of each system moving as a whole. I wanted to learn it all! I even stretched into areas like molecular structures, artificial intelligence, philosophy of mathematics, cause & effects of human behavior, sociology, economics, anatomy. ANYTHING that incorporated mathematical thinking in some form or fashion. But by the time I was a sophomore I barely had any energy left to breathe! So I made it by doing the bare minimum for a while. Perhaps it’s not the best decision to present your faults, but from time to time it shows us all that we are just…human.
So what happened?
Well, all my interests has one underlying pattern: analyzing theoretical possibilities at its most fundamental level. AKA, the infinite! No wonder I was drowning. I love exploring possibilities, but I am not a theorist. Maybe one day, but not yet.
Now I still consider myself a child of wonder when it comes to mathematics, but its not all rainbow daisies and frolicking unicorns. There’s hard work involved. There’s pain and sweat and tears involved. My relationship with mathematics was on the rocks and I thought about breaking up with it for an insane moment there. So I took some time away from the usual, I took time to myself for awhile and not what mathematics demanded of me. I matured more during that time, this time, than I ever had/have before. I am pleased to say that our relationship is so strong that I couldn’t possibly live my life without it. So we made up (good thing too because that graduation date is approaching), but there were conditions. Instead of beating myself up for not being the theoretical analyst I thought I was preparing to be in grad school, I took the path as a mathematical communicator. I am a mentor for those who cannot see the connections I can; I have a gift of visually expressing these fundamental concepts that are so important in logic. Sometimes all it takes is a new approach, a different way of looking at something that was seemingly so complicated to understand why it is what it is. I am just another piece of the puzzle, but an important one. We have the ability to connect ideas in milliseconds with our advancements in technology, but we need to know how to use our tools. Like my crazy professor says, “The day they make a calculator that can do mathematics I want to be first in line.” Anyone can compute, but it takes a special care to do MaThemaTics.
by Nicole Dowling | Feb 25, 2015 | Math, Teaching and Learning
Have you ever thought about your teaching from a student’s point of view?
Now that I am a senior at Kennesaw State University, I have experienced a plethora of time in the classroom. What makes a class period more enjoyable? How do we, as students, get involved EVERY time we enter the room? As many of you have probably noticed, there are effective ways that teachers teach…and not so effective ways. Lets be honest, when the teacher isn’t engaged in the topic he or she is monotonously speaking about, then why should the rest of the class be? What a waste of time for everyone. And let me share a secret: learning takes time.
How many have been frustrated with a teacher that doesn’t teach well, or worse, doesn’t teach at all??? Responsible students are, then, forced to be both the teacher and the student. And we’re not even the ones getting paid! Math that. Or how about this scenario: learning something wrongly and then having to go through the painstaking process of breaking that bad habit by UNLEARNING that something, only to have to go through even MORE practice to set the new good habit in its place. Whew! That’s an awful lot of work that could have been avoided simply by effective teaching.
Please enjoy the following experience from a fellow student, me:
I loved going into my Real Analysis course because my professor would get so enthusiastic about his subject that he spat everywhere when he went off on a tangent. (Ha, math jokes.) Sitting in front of his laptop, he would write his lesson right before our eyes. “Mathematics should come from the heart,” he would nearly whisper as he was deciding on which way to prove something. He emphasized that great mathematical writing comes from good grammar: “Mathematics is hard enough when written correctly. Proper grammar makes the math easier.” His tips on proof-writing bleed into all areas of my life because quite frankly I have little skill in the matter. But I digress.
Some days I just wanted to listen to a good story after a long day of rigorous studying, and his class was just the relaxing break I needed. As it always should be with learning: first, soak in new material; then, play around with it in one’s personal time. “The learning happens outside of the classroom, when you DO the mathematics” as my professor would say. For this reason he records all of his lectures with a video & audio software. Actually, he requires that NO ONE take notes.
How many have tried to fiercely write all of what the teacher said or wrote on the board? So much room for error: what with constantly looking up and down and up and down, or comments meshing into each other, or not hearing everything because you were too worried about writing down the previous thing… Whew. My hand hurts just thinking about it. However, with a recording you have the notes you need, VERBATIM. A recording also sets the standard for the teachers to be professional and not slack off, because no one wants to look bad on camera 
~Now teachers here is a question for you: Would you want to be a student in your own class? ReadRebecca’s blog for more on this topic.
