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 MaThCliX | Oct 20, 2015 | Math
I come across students daily who struggle in their academics, particularly in mathematics. For the students who are not trying or do not care, this message isn’t for you. Unfortunately, there is not much anyone can do to help someone who simply will not try or does not care. For all others, I see many students quietly failing their class as time continues to pass by. One bad grade after another accumulating. At some point, it may possibly be too late to undo! As I begin to understand and discover why this continues to happen to students who are “doing all the homework and still failing the tests”, what I have learned is the simple fact, you can do it, not always alone.
So, students, here are a few tips for you to win your math struggle…
- Do ALL of your homework. And furthermore, write it out on paper, step by step, even if that means you have to kill trees. Learning math means doing math and the paper is worth your learning!
- ALWAYS use answer keys while doing your homework. First, do the problem, then check your solution. Instant feedback is essential to correcting mistakes.
- After doing your homework, if there were types of problems that you didn’t understand or missed a lot of, ask for more practice problems from your teacher or find some on the internet or textbook.
- Okay, so you can do all of your homework? Does that really mean you are ready for that quiz or test? Not necessarily! Create a similar pre-quiz or test and TIME yourself and GRADE yourself. If you are going to make mistakes, make them on your practice quiz/test, not on the real one!
- Build a good working relationship with you teacher/professor. Visit tutorials and office hours. Let them know who you are and show them you are trying. If you are a college student, see if your university offers tutoring or a math lab.
- Create study groups with other students who care and are willing to work hard and want to succeed. Work problems together and check each other as you go.
- Write note cards with important notes/formulas as you go to keep everything in one place.
- And if you still need support, that’s okay! There’s MaThCliX!
You don’t “study” math, you DO it! Bottom line. Know your resources and use them. Others have gone before you and no one becomes a huge success all alone.
by Amanda Fagan | Oct 13, 2015 | Math
Change Your Mind, Change Your Grades
The other day, I was speaking to a friend who recently graduated from college with an Engineering degree. He worked full time during his whole time in school and still managed to graduate with exceptional grades. When I asked him how he did it, he said, “All of those people failed because they told themselves it was hard.” This made me think of all of the difficult classes I have taken over the years—and my attitude while I took them! I confess, I often did tell myself the work was difficult. I would find myself thinking, “This is too hard. I don’t feel like studying right now,” when in reality, those were the times when I needed to work even harder. How much better would my grades have been in those classes if, instead of feeling discouraged, I approached my work with feelings of excitement about the opportunity to learn so many new things? However, sometimes being positive is difficult. Here are some ways to make it a little bit easier:
- Set small goals.
Learning is a process. Sometimes the end goal seems completely unattainable. Try to take it one step at a time. Setting smaller, more attainable goals will give you the confidence boost you need to keep going. As you move towards each goal, you will find yourself improving more than you ever thought you could!
- Gain some perspective.
Try transforming all of your negative thoughts into positive ones. Instead of thinking, “This is too hard!” or “I just can’t do it,” try thinking about all of the progress you have made. Think of your mistakes as opportunities to learn even more. We learn far more from mistakes and failure than we do from doing everything perfectly the first time. Difficulty is simply another opportunity to practice. Try thinking about your homework as an exciting opportunity to learn and practice instead of a trial you must suffer through. Don’t ever think that you cannot do something. Instead try thinking something like, “I can do this, but it might take a little bit of work.”
- Never give up!
The most important thing to remember is to never give up. As long as you keep trying and keep working, eventually you will realize you learned something! Even if the steps you are taking are small, you will eventually have big results. Remember that every bit of practicing gets you a little closer to where you need to be. If you keep trying, you always have the possibility of someday reaching your goals. If you quit now, you have the certainty that you will not.
Remember that your thoughts and attitude have an enormous effect on your studies! Learning becomes much easier and much enjoyable when you treat it as something that can be fun instead of as a chore. It may be difficult at first to have a completely positive outlook. Start small. If you don’t honestly believe what you’re doing is fun, try at least pretending it is. You’ll be amazed what a difference your thoughts make!
by Ty Hilton | Oct 6, 2015 | Math
Dealing with Stress
Throughout my academic career, I have dealt with plenty of stress, whether it was from presenting a project, writing a paper, studying for a test, or writing a blog for work. In any of these situations, it is imperative to not succumb to the stress. Procrastination is a result of stress, but not the only one. Stress can motivate us to accomplish goals, but only if we have the right mindset. The difference between procrastination and completing an assignment promptly is positive thinking. Instead of overwhelming yourself with how much you have to do, try being thankful that at least you have something to do. Having free time is nice, but it can get really boring. Also, try to see the benefits of what you are doing. Try to see what you can learn from doing the assignment. In addition, Focus on how you will feel when you are done, and let that drive you. For example, walking away after completing a test is the best feeling for me. I look forward to that while studying for the test, and it usually motivates me. Lastly, if for any reason you procrastinate on an assignment, and find that you have more work to do than time, then I have one final piece of advice for you. Wing it, and don’t give up. Procrastination may be bad, but it can’t prohibit success. Do your best in any situation and you’ll probably do fine.
by Kenneth Griner | Sep 23, 2015 | Math
The Adult Struggle with Math…
I came across a video the other day that really got me thinking about the generation before me and where my generation might be heading. The video was entitled “Adults Try 5th Grade Math”. Since I enjoyed watching the show “Are You Smarter than a Fifth Grader?” I figured this would be an enjoyable video as well. As I sat there and watched all of the grown-ups who claim to have taken Calculus and other high end math courses that you need for a Master’s degree in some fields struggle adding fractions I began to laugh. This is no laughing matter though. There are adults all around us who cannot do math some elementary school kids find simple. So, why do adults struggle with 5th grade math?
These struggles make you wonder why. Why is it that when school is over with and in your rear view window you forget what you have “learned”? Is it because you focus on what you need for your career and forget the other things you saw as unimportant and not needed? Is it because you get so used to doing derivatives and solving some ridiculous equation that looks like hieroglyphics to others? I believe that it is because nobody has really been taught the lessons that elementary students are learning now.
Think about it. When you were in school it was all about making a good grade; not actually learning what was on that test. I know I have walked out of a test and not remembered how to do a single thing on it but that piece of paper had a big fat A on it. This brings up a whole issue I have with school systems and grading that I will have to bring up another time. Math is like a house. It starts with numbers and basic adding and subtracting which is the foundation and builds up from there. My wish is for the generations before me, my generation and the ones after mine to know that and not forget it. It is nice that you can find the derivative but if you cannot add fractions it is like owning just a window of the second floor of a house that is missing its ground floor. Go back and learn why something happens and you will truly understand it. If you cannot understand why you are already behind.
by Hanna Fleeman | Sep 22, 2015 | Math
We all know how important it is to learn good organizational skills and get organized. Imagine discovering a new law of physics and not remembering where you left your proof! Keeping organized is one of the most important ways to DO MaTh and DO it successfully, so here are some tips to keep you or your student organized.
- Keep everything together.
All of your MaTh assignments, notes, quizzes, tests, and homework should be kept together! Get a binder, and fill it up! Use dividers to separate different categories of assignments, and keep your assignments in order. All of this will help you to easily reference your materials, just in case you need to go back and find something.
2. Keep your notes in order.
A lot of MaTh builds on top of itself. Something you learned in chapter one could be used in chapter three. Keeping things in order will make it easier to go back and reference later, just in case you need to.
3. Organize the work itself.
We all know those lines on the paper are supposed to be used for something. Show your work step by step on your paper! This way, if you get an answer wrong, you can easily go back and see which step you messed up on. This is especially helpful when using order of operations. Finish a step? Write the next one on a new line. Simple, easy (and also very helpful to your teacher who is grading your work).
4. Keep your homework separate and check those answers.
Homework is the most important part of learning MaTh because it’s where you DO MaTh. Check your answers as you work. Did you get one wrong? Go back and explore to see why (look at those steps you’ve been writing out line by line). Ask questions! Whether in class or at MaThCliX, your teachers or tutors are there to help! It’s our job and we are thrilled to do it!
5. Organize your space and make time.
We all know MaTh takes time! And practice makes perfect, so be smart with the time you have. MaThCliX is a great place for you to come and DO MaTh, but it can’t all be done here. Set aside time every day to DO MaTh. Just 30 minutes a day can greatly improve your understanding and techniques. You should also have a specific space in which to DO MaTh. This space needs to be organized, neat, and distraction free. This will help you focus on your work and learn effectively.
Becoming more organized will help you or your student become less stressed and more confident in facing problems head on. Using these tips, make a plan. Organize a space, make time, work it out, and come to us with questions. Have your work neat so we can see exactly what you are doing, and overall be confident! Everyone can DO MaTh, and being organized is just one way to make it easier.
