The world is full of change. In fact, some might say that’s all existence consists of.
Perhaps this is why humans are so concerned with change. It seems we can’t get the idea of it off of our mind. Everybody seems to want a bit of change. Yet, when we get change, we still seem to want more change, new change, or old change. Rarely do we not want change at all.
What’s fascinating about math is that it offers advanced tools to measure change. In particular, when change can be described using functions, math is really effective at analyzing these functions and the various properties they have. These properties yield important information concerning whatever system is being studied.
In this article, we discuss concavity calculators, an online tool that many students use to analyze how functions change over time. First, we offer a brief overview of a concavity calculator, emphasizing its benefits and drawbacks. Then, we offer some tips on how to use a concavity calculator, ending with our final thoughts on the future of concavity calculators.
AN OVERVIEW OF A CONCAVITY CALCULATOR
A concavity calculator is any calculator that outputs information related to the concavity of a function when the function is inputted.
Inflection points are often sought on some functions. These are points on the curve where the concavity of the function changes. Functions can either be concave up or concave down at any point on the curve.
In another sense, inflection points represent where the second derivative of the function is 0. In other words, where the rate of change of the rate of change is 0. Inflection points don’t always exist, as some functions don’t experience any change in concavity.
The best way to access information related to the concavity of a function is to use an online graphing calculator like the ones offered by WolframAlpha or Symbolab. These online calculators allow you to input a function in standard form, providing many different kinds of data concerning this function.
These platforms allow you to customize what data is displayed for each function. If you want to determine the inflection points of a function, then you just simply write “inflection points” before typing in the function.
Depending on which online concavity calculator you use, you will get vastly different results.
Some will only offer you the coordinates for the inflection points, not even offering a graph. Others will offer the inflection points, a graph, and some basic analysis. The best concavity calculators will offer all this with step-by-step guides as to how each value was calculated. Also, the best concavity calculators often offer practice problems and connections to real-world applications.
Most online concavity calculators are free. While you can purchase online concavity calculators, these are usually just gimmicks. Free versions are almost always available with comparable functionality. The only exception would be a concavity calculator that offered high-resolution graphs that other free applications simply didn’t offer.
Concavity calculators that are housed in the hardware of a more general graphing calculator, however, cost money.
The TI-84 Plus, for example, costs $108.00. It allows you to graph functions and analyze important points on the curve. However, in order to find the inflection points and determine where the concavity changes, some knowledge of basic calculus is needed.
THE BENEFITS OF A CONCAVITY CALCULATOR
The main benefit of a concavity calculator is that it makes finding some inflection points easier.
Students are first taught how to find inflection points in a first-semester Calculus course. Students are taught to calculate the second derivative and set it equal to zero in order to determine the x-coordinate of the inflection point.
Sometimes students are introduced to the concept of concavity and inflection points in earlier courses, though almost always on a cruder level. Calculus teaches you how to determine these points using derivatives, showing what these points say about the rate of change of the function.
However, calculating the second derivative can be a tedious process depending on the function. For some functions, calculating the second derivative might be almost impossible or practically unfeasible. This is especially the case for more advanced students who are often studying complicated functions. In these cases, a concavity calculator that can quickly calculate the inflection points is ideal, as this allows the student to avoid taking the derivative by hand.
Many graduate students also have to compute large amounts of data for some research they are engaging in. Instead of calculating the inflection points over and over again, these students can use the concavity calculate to expedite the process. This makes their work more productive.
In addition, a concavity calculator also allows students to be more creative with their work. If you’re able to analyze almost any function, then you don’t have to worry about picking a function that easily yields inflection points. Instead, you can study a funkier function with some unique properties, relying on the calculator to compute the approximate coordinates and show you an accurate graph.
THE DRAWBACKS OF A CONCAVITY CALCULATOR
The main drawback of a concavity calculator is that it encourages you to avoid exploring functions by hand.
Yet, this can be a beneficial activity from time to time, especially if you are pursuing a career in math. You will likely need to be able to graph functions and calculate first and second derivatives in the future. Many students anticipate this and ensure that they graph some basic functions by hand from time to time and analyze their properties.
By relying on a concavity calculator constantly, you can lose touch with these basic skills, which could impact your performance in later math classes.
Even further, understanding what an inflection point represents on a more practical level will be useful if you’re pursuing a highly mathematical career. If you use a concavity calculator every time you need to analyze the concavity of a graph, then you might lose touch with what computations you are even performing. This could be costly in a more practical career in which interpreting abstract information in relevant ways is the basis for your work.
TIPS FOR HOW TO USE A CONCAVITY CALCULATOR
The most important tip for using a concavity calculator is to set strict limits when it comes to using other websites. You want to concentrate your focus on the calculator only, as this will allow you to connect with the functions you are exploring on a deeper level.
Many students approach the use of online calculators with amazement, surprised that they are able to use the internet for school purposes. It’s not long before these students are distracted by some social media post or trending article.
The best way to prevent this from happening is to set clear intentions before you begin. Make it known that you intend to use the internet for school purposes only.
Some students find it effective to make simple compromises when negotiating with themselves. This can be as easy as promising yourself more personal computer time later after you’ve finished your schoolwork.
Some students try rewarding 30 minutes of schoolwork with 10 minutes of free internet time. We don’t recommend this approach, as we think the internet is a rabbit hole. Even if you’re able to follow your initial parameters, it’s likely that your attention will still be divided between the concavity calculator and your anticipated internet usage. This affects your level of concentration and productivity, preventing you from maker deeper connections with the information you are studying.
OUR FINAL THOUGHTS ON THE FUTURE OF CONCAVITY CALCULATORS
We think the future of concavity calculators entails integration with virtual reality headsets. We think these headsets will usher in a new era of education that is personal yet social and expansive at the same time.
We envision future students as no longer observing graphs of various functions from afar but instead sliding down their curves, climbing up their mountains, and probing their inflection points like a scientist taking data. This interactive form of education will provide our memories with richer material, allowing our recall to be sharper and quicker.
We think as technology evolves even further, computers will be integrated into human brains in order to boost our cognitive performance. This might make us smarter than any concavity calculator, though it’s not clear if we would be considered separate from a calculator or the same as one. Once we start inserting chips into our bodies, we’ve arguably fulfilled the prophecy of man becoming machine.
Some futurists think our brains are already the most advanced calculators in the universe. These visionaries think that rather than looking for guidance from outside of ourselves in the form of products or devices, we should be looking within. According to this theory, we are just as powerful as any machine we’ve created or can create. The key is to unlock this potential.
There are many ideas concerning how to unlock this potential. Ascetic study of math is certainly one way to unlock the deep abstract intelligence inherent within us all. Some gurus, however, teach meditation and yoga as another method of unlocking our potential.
All of this is to say that, while concavity calculators are important tools in the pursuit of knowledge, it should never replace our personal quest to manifest our innate intelligence.
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