Understanding Linear Functions in Calculus

Introduction Functions are used to describe mathematical things and can be difficult to define. The basic definition of a function can be said to be – a collection of ordered pairs of things, where the first members are fundamentally different in the pairs. A simple function can be as follows: [{1, 2}, {2, 4}, {3, … Continue reading "Understanding Linear Functions in Calculus"

Introduction

Functions are used to describe mathematical things and can be difficult to define. The basic definition of a function can be said to be – a collection of ordered pairs of things, where the first members are fundamentally different in the pairs.

A simple function can be as follows:

[{1, 2}, {2, 4}, {3, 6}, {4, 18}, {5, 10}]

The above function has five pairs where the first members are 1, 2, 3, 4 and 5.

Functions usually have alphabetical letter as their names. So if we term this function ‘f’, which is the most common letter used for functions, then it will be properly written as:

f(1) = 2, f(2) = 4, f(3) = 6, f(4) = 8, f(5) = 10

Here are two definitions to keep in mind:

The entire set of first numbers in the function is called a domain and the first members are called arguments. In this particular example, the domain has 5 numbers and the numbers 1, 2, 3, 4 and 5 are the arguments of the function.

The whole set of second numbers in the function is called the range and the second members are called the values. Going back to the above function, the range also has 5 numbers and the numbers 2, 4, 6, 8 and 10 are the values of the function.

As mentioned before, the standard naming of a function is f. Thus we can explain this function in a sentence as follows:

The value of the function (f) at argument 1 is 2, its value at argument 2 is 4, its value at argument 3 is 6, its value at argument 4 is 8 and its value at argument 5 is 10.

Therefore a function can also be defined as a set of assigned values (the second numbers) to arguments (the first numbers)

This can be expanded further to say that the condition is that the first member of every pair is different; therefore each argument of the domain of function ‘f’ gets an exclusive value in its range.

The linear function and its importance to calculus

The linear function is the basic and essential function, on which calculus is based upon. This is a function that has a straight line running through the domain of its graphs.

Such a line can be determined by two points that lie on it. Look at the function [a, f{a}], [b, f{b}]. You can pick an “a” and “b” in the domain and determine this line defined by the two values f{a} and f{b}.

Let’s look at the formula for such a function.

It is possible to determine the linear function for the two values mentioned above by using the following formula.

f{x} = [f{a} x – b/ a – b] + [f{b} x – a/ b – a ]

Effectively this means that the first term is 0 when x is equal to b, and it becomes f{a} when x equals a. The second term is 0 when x is equal to a and it becomes f{b} when x equals b.

Another important aspect of a linear function is its slope.

This is defined as the ratio of the change of function f between x = a and x = b the change in x between the two arguments. The y-intercept is the point at which the line passes the y-axis.

The intercept of the line on the y axis is also an essential part of the linear function.

As we have seen, a linear function can be defined one that has a graph with a straight line, and can be described by its slope and y-intercept.

Special linear functions are often useful and they all have an important and unique property – they all have linear functions whose y-intercepts go through the point 0. Their graphs pass through the origin of the x and y axes. They are aptly called homogenous linear functions, and they all share the same property which is:

Their value at any permutation of two arguments is equal to the same permutations of their values at those arguments.

This can be explained by the following formula:

F{ak + bc} = af{k} + bf{c}

The above property is called the “property of linearity”.

NOTE: not all linear functions have this property of linearity. The property implies that once you know the value of a linear function and any two distinct arguments, then you can find the value at any other point or pair of arguments. This is not always true.

Practical applications of the linear functions

There are several real life applications of calculus linear functions. Remember that this is the most basic function on which other functions are based upon. The function is applied in various fields, such as meteorology, pharmaceuticals, engineering, and a lot more.

Whenever you have to create a graph in a straight line, no matter what the slope or y-intercept is, you are applying this basic principle.

NOTE: One should not confuse linear functions in calculus to linear equations in algebra. They have different properties even if sometimes their graphs can be identical. You can find a graph for a linear equation of algebra having the same slope and y-intercept as a graph for linear function of calculus, but they do not

Conclusion

Starting off by understanding this basic formula of calculus will make it very easy for you to move on and understand the deeper functions or integration and differentiation. Calculus should not be a behemoth to be feared but a friend to be understood. Try out some basic exercises on the linear functions in calculus and you will get a better grip on the topic.

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Enhance Your Calculus Capabilities with These Brain Boosting Tips

Improve your problem-solving skills and calculus capabilities with our 11-best brain-boosting tips. Get an A in calculus with these simple tips.

Calculus has the reputation of being one of the most challenging subjects in school, even when compared with other advanced math classes. That’s because calculus is usually the first exposure students get to a version of math that requires more than just memorization to succeed.

Many calculus classes have specific pre-requisites, such as an understanding of advanced algebra, geometry, and trigonometry. After all, it’s essentially another language. However, you’ll need more abstract qualities as well.

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A well-developed imagination will make it much easier to understand calculus. Since calculus works with physical, real-world concepts, the ability to visualize these things in your mind is crucial to your ability to understand and solve calculus problems.

You also need to learn how to attack problems from different angles. There are specific rules in calculus that you’ll have to memorize, but not all calculus problems will follow a pre-set formula. You need to learn how to solve complicated problems without following a specific set of actions you’ve memorized.

Overall, you’ll need advanced problem-solving skills if you want to succeed in calculus. To help you learn or build upon these skills, we’ll teach you to enhance your calculus capabilities with these brain-boosting tips.

General Brain-Boosting Tips

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Even before you start your first day of calculus, you can work on improving your ability to learn and process advanced concepts. And it’s easier than you may think.

Dance Like Nobody’s Watching.

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It may be an overwrought cliché, but this advice not only works for improving your mood, but it can also develop your ability to solve problems.

Studies have found that dancing can improve your brain’s neural processing abilities. Structured dancing, such as tap and ballet, helps with convergent thinking—your ability to find a single, absolute answer to a problem.

On the other hand, more improvised styles of dance, like hip-hop or EDM dancing, improves divergent thinking. That means you’ll have an easier time finding multiple possible answers for a complex problem.

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Since calculus requires both types of thinking, you should consider learning multiple styles of dance. Many schools offer dance classes as an alternative to physical education, or as an elective. If you don’t relish the idea of dancing in front of other people, you can also use Youtube to find dance tutorials and practice in the privacy of your home.

Play Games.

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Over the last decade or two, there has been a rise in the popularity of logic and puzzle games like sudoku. Electronic games designed to work out your brain can be found on nearly any video game platform as well.

These types of games are not just a fun way to engage your mind when you’re bored; they’re also a great way to improve your problem-solving skills. Many logic or puzzle games require you to work a problem from multiple sides and have numerous ways of finding the solution.

For many problems, both in games and in calculus, you’ll need to approach the problem backward, not forward. By practicing this strategy with fun games in your free time, you’ll find that you have an easier time working through your calculus problems as well.

Stay Healthy.

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Obviously, exercise and a proper diet will benefit your body; however, science also proves that a healthy body leads to a healthy mind. People who regularly work out and eat wholesome, healthy food, have more energy, which helps their brain work faster and harder.

Researchers at Harvard found that regular cardio exercise can actually increase the size of your hippocampus, which is the part of the brain responsible for verbal memory and learning. It doesn’t take much, either—simply walking at a brisk pace for about two hours per week is enough to boost your mental function significantly.

A similar Harvard study also discovered a link between unhealthy food and memory problems. Scientists studied the effects of a diet high in LDL (“bad” cholesterol) and saturated fats on memory and found that participants performed much worse on tests of thinking and memory than people with a healthier diet.

Since diet and exercise have positive effects on both your body and your mind, they’re likely to enhance your calculus capabilities as well. They can help relieve stress for keeping focused on your work.

Sleep Well.

We’ve all experienced the adverse effects of sleeplessness—foggy brains, irritability, and lack of energy, to name a few.  Not getting a good night’s sleep can have a catastrophic impact on your ability to learn, as well as your problem-solving skills.

REM sleep, in particular, is vital to your ability to form connections between unrelated ideas. That means your brain works while you’re sleeping to process everything you’ve learned and allowed you to see the bigger context of different subjects and ideas.

REM sleep also enhances creative processing, your brain’s ability to form new ideas and ways of thinking about concepts you’ve learned in the past.

Advanced Problem-Solving Tips

While the above tips will prepare your brain to learn calculus, you may still need some extra help when it comes time to work through calculus exercises. If you find yourself struggling with a problem, try some of these tips.

Mind Maps

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A mind map is essentially a sketch of the problem and all the potential solutions. You can use this technique with any kind of question. Think of it as another way to brainstorm. By drawing out the issue, your mind may be stimulated to generate more ideas.

Basically, you start with your problem in the center of the page. As you brainstorm possible solutions, add branches off the center containing these answers. You can divide these main branches into sub-branches, as you plug in your potential answer to the problem and work through it. If one solution doesn’t work, you move on to the next one and continue the process until one of them works.

Music and Exercise

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We’ve already discussed how exercise can help stimulate your brain in general. However, the addition of music can further push your mind to sort through everything you’ve learned and organized it in ways that may lead to finding a solution to your problem.

Idea Journals

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Many studies have shown that the act of writing types of cement not only new ideas into your brain, but also generates new connections and creative ideas. The purpose of an idea journal is to provide a place for you to write out your problem and any subsequent ideas you generate over time.

As we’ll discuss in a little bit, sometimes your brain will come up with solutions to a problem when you least expect it, while you’re showering or watching television or hanging out with friends. Keeping an idea journal with you at all times allows you to write out these solutions as soon as they pop into your head.

Talk it Out

For many people, the act of talking through a problem helps their mind sort through ideas and generate new solutions. Whether you’re talking to yourself, a friend, or a classmate, merely verbalizing the problem may lead you to blurt out the answer without realizing you knew it all along.

One common strategy is to talk through the problem as if you were teaching the subject to someone else. By explaining the concepts involved in the question (to yourself or others), you could unlock the answers already hidden in your mind.

Create a Sketch.

As we mentioned earlier, most calculus problems can be represented physically—for instance, the shape of a curving slope, or the speed of a moving object. Sketching a visual representation of the problem could help you understand exactly what it is you’re trying to solve.

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Focus on Something Else.

You’ve probably heard the story of Archimedes, who found the answer to how to measure the volume of a king’s crown while taking a bath. While the anecdote may or may not be true, the idea of the brain finding solutions to problems while you’re focusing on something else is solid.

By working on something else, particularly something that doesn’t require too much mental strain, you increase the psychological distance between yourself and your problem. That allows your brain to think more abstractly and creatively on the issue at hand while your attention is focused somewhere else.

Get Back to Basics.

If you’re still having trouble coming up with a solution, try going back to the beginning. Focus on the parts of the problem that you already understand, and work through as much of it as you can. In the process of rehashing familiar concepts, you may trick your brain into forming the complete solution.

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Final Notes

You don’t need to be a math genius to succeed at calculus. Even more than mathematical ability, calculus requires creativity, adaptability, and excellent problem-solving skills. If you work hard, take care of yourself, and use our guide to enhance your calculus capabilities with these brain-boosting tips, there should be nothing stopping you from earning that A.

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