Understanding Linear Functions in Calculus


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 represent the same properties.


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.

The Best Strategy For Passing AP Calculus Tests

You may think that I’m here to offer study techniques around memorizing equations and graphing, but I’d like to offer you something different.  It’s hopefully a technique that you have been told before, but I am here to re-iterate it to you. In fact, the entire strategy is around one simple idea.

Sleep.  That’s right, good ‘ole fashion sleep.  One of the biggest mistakes that you can make going into your exam is just getting out of a late night cram session.  A sleep-deprived person cannot focus and therefore cannot learn or apply anything efficiently. Secondarily, being chronically tired to the point of exhaustion means that you are less likely to perform well.  Which means that in the weeks leading up to your exam, you should be training your body to sleep better.

So here is my suggestion for you.  If you are often too tired to stay awake for an early morning class, then take at least 10 days to re-adjust your sleep pattern so you can fall asleep earlier in the night.   Don’t try to change your pattern in just one night… it won’t work.

Slowly change your routine so that on the night before the exam, you can fall asleep easier and be well rested.  My second piece of advice?  Wake up a little bit early and do some exercise.  Running, jumping jacks, sit-ups, anything to get the blood pumping.  About 20 minutes of exercise can help increase your mental energy and get you pumped.  Just make sure you shower after so you don’t distract your fellow test takers.

Remember, sometimes you can do all the studying in the world but you leave out the essentials.  Don’t be like these students:

The Difference Between First and Second Derivatives

First off, we are proud to announce that Newor Media has acquired Calculus-Help.com.  Newor Media is owned and operated by teachers, so you can continue to expect some great Calculus content in the future.  If you have any suggestions for us, we would love to hear them, and we may include some fun puzzles as well.

Now on the question at hand.  First, some quick references for you. The first derivative of the function f(x), is the slope of the tangent line to the function at the point x:

If the first derivative f’ is positive, then the function f is increasing (pointing upwards)

If the first derivative f’ is negative, then the function f is decreasing (pointing downwards)

As for the second derivate, the second derivative of a function is the derivative of the derivative of that function:

If the second derivative f” is positive , then the function f is concave up (looks like a U shape) .

If the second derivative f” is negative , then the function f is concave down (looks like an upside down U shape)

The second derivative is like the movie Inception.  The first derivative tells us whether or not the function is increasing or decreasing.  The second derivative shows us whether or not the first derivative is increasing or decreasing. So the second derivative plays directly off of the first.

If the second derivative turns out to be positive, then the first derivative will be increasing. The slope of the tangent line to the function is increasing as x increases.  If the second derivative turns out to be negative, then the first derivative is decreasing.  The slope of the tangent line to the function is then decreasing as x increases

Here is some more in-depth information, but this should be a good start.  If you would like to see it in action, take a look:


The Humongous Book of SAT Math Problems: On Sale Now!


Are you getting ready to take the SAT? Have I got the book for you! Presenting The Humongous Book of SAT Math Problems, your one-stop show to a metric ton (give or take) of math practice for the SAT Test. The book contains all of the following, except for one. Try to figure out which is not included:

  • Test-taking tips, including specific advice for multiple-choice and grid-in questions
  • In-depth mathematical tutoring on all of the major topics included on the SAT
  • Practice problems for all four categories of mathematical SAT questions
  • The math sections of three full-length practice SAT tests
  • A bronze bust of former president William Howard Taft

Set your math phasers to “excited”! Click here for ordering info.

Errata for the Humongous SAT Book

A chart should appear above Problem 5.23 and it is missing. This makes an already hard problem infinitely harder 🙂

Click here to download and print the chart. Alternately, you can right-click and select “Save Image” to save it to your computer.

Note that the chart opens in a new window. If you can’t see it, you may need to temporarily disable pop-up blockers.

Calculus-Help.com Fund Raiser: Buy a Signed TI-nSpire Calculator to Support the Website


Up for sale is a TI-spire calculator owned and signed by by W. Michael Kelley, the author of The Complete Idiot’s Guide to Calculus and The Humongous Book of Calculus Problems. Is Kelley famous? No. Does the signature actually add any monetary value to the auction? No. In fact, like graffiti, it may actually devalue the sale. That’s a chance we’re willing to take in a fund raiser to reboot Calculus-Help.com for the new school year: http://www.ebay.com/itm/140846256145.

Every day Mike gets a ton of email from people practically begging to buy advertisements on his site. Actual begging. We feel sorry for these people, because what they don’t know is that Mike made a (probably very short-sighted) pledge to never sell ads on the site for any reason. “But can’t we just throw money at you?” they ask, usually in the form of impersonal spam. “No,” Mike says. “I must hold to a promise, a sacred covenant, that I sometimes really regret making.” For those of you thinking, “Sure, there’s no advertising, but practically every page is an ad for Mike’s books. Doesn’t that count as advertising?” understand that I am staring at you with a squinty stare, telepathically asking you to kindly put a sock in it, because you’re embarrassing me in front of my new friends–friends that could potentially buy the calculator I ruined by signing it with a silver Sharpie.
To help keep the site free, Mike is auctioning off this calculator, which is lightly used, but it contains all the important pieces (like connection cables, the manuals, an alternate TI-84 face plate (let your dork flag fly) and is in incredible shape. It works like a charm, as you can see by the mundane and meaningless calculations I took a picture of. In case you were wondering if this complicated piece of machinery can add, the answer is a resounding YES! (I may or may not have utilized the full power of this guy.)


Motion, Position, Definite Integrals, and Bears

Photo by Mackenzie and John
Photo by Mackenzie and John

Unless you live in Chicago, the new Problem of the Week may be unsettling because it is rife with bears. Not the cute kind of bears that get stuck in a rabbit’s tree house because they may or may not be morbidly obese. (“Oh, bother.”) No. These are the “I’d rather eat you, bones and all, than pretend that I am a cloud to hustle some bees out of their honey” kind of bears. The dangerous kind.

Are you thinking, “This must be an optimization problem, where you are asked to calculate the bear minimum”? If you did think that, please know that I am, as you read this sentence, smiling approvingly and giving you polite golf applause for your clever pun. However, this problem is all about definite integrals, so there is no time for joking around.

Steel yourself for the battle of your life! Attack this problem with your bear hands! But please, before you do, I would like the polite golf applause reciprocated.