The language invented for the specific purpose of describing the dynamic nature of our universe. To put it simply, calculus the maths of motion and change.

The word calculus originates from the Latin word meaning pebble. The Romans use pebbles to perform calculations on an abacus and the word became associated generally with computation just like the word calculator.

The beauty of calculus is not just in the maths alone. It’s in the way that calculus can form a connection or relationship and a language to describe the dynamic nature of our world. There are unlimited uses and benefits of calculus in any field. Calculus the language of motion and change.

And by using calculus we have the ability to find the effects of changing conditions on a system like the weather for example. In the atmosphere, we have a changing temperature and changing pressure. So by using differential equations meteorologists can indicate and predict the weather for our benefit.

Calculus holds incredible power over the physical worlds by modeling and controlling systems. It’s the language of medical experts, scientists, engineers, statisticians, physicists, and economists. If a quantity or a system is changing we can use mathematical modeling of calculus to:

• analyze a system,
• find an optimal solution,
• and predict the future.

Motion, electricity, heat and light, harmonics and acoustics, astronomy, radioactive decay, reaction rates, birth and death rates, costs and revenue…All of these can be modeled beautifully using calculus.

In calculus, we have two different branches. The first branch is differential calculus and this involves the concept of the derivative of a function. This branch of calculus studies the behavior and rate at which a quantity like distance. For example, changes over time.

When we use the process of differentiation we are essentially analyzing the changing rate of a quantity and making predictions about its behavior. So by finding the derivative, we can find the exact instantaneous rate of change at any point we like. If a function has a constant rate of change we get a straight line and it’s easy enough to just find the rate of change using rise over run. However, when a function changes its rate a multitude of times by using differentiation we can find exactly what its instantaneous rate of changes at any and every point in time.

The second branch of calculus is integral calculus. Integration is the reverse process of differentiation. Sometimes called antidifferentiation. With integration, we can describe the area of a 2D region with a curved boundary or the volume of a 3D object with a curved boundary. We integrate by breaking the region apart into thin unlimited vertical rectangles of equal width until the width of the rectangles virtually become 0 which is called a LIMIT. This limiting process allows us to calculate areas and volumes with exact precision. If we differentiate a function and then integrate it, it will always take us back to where we started.

Both these branches differentiation and integration are connected together by something called the fundamental theorem of calculus. This theorem created by Newton and Leibnitz states at differentiation and integration are inverse operations or opposites. Just like yin and yang, black and white, or matter and anti-matter.

Take the square root for instance. The opposite of taking the square root is squaring a number. Just like differentiation is the opposite or inverse of integration.

Now we know what calculus is. Wouldn’t it be interesting to see how it can be used in aerospace to describe a rocket launch? If an object is in motion like a rocket we can use calculus to model it. The thrust of a rocket into space is based on the calculus of motion. In rocket physics, we are applying Newton’s second and third law to a rocket that has a variable mass. How is the mass variable?

The Rocket’s mass is decreasing over as the fuel propellant is being burned off. As the rocket propellant ignites the rocket experiences a very large acceleration as the exhaust exits out the back of the rocket at a very high velocity. This backward acceleration exerts a push force on the rocket in the opposite direction causing the rocket to accelerate upwards. The force acting on the rocket called the thrust is the rate of change of momentum which is the first derivative of momentum.

Using calculus momentum or the amount of motion of the rocket:

P = mass x velocity. And so the rate of change of momentum:

P’ = dmv/dt, the thrust of the rocket.

We can also write this as a physics equation: F =ma, Newton’s second law. And rewriting this from a calculus standpoint:

F =M x dmv/dt.

To put it simply the thrust of the rocket during a launch is the first derivative of momentum. Rocket propulsion also employs Newton’s third law, conservation of momentum. This dictates that if a material is ejected backward like the exhaust in a rocket launch, the forward momentum of the remaining rocket must increase because an isolated system can’t change its net momentum. In other words for every action, there is an equal and opposite reaction — Newton’s third law —

After launch to achieve the desired final orbital velocity around the earth or to escape from Earth’s gravity the mass of the rocket must be as small as possible. And so the rocket sheds mass by using different rocket stages separating its parts such as the rocket boosters.

Now that we’ve seen calculus applied in a physics of aerospace, let’s see the benefits of calculus in the world of economics. A lot of people dream about running their own business. Wouldn’t it be great if you could work out exactly how to maximize your business profits and help build a thriving company?

Calculus can be used to maximize profits and revenue for any business. In actual fact, calculus provides the language of microeconomics and the names by which economists can model and solve financial problems.

Let’s see how we can apply calculus to maximize your profits in your theoretical video game business, Pow Pow.

Revenue function is given by R(x).

Marginal revenue R’(x) is the first derivative of revenue.

This is the increase in revenue generated when producing one additional video game. Change in revenue divided by changing the number of video games.

So what this tells us is exactly how many units you should sell to maximize your revenue. So Pow Pow doesn’t lose any money by producing too many units. This also takes into consideration the fixed cost of producing a big batch of video games.

But this is less than the price that you wanted to charge for an additional video game. As you can see we have a financial problem here and we need to model the revenue here using calculus to find the optimal quantity of games to maximize your revenue.

Let’s say we model the revenue for Pow Pow and produce the revenue function as

R(x) = 100x — 1/2 x², where R is the revenue and x is the numbers of video games sold.

If we graph the revenue function we get a concave down parabola. Marginal revenue is the first derivative of revenue. Differentiating the function we get

R’(x)=100-x

R’(x) is the gradient function of R(x), so the change in the rate of revenue which is called marginal revenue. If we found the maximum revenue from the first derivative algebraically, we need to let the first derivative equals zero to find the maximum x point or the maximum number of video games first.

x=0,

100-x =0,

x = 100.

Substituting x = 100 back into the revenue equation to find the actual revenue for pow-pow, your revenue is

R(100) = 100(100) — 1/2 (10⁰²) = $5,000.

This means that the rate of production resulting in maximum revenue occurs when the number of video games sold is a hundred resulting in total revenue of $5,000. As you can see we can easily maximize your business profits by using the first derivative of revenue marginal revenue.

It’s also known that a company produces best results when production and sales continue on and to a marginal revenue equals marginal cost. So now that we’ve seen the benefits of calculus in aerospace and economics, let’s now see its benefits in medicine.

Let’s say that you’re a doctor, Dr. Cure and you would like to observe the progression of a tumor in one of your patients, John. John has a small early onset tumor and you would like to see whether it’s responding to a new innovative drug, arrivederci tumor which has a no side-effects. As a doctor, you would like to model the growth of John’s tumor using calculus to analyze the progression or reversion of his disease. The function you have created to model the progression of growth of John’s tumor is an exponential function with respect to time.

Differentiating this the first derivative of the tumor volume will be the change in tumor volume over time and in medicine, this is called the specific growth rate, SGR. The first derivative V’(t) gives you important information about whether John’s tumor is growing or shrinking and the rate at which it’s doing so.

If the tumor has a higher SGR, Dr. Cure can interpret this as a rapidly growing tumor. And then he can make decisions about the form of therapy or change in therapy to cure the tumor and get John back to good health again. If the SGR is low then Dr. Cure can assume that the new innovative drug arrivederci tumor is working, the tumor is shrinking and discontinues John on the current regime.

As we’ve seen the beauty and benefits of calculus can be applied in any scenario of change or motion whether it be aerospace, economics, medicine, and more. The benefits of calculus are endless and if we have any problem in any dynamic situation that involves either change or motion you can be sure that we can turn to calculus as a tool to model the problem and provide us with the answers.