"I don't believe in statistics. I believe in calculus." -Ben Horowitz

Some of the world's most brilliant minds stand behind calculus and claim it to be one of the best academic disciplines ever developed. Extremely logical and convenient, calculus has saved much time when calculating aspects of the moving/changing world.

As a student of calculus, whether at the end of secondary school or university, it is of the utmost importance to review basic equations and examples to not lose practice. 

Therefore, without further delay, in today's article, we shall examine some essential problems of calculus that can easily be fixed!

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Examples and Equations of Calculus' Main Principles

test answers for calculus
Making mistakes is all part of the process; however, don't let errors prevent you from practising calculus regularly. (Source: Unsplash)

Since the mathematic topic of the calculus was invented by Isaac Newton in the late 1600s, it has been frequently used by academics and students alike to make solving essential aspects of the moving world more attainable and efficient. 

Fun fact: though primarily developed by Isaac Newton, the essential aspects of calculus studied today were pioneered by Gottfried Leibniz. 

It is important to state that calculus is used in a wide variety of professional sectors such as medicine, natural science, economics, and engineering. 

Nevertheless, for those who are not familiar with the basic principles of calculus, we shall consider some useful examples in major subgenres of calculus such as limits, derivatives, and integrals. 

The best information about calculus can be found on reputable online resources.


In the study of maths, limits are used frequently and must be comprehended. A limit is a value that a function approaches as the input approaches some value.

It is important to mention that limits are necessary for calculus to describe and define continuity, derivatives, and integrals. 

To become familiar with limits and get a feeling as to what they are trying to tell us, one-sided limits, limit properties, infinite limits, continuity, and limits at infinity need to be studied and problems effectively reviewed.

Since the internet is a fantastic place full of valuable information, the following is a simple example that might be found in a limits section of calculus:

For the function, (a) f(x)=8−x3x2−4 answer each of the following questions.

Evaluate the function of the following values of x compute (accurate to at least 8 decimal places).

  1. 2.5,
  2. 2.1,
  3. 2.01,
  4. 2.001,
  5. 2.0001,
  6. 1.5,
  7. 1.9,
  8. 1.99,
  9. 1.999,
  10. 1.9999

Use the information from (a) to estimate the value of limx→2⁡8−x3x2−4.

Or better yet, to see equivalent examples about calculus limits, visit this webpage.


learning the basics of calculus
By going through calculus examples and equations regularly, students avoid cramming and stress before an exam. (Source: Unsplash)

Encompassed under the differential type of calculus, derivatives are essential to understanding important mathematical topics. For example, derivatives of a function can be defined as a tool to measure the sensitivity to change the function value with respect to a change in its argument.

Thanks to the discovering of derivatives, things such as learning the derivative of a moving object with respect to time is the object's velocity can be acquired without implementing the previous archaic methods.

When analysing derivatives in calculus, different subgenres such as product and quotient rule, derivatives of trig functions, chain rule, and differentiation formulas can be learnt. 

Since we are providing examples for major branches of calculus, the following are some basic derivative examples/formulas to acquire how knowing the definition of derivatives can actually help to work through a problem and find out how to compute the derivative of a function:

  1. f(x)=6
  2. V(t)=3−14t
  3. g(x)=

By working through the previously mentioned equations, with the overall definition of derivatives in mind, students become skilled in some of the most important aspects of calculus.

If you find the equations of derivatives look more complex than Mandarin Chinese symbols you're not alone! That's why we recommend looking at how to find the solution to derivative problems online or consulting with a private maths tutor.


In mathematical terms, integrals can be used to find areas, volumes, central points, and many other essential things. Since integral calculus is one of the two main pillars of calculus, it is of the utmost importance that pupils become familiar with the fact that this part of calculus aims to find the quantity where the rate of change is best known.

Integral calculus draws much attention to aspects such as slopes of tangent lines and velocities. 

When analysing the examples and equations of integrals in calculus, pupils become familiar with indefinite integrals, area problems, computing definite integrals, and the substitution role.

The following are some problems that students may come across when revising indefinite integrals at an A-Level:

  • ∫6x5−18x2+7dx
  • ∫6x5dx−18x2+7

By checking out the Tutorial Math Lamar site, where the previously mentioned equations are encountered, the solutions to the mathematical problems can be found and many examples can be revised to master integrals in calculus before the school year even begins!

Tips to Excel at Calculus

learning calculus through examples
While not necessarily an academic discipline that is adequate for children, calculus doesn't have to be as difficult as everyone says. (Source: Unsplash)

Even if you're sailing by and getting decent test marks, there is always room to improve when analysing a mathematical subject such as calculus. Tips, tricks, and pieces of advice from others make learning calculus more endurable.

Nevertheless, since there is a surplus of mediocre information on the internet rather than effective resources, it may seem almost impossible to find suggestions that will launch students to success. 

Have no fear, Superprof is here! As a result of the thousands of tutors offering successful classes in a wide number of topics, Superprof has become one of the leading experts in the online field of education; the advice we provide is always guaranteed to help students.

Therefore, without further ado, the following are the most useful tips to become a pro at calculus:

  • Don't Neglect Homework Assignments: we know that homework can be a complete bore; nevertheless, by doing their homework examples, calculus students prepare themselves for success by having reviewed problems, equations, and examples. Don't think that you can only attend class and be an expert; don't get behind on assignments, stay up to date!
  • Work in Small Groups: many calculus students at an A-Level or university grade have found it useful to create study groups with other like-minded pupils. By doing this, strengthens of others are used to balance the weaknesses of some; an interchange of academic encouragement is shared.

There are many more valuable tips that can be put into practice when consulting with more experienced individuals in the field of calculus. It's important to remember that calculus doesn't have to be difficult; it can be quite a fascinating learning experience!

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