Calculus is an integral section of mathematics and uses more advanced techniques to solve issues dealing with continuous change.
The advanced math discipline was first termed "the calculus of infinitesimals"; hence, it is sometimes referred to as infinitesimal calculus.
Traditionally, calculus was known to act as a study of the continuous change of functions. As a result, the two primary calculus bases are integrals and derivatives.
A function's derivative measures its rates of change, whereas the integral measures the area under the function's curve.
Furthermore, the integral works by accumulating discrete values, while the derivative explains the function.
Now that you know what the bases of calculus are, let's explore them in more detail:
What Is Calculus, And How Does It Work?
Calculus is used within mathematical models to help obtain optimal solutions. Therefore, it helps us understand the changes between values linked by a specific function.
Calculus was developed by mathematicians Leibniz and Newton in the 17th century and focuses on critical topics within math such as:
These are used to explain how the rate of change works. Hence, calculus can be classified in two sections, broadly:
- Integral Calculus
- Differential Calculus
Both sections work as the foundation of another more advanced concept in mathematics, "Analysis." It deals with how slight changes impact dependent variables, which leads the function reading to zero.
The Basic Topics Within The Subject Of Calculus
There are some essential topics within calculus that all students need to learn; these are:
Introduction To The Concept Of Functions
Within this topic, students are introduced to linear and differentiable functions.
What Are Functions?
This topic talks about the transformation of functions and integral.
In this, students learn about integration and the differential equation.
Functions are a representation of the relationship between two variables, namely the dependent and independent variables.
The Two Types Of Calculus
As stated earlier, there are two main sections in which calculus is divided: differential and integral. Let's take a look at each of them individually:
Differential calculus is used to solve problems when you need to find the rate of change of a function concerning other variables.
Therefore, to calculate the minima and maxima values of a function to obtain the right solution, derivatives are used.
Differential helps when studying the limits of quotients that deal with:
- Variables like y and x
- The resultant changes in those variables.
- Differentials and noted by dx and dy
Therefore, differentiation is the process through which derivatives are found. A function's derivative, y in relation to variable x, is shown through the following: f' (x) or dy/dx.
A limit aids students when they calculate the degree of closeness to the approaching term or value. There is a limit formula that represents this:
limx→cf(x) = A
What this expression means is "The limit of f of x as x approached c equals A"
A derivative represents the sudden rate of change of quantities in relation to one another.
The representation of a function's derivative is:
limx→h[f(x + h) − f(x)]/h = A
The f(x) or function is known to continuous at a specific point x= a, only if these conditions are granted:
- A defined f(a)
- The existence of the limx→af(x) exists
- limx→a− f(x) = limx→a+ f(x) = f(a)
Differentiability And Continuity
If the function is differentiable at any given point, it will always be continuous, whereas if the condition was opposite, it might not be so.
In integral calculus, one studies integrals and their properties. This helps with:
- The calculation of f (the function) from f' (which is the derivative.) If the function is differentiable at any interval, the derivative will be defined during the interval.
- The calculation of the area under the curve of the function.
Integration is simply the process through which integrals are found, and it works contrary to differentiation.
The latter divides one part into smaller parts, whereas integration is a collection of many small parts used to form a whole. It is used primarily when calculating areas.
Definite integrals have specific boundaries or limits to calculate a function. The lower and upper limits of the independent variable of the function, in this case, are specified. Mathematically, a definite integral is denoted as:
∫ab f(x).dx = F(x)
Indefinite integrals do not have specific boundaries, which means the lower and upper limits are not defined.
This is why the constant value – depicted as C – always accompanies the integration value, which is represented as:
∫ f(x).dx = F(x) + C
Example Of Calculus Equations
Often, examples of calculus equations can provide high value for students studying for a calculus test. Let's look at two solved examples:
Let g(y) = ey and f(y) = y2. Apply the chain rule when calculating h′(y) where h(y) = f(g(y)).
The solution to this will be:
Since, f(y) = y2 and g(y) = ey. First derivative above functions are f'(y) = 2y and g'(y) = ey
You are finding h′(y)
h(y) = f(g(y))
h'(y) = f'(g(y))g'(y)
h'(y) = f'(ey)ey
When you substitute the values, you get the answer:
h'(y) = 2e2y or
h'(y) = 2ey × ey
In this instant, y is a function of x, the function being defined as: y = f(x) = 1/(1 + x2).
Calculating the functions output values for for x = 0, x = −1, and x = √2.
The solution to this would be:
f(0) = 1/[1 + (0)2] = 1/1 = 1
f(−1) = 1/[1 + (−1)2] = 1/2
f(√2) = 1/[1 + (√2)2] = 1/3
How To Pass Calculus
Calculus a necessary cog in most technical fields relating to science and engineering, but some perceive it to be too difficult to handle and drop out.
To better explain this problem, here is a stat: One student out of every three is noted to either flunk calculus or drop out altogether.
However, it can be a gratifying and beautiful subject if you master it. Still, the fear overwhelms most students, and they end up quitting instead.
This is why students should be urged to understand calculus formulas instead of mindlessly memorizing them.
Here are a few things to keep in mind when studying calculus:
- Take the time and make an effort to understand what the course is all about. Your syllabus will act as a guideline describing which topics to focus on and to what extent. You will also be given details on which subject carries how many marks so you can schedule your study time accordingly
- Do not miss out on classes. Even just missing one or two lectures can have you lagging and make it more challenging for you to catch up and pass the course. However, it would be best if you considered hiring a private calculus tutor in such an instance as they will keep you up-to-date and help you understand the topics you have missed out on
- Besides studying calculus in class and on your own, consider joining a study group. It can be quite an eye-opening experience, one that exposes your weaknesses and strengths while you learn from others
- Keep on practicing!
- Do not forget to take a break. If you burn out, all your hard work will go to waste. Remember that your mind will work best if it is open to learning. Take breaks between study sessions to free up your mind and untangle any concepts which may confuse you
How To Find A Calculus Tutor
Even if you find it challenging to understand calculus's basic concepts and formulae, it does not mean you can never get the hang of it.
Everyone learns differently, and some people take a bit more time than others, which is why you should not be ashamed of hiring a calculus teacher.
If you have decided on getting some extra study help for calculus, these are ways you can go about finding a calculus tutor:
Whether you are in college or high school, you can find peer tutors to give you a helping hand. These are students who have a good grasp of the subject and are willing to share their knowledge.
You can ask around or go to your school's website or administration if you need extra help. However, the best way to find one will always be through social media.
Asking A Teacher To Tutor You
Teachers often agree to teach students outside of class hours if they feel it is needed. Ask a professor you trust if they will be able to tutor you outside school hours.
Although, you should keep in mind that some institutes frown upon teachers and students interacting outside of the classroom; hence, you should get clarity on the rules before being vilified!
Superprof is a surefire way of getting the help you need in calculus. They have a long list of calculus tutors in several parts of the country.
Their qualified instructors can teach you in person or take sessions online if that is more convenient for you. They are flexible with their timings and will work according to your schedule and learning style.
Nailed The Basics!
Now that you have a good understanding of the basics, you can start practicing more and more calculus problems and learn more advanced concepts.
Remember, anything is possible, and even if it gets complicated, you should not give up.
If you find yourself confused, refer to one of these calculus resources, or hire a tutor from Superprof – the best tutoring platform in America!