Calculus

Calculus has two main branches, Differential Calculus, and Integral Calculus. Differential Calculus is the branch of calculus that deals with limits, continuity, derivatives, and differentiability. Continuity is defined as the property of a function to have a non breaking graph i.e. limit of the function exists at all the points in the domain and differentiability is the property of the function to have a derivative at all the points in the domain. Whereas the Integral calculus is the branch of calculus that deals with integration and the area under the curve of a function. Integrations are the exact opposite of differentiation, and the integration of a function in defined limits gives the area under the curve of the function.

In this tutorial, we are going to learn about calculus, its branches and types, and its uses in real life.

Differential Calculus

In Differential Calculus, the topics of Limit and Continuity as well as Differentiation and Differentiability, are studied.

Limits − The limit of a function is defined as the value the function approaches as the argument approaches a particular point in domain.

$$\mathrm{\lim_{x \rightarrow x_{0}}f(x)}$$

If the value of the limit approaching from the left and right of the function are equal and equal to the value of the function at the point, then the limit of the function exists.

$$\mathrm{\lim_{x \rightarrow a^{-}}f(x)\:=\:f(a)\:=\:\lim_{x \rightarrow a^{+}}f(x)}$$

Left hand Limit = Value of function = Right hand Limit

Continuity − If the limit of the function exists throughout the domain of the function, then the function is said to be continuous.

Derivatives − The derivatives are defined as the rate of change of the function/dependent variable with respect to the argument/independent variable.

$$\mathrm{f'(x)=\frac{d(f(x))}{dx}=\lim_{\Delta x \rightarrow 0}\frac{\Delta f(x)}{\Delta x}=\lim_{h \rightarrow 0}\frac{(x+h)-f(x)}{x+h-x}=\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h}}$$

Note: This equation is known as the first principle of derivatives.

The derivative of a function is said to exist if the left hand derivative and right hand derivative of the function are equal.

Differentiability − If the derivative of the function exists throughout the domain of the function, then the function is said to be differentiable.

Integral Calculus

In Integral Calculus, the topics of Definite and Indefinite integrals are studied.

Definite Integrals − If the integral of a function is calculated in a given finite interval, then the integral is said to be Definite Integral.

$$\mathrm{\int_a^bf(x)=F(b)-F(a)\:\: (A\: fixed\: numerical\: value)}$$

Indefinite Integrals − These integrals are mathematically the exact opposite of the derivatives. They are not in any particular interval and result in a function whose derivative results in the original function.

$$\mathrm{\int f(x)=F(x)+C}$$

Note: A constant C is added to the solution to compensate for any constant that might have been in the original function whose derivative is the given function since the derivative of a constant is 0.

Univariate Calculus

Univariate Calculus is the study of calculus where there is only one dependent variable and one independent variable.

Note: 1 independent variable is the reason why this is called Univariate Calculus.

All of the formulae discussed above are under Univariate Calculus.

Multivariate Calculus

In multivariate calculus, there are more than 1 (multiple) independent variables.

The formulae in multivariate calculus change slightly from the univariate calculus.

• Limits − The limit in multivariate calculus is defined by the value that the function approaches when the variables reach a coordinate point in a multidimensional space rather than a number of an axis.

  Example: For 2 independent variables,

  $$\mathrm{\lim_{x,y \rightarrow x_o,y_o} f(x,y)}$$

• Derivatives (partial) − Derivatives change into partial derivatives since the change in one independent variable changes the dependent variable differently than the other independent variable, so there are different partial derivatives for each independent variable.

  Example: For a dependent variable z and 2 independent variables x and y

  PD of z wrt x = $\mathrm{\frac{δz}{δx}}$

  PD of z wrt y = $\mathrm{\frac{δz}{δy}}$

  PD of z wrt both x and y = $\mathrm{\frac{δ^2 z}{δxδy}}$

  PD of z wrt both y and x = $\mathrm{\frac{δ^2 z}{δyδx}}$

  Note: $\mathrm{\frac{δ^2 z}{δxδy}}$ and $\mathrm{\frac{δ^2 z}{δyδx}}$ are not the same thing, the first one is the PD of z wrt x and then PD of the resultant wrt y, whereas the second one is is the PD of z wrt y and then PD of the resultant wrt x.

• Integrals − Similar to the Derivatives the Integrals are also not complete in multivariate calculus. But the most common is,

  $\mathrm{\int \int f. dx.dy}$ (Integrating first with x and then y) which is different than,

  $\mathrm{\int d[\int f.dy.]dx}$(Integrating first with y and then x)

  Note: Limits can also be used to calculate definite integrals in multivariate integrals, a small thing to remember is that the first definite integral has it’s limits in the second integrals’ variable.

Applications of Derivatives and Integrals

Applications of derivatives − Derivatives are majorly used in geometry, economics, study of functions (algebra) and optimisation problems.

• To find the rate of change of one parameter with respect to the other.

• To find the increasing and decreasing nature of a function.

• To find approximations.

• To find the equation of tangents and normal to the equation.

• To find the maximum and minimum value of a function as well as the value of the argument where these maximum/minimum values occur.

• To optimize the output by utilizing the input.

Applications of integrals − Integrals are only mainly used to find the area under a curve of a function or to solve differential equations, which are a major tool in various real-world scenarios, such as population problems, radioactivity problems, growth and decay problems, etc.

Conclusion

In this tutorial, we learned about Calculus, its branches differential calculus and integral calculus, different terms and aspects of differential and integral calculus, and applications of differential and integral calculus in real life.

FAQs

1. What is calculus?

Calculus is the branch of mathematics that deals with the study of the relationship between functions and their argument or dependent and independent variables.

2. What are Derivatives?

Derivatives are defined as the rate of change of a function or a dependent variable with respect to the arguments or the independent variables.

3. What do you mean by continuity of a function?

Continuity is defined as the property of a function that the graph of the function is continuous without any breakpoint or hole in any point in the domain of the function.

4. What is the difference between Definite and Indefinite Integral?

Indefinite integrals are defined as the mathematical opposite of derivatives. They result in a function, the derivative of which will result in the original function.

Definite integrals, on the other hand, are defined as the area under the curve of the function in a given interval. It results in a real number value.

5. Find the derivative and Integral of $\mathrm{f(x)=3x^2-sin x+\frac{4}{x}}$ with respect to x.

Derivative,

We know that the derivative of xⁿ is nx^n-1, and of sin x is cos x.

$$\mathrm{f'(x)=\frac{d}{dx}(f(x))=6x-cos x-\frac{4}{x^2} }$$

Integral,

We know that $\mathrm{\int x^n dx=\frac{x^{n+1}}{n+1}, \int \frac{1}{x} dx=log x, and \int sin x⋅dx=-cos x }$

$$\mathrm{F(x)=\int f(x)⋅dx=x^3+cos x+4log x+C }$$