Chapter 4 – Calculus

Whenever we are talking about Differential Calculus, we can’t ignore derivatives. If you want to demonstrate the rate changes, you can do so using the derivative function. Another name for the derivative is the slope. The Derivative is used to measure the steepness of the graph of a function. It shows the change ratio within the value of a function with respect to the change in the independent variable.

If you want to express y’s derivative in regards to x, you can do so like this: dy/dx. This is called the Derivative of a Function.

In this topic, you will understand the integrals as well as their properties. Integration is a crucial topic, and its process is just the opposite of differentiation.

Here are a few basic formulas for Integral Calculus:

1. kƒ(x) dx = k ƒ(x) dx
2. [ƒ(x)g(x)] dx = ƒ(x) dxg(x) dx
3. k dx = kx + C
4. xⁿ dx = + C, n-1
5. e^x dx = e^x + C
6. a^x dx = + C, a 0, a

The essence of calculus is Differentiation. The definition of derivative can be explained in a way that it is an instant range of change in functionality that is based on one of the variables. It can be compared to locating a tangent’s slope to the function at a certain point.

Some of the basic formulas for Differentiation include:

• dk/dx = 0 where k is constant
• d(x)/dx = 1
• d(kx)/dx = k where k is constant
• d (xⁿ)/dx = nx^n-1

One of the essential aspects of Calculus is Limits. Limits are useful for defining integrals, continuity, and also the derivatives. Let’s discuss the limit of a function:

If we take a function “f” which can be defined as some open interval and consists of numbers such as “a” or might be at “a” itself. In that case, you can write the limit of a function like:

lim x → af(x) = L, if given e > 0, there exists d > 0 such that 0 < |x – a| < d implies that |f(x) – L| < e. This means that the limit f(x) as “x” reaches “a” is “L.”

Limits and Continuity:

They are one of the most important topics as far as Calculus is concerned. Since the limit is already discussed, let’s take a look at what continuity is all about.

The topic of continuity is interesting and important too. A very easy way for testing continuity is to find out whether a pen can follow the graph of a function without you taking it off from a paper. While studying both precalculus and calculus, you need to understand the conceptual definition; however, moving ahead, a technical explanation is also necessary. With limits, the way for you to define continuity will be simple.

Continuity and Discontinuity:

As we already have understood that continuity can be defined as a pen following the graph’s function without lifting it up from a paper, here is what discontinuity is all about.

Discontinuity is of four types: Removal, Infinite, and Jump.

• Removal Discontinuity is expressed as: f(a) = lim x → af(x) f(a) = lim x → af(x)
• Infinite Discontinuity is expressed as:
  lim x → 0− f (x) = lim x → 0 − xsin1x = 0 [Since -1 Similarly, lim x → 0 + f(x) = lim x → 0 + xsin1x = 0, [f(0) = 0]. Thus, lim x → 0−f (x) = lim x → 0 + f(x) = f(0).
• Jump Discontinuity is expressed as: lim x → a + f(x) ≠ lim x→a − f(x)

1. 1.  What is calculus in simple terms?
      Ans. Calculus is a branch of mathematics that helps in understanding different changes in values, all of which are concerning a function. Calculus mainly has two types; they are differential and integral calculus.
   2. What are the 4 concepts of calculus?
      Ans. The significant calculus concepts include:
      • Limits
      • Continuity
      • Derivative
      • Integration
   3. How do you explain limits in calculus?
      Ans. Limits are an integral and vital part of calculus as they are used in describing the integrals, continuity and derivatives too.
      If we take a function “f” which can be defined as some open interval and consists of numbers such as “a” or might be at “a” itself. In that case, you can write the limit of a function like:
      Limx → af (x) = L, if given e > 0, there exists d > 0 such that 0 < |x – a| < d implies that |f (x) – L| < e. This means that the limit f (x) as “x” reaches “a” is “L.”
   4. Who is the father of calculus?
      Ans. Sir Isaac Newton invented Calculus.
   5. What is the most difficult type of math?
      Ans. According to different people, calculus is the toughest math to crack.