Calculus

Many students find class 12 Calculus a hard subject. Due to the unfamiliar and challenging concepts they are learning, many students find it difficult. The study of rates of change is calculus. There are two major branches of calculus. The first is differential calculus, which determines a quantity's rate of change. The second is integral calculus. It finds the quantity where the rate of change is known. Calculus explains how values change for any function, both on a small and large scale. Applications of calculus can be found in science, economics, and engineering. It can address a wide range of issues that algebra alone cannot.

What is Differential Calculus?

This is related to the problem of determining how quickly a function changes with other variables. Differentiation is the process of identifying the derivatives. The function's derivatives are generally expressed by $\dfrac{\text{d}y}{\text{d}x}$ or $f\prime(x$). The function is therefore the derivative of y to the variable x.

What is Integral Calculus?

This type of calculus is the exact opposite (reciprocal) of differentiation. Differentiation can be explained as breaking a large object into numerous smaller ones. Correspondingly, we can define integration as the assembling of numerous minor components to create a larger whole.

$\int f(x)dx$ which means Integral of f with respect to x.

Limit and Continuity

The core aspects of many calculus concepts are limits. Limits were developed to solve the mathematical problem of division by zero. Instead, we use limits to think about what happens when we divide by a very small number that is nearly equal to, but not exactly, zero.

Limit: In general, we define the limit for any function f(x), if f(x) approaches a value L as x approaches a, we define the limit of f(x) as x approaches c as L. It is written as:

$\lim_{x \to c} f(x) = L$

1. Left-Hand limit of f(x) at c: $\lim_{x \to c-} f(x)$ is the expected value of f at x = c given the values of f(x) near x to the left of c.

2. Right-Hand limit of f(x) at c: $\lim_{x \to c+} f(x)$ is the expected value of f(x) at x = c given the values of f(x) near x to the right of c.

3. $\lim_{x \to c} f(x)$ means the right-hand and left-hand limits coincide and have the common value as the limit of f(x) at x = c.

Continuity: The possibility to trace a function's graph with a pencil without taking the pencil off the paper is a feature of many functions. These are referred to as continuous functions. A function is continuous at a particular point if its graph does not break at that point. The following three conditions must all be fulfilled for a function to be considered continuous at a specific point.

1. The function f(a) is defined

2. For the function f(x); $\lim_{x \to a} f(x)$ exists

3. For the function f(x); $\lim_{x \to a+} f(x)=\lim_{x \to a-} f(x)=f(a)$

Derivative

The derivative is the most effective component in differential calculus. To display the rate of change, use the derivative. It is useful to illustrate how much the function is altering at each point. The derivative is referred to as a slope.

Slope of a Tangent

On a graph, a derivative is defined as the slope of the tangent, which either meets the curve at a certain point or yields a derivative at that point.

Solved Examples

Example 1: Evaluate $\lim_{x \to -2} 3x^2+5x-2$

Solution: Substitute x as -2; we get

$\lim_{x \to -2} 3x^2+5x-2$

$=3(-2)^2+5(-2)-2$

$=3(4)+10-2$

$=12-10-2$

$=0$

Example 2: Check the contiuity for the function f(x)=2x+3 at x=1.

Solution: It is given that the function f(x)=2x+3 is defined at x=1 which means

f(1)=2(1)+3=5

Now limit of the function at x=1,

$\lim_{x \to 1} f(x) = \lim_{x \to 1} f(2x+3) = 2(1)+3=5$

$\lim_{x \to 1} f(x)=5=f(1)$

Therefore, f(x)=2x+3 is continuous at x=1.

Practise Problem

1. Find the derivative of $g(x) = \sin x^2$.

1. $ -\sin x^2$

2. $2x \cos x^2$

3. $ -2x \cos x^2$

4. $-2x \sin x^2$

Ans: B.

2. Find derivative of $\tan (x+7)$.

1. $\sec^2 (x+7)$

2. $7 \sec^2 (x+7)$

3. $7x \sec^2 (x+7)$

4. $\sec^2x$

Ans: A.

Conclusion

This article covers the study of rates of change called calculus. Differential calculus and integral calculus are the two main subfields of calculus. Calculus discusses the small- and large-scale value changes for any function. The concepts of limit, continuity, derivatives, differentiation, and integration are also covered in this article. You can better comprehend calculus by using the examples that have been solved.