# Consider the following statements

Statement-1: Differentiation and integration are processes involving limits.

Statement-2: The process of differentiation and integration are inverse of each other.

Choose the correct option.

(a) Statement-1 is true

(b) Statement-2 is true

(c) Both statements are true

(d) Both statements are false

Answer

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Hint: To find the correct option, write the basic formulas involved with the integration and differentiation of any function and then check the validity of the options.

To solve this question, we will check the validity of both the statements.

We will begin by writing the formulas involved by integration and differentiation of any function to check the first statement.

Consider any general function \[f\left( x \right)\].

To find the differentiation the function \[f\left( x \right)\]at\[x=a\], we will use the formula \[f'\left( a \right)=\underset{h\to \infty 0}{\mathop{\lim }}\,\dfrac{f\left( x+h \right)-f\left( x \right)}{h}\].

Thus, we observe that differentiation of any function involves the limit.

To find the integral of the function \[f\left( x \right)\] over an interval \[[a,b]\], we will subdivide the interval \[[a,b]\]into\[n\] subintervals of equal width, \[\vartriangle x\], and from each interval choose a point \[{{x}_{i}}^{*}\]. Then the integral of the function is\[\int\limits_{a}^{b}{f\left( x \right)dx}=\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{f\left( {{x}_{i}}^{*} \right)}\vartriangle x\].

Thus, we observe that the integral of any function involves limit.

Hence, we observe that Statement-1 is correct.

Now, we will check the second statement which says that the process of integration and differentiation are inverse of each other.

One must know that differentiation forms an algebraic expression that helps in the calculation of the gradient of a curve at any point. While, integration is used to calculate the area under any curve.

Hence, we observe that differentiation tries to lower down the function into simpler parts from which it has originated. While, integration tries to combine the smaller fragments from which the function is obtained.

Thus, integration and differentiation are opposite processes.

Hence, statement-2 is correct.

So, both statements are correct.

Option (c) is the correct answer.

Note: One must clearly know the basic formulas involved with the differentiation and integration formulas.

To solve this question, we will check the validity of both the statements.

We will begin by writing the formulas involved by integration and differentiation of any function to check the first statement.

Consider any general function \[f\left( x \right)\].

To find the differentiation the function \[f\left( x \right)\]at\[x=a\], we will use the formula \[f'\left( a \right)=\underset{h\to \infty 0}{\mathop{\lim }}\,\dfrac{f\left( x+h \right)-f\left( x \right)}{h}\].

Thus, we observe that differentiation of any function involves the limit.

To find the integral of the function \[f\left( x \right)\] over an interval \[[a,b]\], we will subdivide the interval \[[a,b]\]into\[n\] subintervals of equal width, \[\vartriangle x\], and from each interval choose a point \[{{x}_{i}}^{*}\]. Then the integral of the function is\[\int\limits_{a}^{b}{f\left( x \right)dx}=\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{f\left( {{x}_{i}}^{*} \right)}\vartriangle x\].

Thus, we observe that the integral of any function involves limit.

Hence, we observe that Statement-1 is correct.

Now, we will check the second statement which says that the process of integration and differentiation are inverse of each other.

One must know that differentiation forms an algebraic expression that helps in the calculation of the gradient of a curve at any point. While, integration is used to calculate the area under any curve.

Hence, we observe that differentiation tries to lower down the function into simpler parts from which it has originated. While, integration tries to combine the smaller fragments from which the function is obtained.

Thus, integration and differentiation are opposite processes.

Hence, statement-2 is correct.

So, both statements are correct.

Option (c) is the correct answer.

Note: One must clearly know the basic formulas involved with the differentiation and integration formulas.

Last updated date: 03rd Oct 2023

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