RD Sharma Class 12 Solutions Chapter 15 - Mean Value Theorems (Ex 15.1) Exercise 15.1

Q: 1. Name the applications of mean value theorem

In Calculus, the Mean Value Theorem is regarded as the most significant technique. This theorem aids in the analysis of function behaviour. The mean value theorem has a wide range of applications. This theorem is most commonly used in:The rule of LeibnizThe rule of L'HospitalSymmetry Of second differentialsWhen f: (p, p) R is differentiable and f'() = 0 for all (p, q), "f" is constant.If function (f) is an open set in Rn and f: X Rm is a continuous partial derivatives function, then "f" is differentiable.

Q: 2. Give a brief introduction to the mean value theorem?

In calculus, the Mean Value Theorem is a crucial theorem. Parmeshwara, a Mathematician from Kerala, India, proposed the earliest form of the mean value theorem in the 14th century. In addition, Rolle proposed an easier version of this in the 17th century: Rolle's Theorem, which was only proved for polynomials and was not part of the calculus. Finally, Augustin Louis Cauchy proposed the current version of the Mean Value Theorem in 1823.The Mean Value Theorem asserts that there is one point on a curve travelling between two points where the tangent is parallel to the secant running through the two points. This mean value theorem gave rise to Rolle's theorem.

Q: 3. What are the corollaries of the mean value theorem?

There are two corollaries of the mean value theorem.Corollary 1: If f'(x) = 0 for all x in an open interval (a, b), then f(x) = C for all x in (a, b), where C is a constant.Corollary 2: There exists a constant C such that f(x) = g(x) + C if f'(x) = g'(x) at each point x in an open interval (a, b).The first corollary establishes that if a function's derivative is zero, the function is constant. The second corollary is that the only difference between graphs of functions with equal derivatives is a vertical shift. In integral calculus, this characteristic is utilized to solve initial value problems.

Q: 4. How is the mean value theorem represented graphically?

The mean value theorem is easier to grasp when the function f(x) is represented graphically. We'll look at two different points (a, f(a)), and (b, f(b)). The secant of the curve is the line linking these locations, which is parallel to the tangent cutting the curve at (c, f(c)). The secant of the curve connecting these locations has the same slope as the tangent at point (c, f(c)). We know that the slope at that location is the derivative of the tangent.

Q: 5. How to prepare for the mean value theorem?

Students need to have a solid foundational theorem to learn about the mean value theorem. It is important to have a strong base to build more knowledge on the foundation and consolidate your knowledge. It is crucial to have a strong grasp over concepts like the rule of leibniz, the rule of l’hospital, corollaries of mean value theorem, slope of the tangent curve, slope of the secant etc. Students should also learn through solving sample papers. Students can access free study materials through the Vedantu app and website.

RD Sharma Solutions

RD Sharma Class 12 Solutions Chapter 15 - Mean Value Theorems (Ex 15.1) Exercise 15.1

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An overview of RD Sharma Class 12 Solutions Chapter 15 - Mean Value Theorems (Ex 15.1) Exercise 15.1

Free PDF download of RD Sharma Class 12 Solutions Chapter 15 - Mean Value Theorems Exercise 15.1 solved by Expert Mathematics Teachers on Vedantu. All Chapter 15 - Mean Value Theorems Ex 15.1 Questions with Solutions for RD Sharma Class 12 Mathematics to help you to revise complete Syllabus and Score More marks. Register for online coaching for IIT JEE (Mains & Advanced) and other Engineering entrance exams.

Overview of RD Sharma Class 12 Solutions Chapter 15

Let us first discuss what is the mean value theorem? It defines that for any given curve in between two ending points, there should be a point at which the slope of the tangent to the curve is almost same to the slope of the secant through its ending points.

If f(x) is a function, so that f(x) is continuous on the closed interval p,q and also differentiable on the open interval (p, q), then there is point r in (p, q) that is, p < r < q in such a way that f’(r) = f(q) -f(p)/ q-p

Lagrange’s Mean Value Theorem or first mean value theorem is a synonym for the mean value theorem. This article discuss about Mean Value Theorem for Integrals, Mean ValueTheorem for Integrals problems and Cauchy Mean Value Theorem

Geometrical Representation of Mean Value Theorem

The mean value theorem graph shows the graph of the function f(x).

Let us look at the point A as (a,f(a)) and point B as = (b,f(b))]

The point C in the graph from where the tangent passes through the curve is (cf(c)).

The slope of the tangent line is almost the same as the secant line i.e. both the tangent line and the secant line are parallel to each other.

FAQs on RD Sharma Class 12 Solutions Chapter 15 - Mean Value Theorems (Ex 15.1) Exercise 15.1

1. Name the applications of mean value theorem

In Calculus, the Mean Value Theorem is regarded as the most significant technique. This theorem aids in the analysis of function behaviour. The mean value theorem has a wide range of applications. This theorem is most commonly used in:

The rule of Leibniz
The rule of L'Hospital
Symmetry Of second differentials

When f: (p, p) R is differentiable and f'() = 0 for all (p, q), "f" is constant.

If function (f) is an open set in Rn and f: X Rm is a continuous partial derivatives function, then "f" is differentiable.

2. Give a brief introduction to the mean value theorem?

In calculus, the Mean Value Theorem is a crucial theorem. Parmeshwara, a Mathematician from Kerala, India, proposed the earliest form of the mean value theorem in the 14th century. In addition, Rolle proposed an easier version of this in the 17th century: Rolle's Theorem, which was only proved for polynomials and was not part of the calculus. Finally, Augustin Louis Cauchy proposed the current version of the Mean Value Theorem in 1823.

The Mean Value Theorem asserts that there is one point on a curve travelling between two points where the tangent is parallel to the secant running through the two points. This mean value theorem gave rise to Rolle's theorem.

3. What are the corollaries of the mean value theorem?

There are two corollaries of the mean value theorem.

Corollary 1: If f'(x) = 0 for all x in an open interval (a, b), then f(x) = C for all x in (a, b), where C is a constant.

Corollary 2: There exists a constant C such that f(x) = g(x) + C if f'(x) = g'(x) at each point x in an open interval (a, b).

The first corollary establishes that if a function's derivative is zero, the function is constant. The second corollary is that the only difference between graphs of functions with equal derivatives is a vertical shift. In integral calculus, this characteristic is utilized to solve initial value problems.

4. How is the mean value theorem represented graphically?

The mean value theorem is easier to grasp when the function f(x) is represented graphically. We'll look at two different points (a, f(a)), and (b, f(b)). The secant of the curve is the line linking these locations, which is parallel to the tangent cutting the curve at (c, f(c)). The secant of the curve connecting these locations has the same slope as the tangent at point (c, f(c)). We know that the slope at that location is the derivative of the tangent.

5. How to prepare for the mean value theorem?

Students need to have a solid foundational theorem to learn about the mean value theorem. It is important to have a strong base to build more knowledge on the foundation and consolidate your knowledge. It is crucial to have a strong grasp over concepts like the rule of leibniz, the rule of l’hospital, corollaries of mean value theorem, slope of the tangent curve, slope of the secant etc. Students should also learn through solving sample papers. Students can access free study materials through the Vedantu app and website.