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D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentiation and integration I\{f(x)\} = integrate f(t) dt from 0 to x Df(x) =f^ prime (x) then (a) D is linear where I is not linear transformation (b) D is not linear while I is linear transformation (a) D and I both are not linear transformation (d) D and I both are linear transformation?
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D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentia...
Linearity of Differentiation and Integration
Differentiation and integration are fundamental operations in calculus. They both involve transforming functions, but their behaviors with respect to linearity differ. Let's analyze the linearity of differentiation and integration separately.
Differentiation (D)
Differentiation is a linear transformation, which means it satisfies the following properties:
1. Additivity: For any two functions f(x) and g(x), the derivative of their sum is the sum of their derivatives.
- D(f(x) + g(x)) = Df(x) + Dg(x)
2. Homogeneity: Multiplying a function by a constant scales its derivative by the same constant.
- D(cf(x)) = c * Df(x)
The linearity of differentiation can be observed by applying the power rule, product rule, and chain rule, which all preserve linearity.
Integration (I)
Integration, on the other hand, is not a linear transformation. It does not satisfy the properties of additivity and homogeneity. The integral of a sum of functions is not equal to the sum of their integrals, and scaling a function inside the integral does not scale the result by the same constant.
For example, let's consider the integral of two functions f(x) and g(x):
- I(f(x) + g(x)) ≠ If(x) + Ig(x)
- I(cf(x)) ≠ c * If(x)
The lack of linearity in integration is due to the accumulation of values over a range, which introduces dependencies between different parts of the function.
Conclusion
Based on the above analysis, we can conclude that:
The correct answer is (a) D is linear where I is not a linear transformation.
- Differentiation (D) is a linear transformation as it satisfies the properties of additivity and homogeneity.
- Integration (I) is not a linear transformation as it does not satisfy the properties of additivity and homogeneity.
It is important to note that linearity is a fundamental concept in mathematics, and understanding whether a transformation is linear or not helps in solving various problems and analyzing their properties.
Differentiation and integration are fundamental operations in calculus. They both involve transforming functions, but their behaviors with respect to linearity differ. Let's analyze the linearity of differentiation and integration separately.
Differentiation (D)
Differentiation is a linear transformation, which means it satisfies the following properties:
1. Additivity: For any two functions f(x) and g(x), the derivative of their sum is the sum of their derivatives.
- D(f(x) + g(x)) = Df(x) + Dg(x)
2. Homogeneity: Multiplying a function by a constant scales its derivative by the same constant.
- D(cf(x)) = c * Df(x)
The linearity of differentiation can be observed by applying the power rule, product rule, and chain rule, which all preserve linearity.
Integration (I)
Integration, on the other hand, is not a linear transformation. It does not satisfy the properties of additivity and homogeneity. The integral of a sum of functions is not equal to the sum of their integrals, and scaling a function inside the integral does not scale the result by the same constant.
For example, let's consider the integral of two functions f(x) and g(x):
- I(f(x) + g(x)) ≠ If(x) + Ig(x)
- I(cf(x)) ≠ c * If(x)
The lack of linearity in integration is due to the accumulation of values over a range, which introduces dependencies between different parts of the function.
Conclusion
Based on the above analysis, we can conclude that:
The correct answer is (a) D is linear where I is not a linear transformation.
- Differentiation (D) is a linear transformation as it satisfies the properties of additivity and homogeneity.
- Integration (I) is not a linear transformation as it does not satisfy the properties of additivity and homogeneity.
It is important to note that linearity is a fundamental concept in mathematics, and understanding whether a transformation is linear or not helps in solving various problems and analyzing their properties.
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D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentiation and integration I\{f(x)\} = integrate f(t) dt from 0 to x Df(x) =f^ prime (x) then (a) D is linear where I is not linear transformation (b) D is not linear while I is linear transformation (a) D and I both are not linear transformation (d) D and I both are linear transformation?
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D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentiation and integration I\{f(x)\} = integrate f(t) dt from 0 to x Df(x) =f^ prime (x) then (a) D is linear where I is not linear transformation (b) D is not linear while I is linear transformation (a) D and I both are not linear transformation (d) D and I both are linear transformation? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentiation and integration I\{f(x)\} = integrate f(t) dt from 0 to x Df(x) =f^ prime (x) then (a) D is linear where I is not linear transformation (b) D is not linear while I is linear transformation (a) D and I both are not linear transformation (d) D and I both are linear transformation? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentiation and integration I\{f(x)\} = integrate f(t) dt from 0 to x Df(x) =f^ prime (x) then (a) D is linear where I is not linear transformation (b) D is not linear while I is linear transformation (a) D and I both are not linear transformation (d) D and I both are linear transformation?.
D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentiation and integration I\{f(x)\} = integrate f(t) dt from 0 to x Df(x) =f^ prime (x) then (a) D is linear where I is not linear transformation (b) D is not linear while I is linear transformation (a) D and I both are not linear transformation (d) D and I both are linear transformation? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentiation and integration I\{f(x)\} = integrate f(t) dt from 0 to x Df(x) =f^ prime (x) then (a) D is linear where I is not linear transformation (b) D is not linear while I is linear transformation (a) D and I both are not linear transformation (d) D and I both are linear transformation? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentiation and integration I\{f(x)\} = integrate f(t) dt from 0 to x Df(x) =f^ prime (x) then (a) D is linear where I is not linear transformation (b) D is not linear while I is linear transformation (a) D and I both are not linear transformation (d) D and I both are linear transformation?.
Solutions for D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentiation and integration I\{f(x)\} = integrate f(t) dt from 0 to x Df(x) =f^ prime (x) then (a) D is linear where I is not linear transformation (b) D is not linear while I is linear transformation (a) D and I both are not linear transformation (d) D and I both are linear transformation? in English & in Hindi are available as part of our courses for Mathematics.
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D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentiation and integration I\{f(x)\} = integrate f(t) dt from 0 to x Df(x) =f^ prime (x) then (a) D is linear where I is not linear transformation (b) D is not linear while I is linear transformation (a) D and I both are not linear transformation (d) D and I both are linear transformation?, a detailed solution for D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentiation and integration I\{f(x)\} = integrate f(t) dt from 0 to x Df(x) =f^ prime (x) then (a) D is linear where I is not linear transformation (b) D is not linear while I is linear transformation (a) D and I both are not linear transformation (d) D and I both are linear transformation? has been provided alongside types of D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentiation and integration I\{f(x)\} = integrate f(t) dt from 0 to x Df(x) =f^ prime (x) then (a) D is linear where I is not linear transformation (b) D is not linear while I is linear transformation (a) D and I both are not linear transformation (d) D and I both are linear transformation? theory, EduRev gives you an
ample number of questions to practice D: P n (R) P n - 1 (R) and I:P n - 1 (R) P n (R) defined b differentiation and integration I\{f(x)\} = integrate f(t) dt from 0 to x Df(x) =f^ prime (x) then (a) D is linear where I is not linear transformation (b) D is not linear while I is linear transformation (a) D and I both are not linear transformation (d) D and I both are linear transformation? tests, examples and also practice Mathematics tests.
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