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The zerolength interval property is one of the properties used in definite integrals and they are always positive. The zerolength interval property is .
= 3(4) – 4
= 8
= – 2(3)
= 6
= 2(4) – 2
= 6
= 7(e^{6} – e^{2})
= – 4
What property this does this equation come under ?
comes under the reverse integral property.
In the reverse integral property the upper limits and lower limits are interchanged. The reverse integral property of definite integrals is
In the reverse integral property the upper limits and lower limits are interchanged. The reverse integral property of definite integrals is .
The adding intervals property of definite integrals is .
What is the reverse integral property of definite integrals?
In the reverse integral property the upper limits and lower limits are interchanged. The reverse integral property of definite integrals is
If the order of the matrix is m×n, then how many elements will there be in the matrix?
The number of elements for a matrix with the order m × n is equal to mn, where m is the number of rows and n is the number of columns in the matrix.
The given matrix has 3 rows and 2 columns. Therefore, the order of the matrix is 3×2.
Does Rolle’s theorem applicable if f(a) is not equal to f(b)?
According to Rolle’s theorem, if f : [a,b] → R is a function such that
Another form of Rolle’s theorem for the continuous condition is _____
According to Rolle’s theorem, if f : [a,a+h] → R is a function such that
The matrix which follows the conditions m=n is called?
A square matrix is a matrix in which the number of rows(m) is equal to the number of columns(n). Therefore, the matrix which follows the condition m=n is a square matrix.
The matrix which follows the condition m>n is called as ____________
The matrix in which the number of columns is greater than the number of rows is called a vertical matrix. There the matrix which follows the condition m>n is a vertical matrix.
Rolle’s theorem is just a special case of Lagrange’s mean value theorem when f(a) = f(b) and Lagrange’s mean value theorem is also called the mean value theorem.
According to Lagrange’s mean value theorem, if f : [a,b] → R is a function such that f is differentiable on (a,b) then the formula for Lagrange’s theorem is f’(c) = .
Function f should be _____ on [a,b] according to Rolle’s theorem.
According to Rolle’s theorem, if f : [a,b] → R is a function such that
What is the relation between f(a) and f(b) according to Rolle’s theorem?
According to Rolle’s theorem, if f : [a,b] → R is a function such that
Another form of Rolle’s theorem for the differential condition is _____
According to Rolle’s theorem, if f : [a,a+h] → R is a function such that
What is the relation between f(a) and f(h) according to another form of Rolle’s theorem?
According to Rolle’s theorem, if f : [a, a + h] → R is a function such that
What are/is the conditions to satify Lagrange’s mean value theorem?
According to Lagrange’s mean value theorem, if f : [a,b] → R is a function such that
Lagrange’s mean value theorem is also called as _____
Lagrange’s mean value theorem is also called the mean value theorem and Rolle’s theorem is just a special case of Lagrange’s mean value theorem when f(a) = f(b).
Is Rolle’s theorem applicable to f(x) = tan x on ?
Given function is f(x) = tan x on
F(x) = tan x is not defined at x on
So, f(x) is not continuous on .
Hence, Rolle’s theorem is not applicable.
Find ’C’ using Lagrange’s mean value theorem, if f(x) = e^{x}, a = 0, b = 1.
Given f(x) = ex, a = 0, b = 1
e^{c} = e  1
Differentiation of the function f(x) = sin(x^{2}) is done with chain rule. First we differentiate sin function which becomes cos and then differentiate the inner (x^{2}) which becomes 2x, hence it comes out to be 2xcos(x^{2}).
We know that derivative of tanx is sec^{2}(x), now in the above question we get tan(x + 4), hence its derivative comes out to be sec^{2}(x + 4), as the inside expression (x + 4) is differentiated into 1.
Therefore the answer is sec^{2}(x + 4).
Value after differentiating cos (sinx) is _________
We differentiate the given function with the help of chain rule so we first differentiate the outer function which becomes –sin and then we differentiate the inner function sinx which is differentiated and comes out to be cosx, hence the differentiated function comes out to be sin (sinx).cosx.
Differentiating on both sides we get 2 + 3dy/dx = cosx.
Given function is y =
Now taking RHS and substituting x = tang in it and then we get,
y = , Now it becomes the expansion of the function tan3g,
Hence the given function becomes y= tan1(tan3g). Which is equal to 3g, now substituting the value of g= tan1x, now after differentiating both sides we get the answer
It is a standard rule for derivative of a function of this form in which the original power comes in front and the value in the power is decreased by one. Therefore the only option of this form is nx^{n1} from the given options.
We know xy = 1, hence we differentiate it on both sides:
We get 1 dy/dx = 0, dy/dx = 1, hence the value of dy/dx comes out to be 1.
Value after differentiating cos (x^{2 }+ 5) is ________
We differentiate the given function with help of chain rule and hence the outer function becomes –sin and the inner function is differentiated into 2x, therefore the answer comes out to be sin (x^{2} + 5).2x.
The derivative of cotx is –cosec^{2}x, as this function has a fixed derivative like sinx has its derivative cosx. Therefore the answer to the above question is –cosec^{2}x.
We differentiate the given function with the help of chain rule and hence
the outer function is differentiated into cos, and the inner function comes out to be a and the constant b becomes 0, which is multiplied to the whole function and the answer comes out to be
⇒ a.cos (ax + b).
What is the mathematical expression for the definition of continuity?
function f defined on (a,b) is said to be continuous on (a,b) if it is continuous at every point of (a,b) i.e., if lim_{x→c}f(x) = f(c) ∀ c ∈ (a,b).
What is the mathematical expression for f is right continuous on (a,b)?
A function is said to be continuous when it is both left continuous and right continuous. Mathematical expression for a function f is right continuous on (a,b) is lim_{x→a+}f(x)=f(a).
Minor matrix is not a type of matrix. Scalar, diagonal, symmetric are various type of matrices.
What is/are conditions for a function to be continuous on (a,b)?
The three conditions required for a function f is said to be continuous on (a,b) if f is continuous at each point of (a,b), f is right continuous at x = a, f is left continuous at x = b.
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