Here you will learn some differentiability examples for better understanding of differentiability concepts. Example 1 : If f(x + y) = f(x) + f(y) – 2xy – 1 for all x and y. If f'(0) exists and f'(0) = -sin\(\alpha\), then find f{f'(0)}. Solution : f'(x) = \(\displaystyle{\lim_{h \to 0}}\) \(f(x+h) – f(x)\over h\) = […]
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Here you will learn some circle examples for better understanding of circle concepts. Example 1 : Find the equation of the normal to the circle \(x^2 + y^2 – 5x + 2y -48\) = 0 at the point (5,6). Solution : Since the normal to the circle always passes through the centre so the equation
Here you will learn what is involutory matrix with examples. Let’s begin – Involutory Matrix If \(A^2\) = I . the matrix A is said to be an involutory matrix, i.e. the square roots of the identity matrix (I) is involutory matrix. Note : The determinant value of this matrix (A) is 1 or -1.
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Here you will learn what is nilpotent matrix with examples. Let’s begin – Nilpotent Matrix A square matrix of the order ‘n’ is said to be a nilpotent matrix of order m, m \(\in\) N if \(A^m\) = O & \(A^{m-1}\) \(\ne\) O. Example : Show that A = \(\begin{bmatrix} 1 & 1 & 3
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Here you will learn what is periodic matrix with examples. Let’s begin – Periodic Matrix A square matrix which satisfies the relation \(A^{k+1}\) = A for some positive integer k, is called a periodic matrix. The period of the matrix is the least value of k for which \(A^{k+1}\) = A holds true. Note that
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Here you will learn what is idempotent matrix with examples. Let’s begin – Idempotent Matrix A square matrix is idempotent matrix provided \(A^2\) = A. For this matrix note the following : (i) \(A^n\) = A \(\forall\) n \(\ge\) 2, n \(\in\) N. (ii) The determinant value of this matrix is either 1 or 0.
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Here you will learn proof of integration of sec inverse x and cosec inverse x. Let’s begin – Integration of Sec Inverse x The integration of sec inverse x or arcsec x is \(xsec^{-1}x\) – \(log|x + \sqrt{x^2 – 1}|\) + C Where C is the integration constant. i.e. \(\int\) \(sec^{-1}x\) = \(xsec^{-1}x\) – \(log|x
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Here you will learn proof of integration of cot inverse x or arccot x and examples based on it. Let’s begin – Integration of Cot Inverse x The integration of cot inverse x or arccot x is \(xcot^{-1}x\) + \(1\over 2\) \(log |1 + x^2|\) + C Where C is the integration constant. i.e. \(\int\)
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Here you will learn proof of integration of tan inverse x or arctan x and examples based on it. Let’s begin – Integration of Tan Inverse x The integration of tan inverse x or arctan x is \(xtan^{-1}x\) – \(1\over 2\) \(log |1 + x^2|\) + C Where C is the integration constant. i.e. \(\int\)
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Here you will learn proof of integration of cos inverse x or arccos x and examples based on it. Let’s begin – Integration of Cos Inverse x The integration of cos inverse x or arccos x is \(xcos^{-1}x\) – \(\sqrt{1 – x^2}\) + C Where C is the integration constant. i.e. \(\int\) \(cos^{-1}x\) = \(xcos^{-1}x\)
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