promath There is a lot of confusion on which is bigger between million and billion. In this post, we will find the answer. Let us now find both billion and million in numbers. 1 billion = 10,000 lakhs 1 million = 10 lakhs Since 1 lakh = 1,00,000 we have that 1 billion = 1,000,000,000 1 … Read more
The Maclaurin series expansion of ex or the Taylor series expansion of ex at x=0 is given by the following summation: ex = $\sum_{n=0}^\infty \dfrac{x^n}{n!}$ = $1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+\cdots$. In this post, we will learn how to find the series expansion of ex. Taylor Series Expansion of ex at x=0 The Maclaurin series expansion of a function … Read more
For any real number c, we have limx→ccosx = cosc and cosx is defined for all real numbers. Thus, the function cosx is continuous everywhere. Now, we will prove that cosx is continuous for all values of x by the epsilon-delta method. We will use the following two formulas: Prove cosx is continuous Let f(x)=cosx … Read more
The simplification or the formula of 1+cot2x is given as follows: 1+cot2x = cosec2x. In this post, we will learn how to establish the formula of 1+cot2x. Proof of 1+cot2x Formula Question: Prove the formula 1+cot2x = cosec2x. Solution: L.H.S. = 1+cot2x = $1+(\dfrac{\cos x}{\sin x})^2$ as cotx=cosx/sinx. = $1+\dfrac{\cos^2 x}{\sin^2 x}$ = $\dfrac{\sin^2 x … Read more
The general solution of the differential equation dy/dx=y/x is given by y=cx where c is any constant. To solve dy/dx = y/x, we will use the separation of variables method. Let us learn how to find the solution of dy/dx=y/x. General Solution of dy/dx=y/x Question: Find the general solution of $\dfrac{dy}{dx}=\dfrac{y}{x}$. Answer: $\dfrac{dy}{dx}=\dfrac{y}{x}$ We can … Read more
The integration of the square root of 1+sin2x is given by ∫√(1+sin2x) dx = cosx-sinx+C where C is an integration constant. In this post, we will learn how to integrate square root of 1+sin2x. Integral of Square Root of 1+sin2x Question: Find the integral of 1+sin2x, that is, find ∫√(1+sin2x) dx. Answer: We will use … Read more
The Maclaurin series expansion of cosx or the Taylor series expansion of cosx at x=0 is given as follows: cosx = $\sum_{n=0}^\infty \dfrac{(-1)^n}{(2n)!}x^{2n}$ = $1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!}+\cdots$ Taylor Series Expansion of Sinx at x=0 Note that the Maclaurin series expansion of f(x)=cosx or the Taylor series of a function f(x) at $x=0$ is given by the following … Read more
The cot(x+y) formula or the identity is given by cot(x+y) = $\dfrac{\cot x \cot y -1}{\cot x + \cot y}$. Here we will prove the formula of cot x+y. Proof of cot(x+y) Formula cot(x+y) = $\dfrac{\cot x \cot y -1}{\cot x + \cot y}$. Proof: Let us recall the following two formulas: sin(x+y) = sinx … Read more
The tan(x-y) is the tangent of the difference of the angles x and y. Tan(x-y) formula/identity is given as follows: tan(x-y) = $\dfrac{\tan x -\tan y}{1+\tan x \tan y}$. In this post, we will prove the formula of tan x-y. Proof of tan(x-y) Formula tan(x-y) = $\dfrac{\tan x -\tan y}{1+\tan x \tan y}$ Proof: We … Read more
The function cosx siny is the product of a cosine function and a sine function. In this post, we will learn how to prove the formula/identity of cosx siny. cosx siny Formula The cosx siny formula is given as follows: cosx siny = $\dfrac{\sin(x+y)-\sin(x-y)}{2}$ Let us now prove the above formula of cos x sin … Read more