While you may have known about various types of functions, their domain and range, it is equally worthy to know about signum function. Signum Function The signum function is defined as $f(x)=\frac{x}{|x|}$ where; $f(x)=-1$, when $x<0$ $f(x)=0$, when $x=0$ and; $f(x)=1$, when $x>0$. It could also be said that the signum function returns -1, 1 … Read more
The General solution of trigonometric equations of the type sin y = sin a, cos y = cos a and tan y = tan a General Solution: The entire set of the values belonging to the unknown angle satisfies the equation. Also, it has all the specific solutions along with the principal solutions. General Solutions … Read more
You may have already known well about what is a Cartesian product of sets. However, here we would consider recalling all of it to reach out better conclusions and discover and learn several more new concepts. You know that we had earlier talked about defining two non-empty sets A and B that are denoted by … Read more
1. Sin 2x = Sin 2x = sin(2x)=2sin(x). cos(x) Sin(2x) = 2 * sin(x)cos(x) Proof: To express Sine, the formula of “Angle Addition” can be used. sin(2x) = sin(x+x) Since Sin (a + b) = Sin(a). Sin(b) + Cos(a).Cos(b) Therefore, sin(x+x) = sin(x)cos(x) + cos(x)sin(x) = 2. sin(x). cos(x) Also, Sin 2x = $\frac{2tanx}{1+\tan 2x}$ … Read more
While you may be clarifying the concepts of functions and relations, considering pictorial and graphical representation of concepts would be of great help to understand things even better. Graphical representation of a function Now when you’re already aware of what are functions, we would straight away move towards its graphical representation. Suppose that a function … Read more
Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation Like many basic results in the calculus, Rolle’s theorem also seems obvious yet important for practical applications. It just says that between any two points where the graph of the differentiable function f (x) cuts the horizontal line there must be a point … Read more
Logarithmic differentiation, derivative of functions expressed in parametric forms. Second-order derivatives The understanding of derivatives involving complicated functions involving products, quotients, or powers can often be simplified by using logarithmic functions. The method used in the following example is called logarithmic differentiation. Example: Differentiate: We take logarithms on both sides of the equation and then … Read more
Exponential functions can be differentiated using the chain rule. One of the most intriguing and functional characteristics of the natural exponential function is that it is its own derivative. In other words, it has solution to the differential equation being the same such that,y’ = y.The exponential function which has the property that the slope … Read more
The function defined by f(x) = bx; (b>0), b≠1) is called an exponential function with base b and exponent x.Here, the domain of f can be explained as a set of all real numbers. Let m and n be positive numbers and let a and b be real numbers. Then, The exponential function y = bx (b> … Read more
Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions and differentiability The derivative does not exist (the function is not differentiable) at any point where the function is not continuous. Therefore, if the function is not continuous it’s also not differentiable. But even if the function … Read more