Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations) Optimization using calculus: A box has a square base of side x cm and a height of h cm.It has a volume of 1 litre (1000 cm3) For what value of x will the surface area of the box … Read more

Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normal, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test is given as a provable tool) To solve practical problems such as engineering optimization, the greatest challenge is often to convert the word problem into … Read more

Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple application Cos (x + y) × cos y + sin (x + y) × sin y = cos x Firstly, we will be using the trigonometric identities. As we know that: Cos (a+b) = cosa × cosb – … Read more

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