Derivatives - Calculus 3.0

Quick fact - Calculus actually means stone in latin. It's interesting how maths calculus makes us wish a latin calculus fell on Newton's head rather than an apple.

huh, interesting indeed. (❁´◡`❁)

Haven't read the other calc blogs? click here!

Calculus__Differentiation

Calculus 2.0 _ Integration

Now there are few terms and theorems in calculus which are used very often, without whose definitions understanding calculus gets tough.

In this blog we will figure out these terms --

What is

Absolute maximum / minimum

Let f be a function with domain D. Then has an absolute maximum value on D at a point c if f(x) ≤ f(c) for all x in domain or absolute minimum if  f(x) ≥ f(c) for all x in domain.

Simple enough rite, if it is the highest value the function can attain then it is absolute maxima, if lowest then absolute minima.

These are also referred to as global maxima/minima.

Local (Relative) Extreme values

A function f has a local maximum value at point c if f(x)  ≤ f(c) for all x belongs to domain lying in some interval containing c.

Local minima works the similar way.

These are basically higher than their surrounding values but not the highest possible in the function.

An example

Critical Point

An interior point of domain of a function where f'(x) - first derivative of function is either zero or undefined.

<The symbol in red shows undefined>

Rolle's Theorem

The official definition -

Suppose that y = f(x) is continuous over the closed interval [a, b] and differentiable at every point of its interior (a,b). If f(a) = f(b) , then there is atleast one number c in (a,b) at which f'(c) = 0

This means if a differentiable function crosses a horizontal line at two different points then there is at least one point between them where the tangent to the graph is horizontal and derivative zero.

Mean Value Theorem

Official definition -

Suppose y = f(x) is continuous over a close interval [a,b] and differentiable on the interval's interior (a,b). Then there is atleast one point c in (a,b) at which

What does it mean?

Take Rolles theorem and slant it, that's it.

Rolles theorem talks about horizontal lines ( bc f(a) needs to be equal to f(b) ), in this it talks about lines in any angle.

The MVT guarantees that there is a point where the tangent line is parallel to the secant line that join A and B

We can also think of the number f(b) - f(a) / b - a as the average change in f over [a,b] and f'(c) as an instantaneous change. Then the MVT says that the instantaneous change at some interior point is equal to the average change over the entire interval.

As usual, a meme to end the blog