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The session will be on Calculus on the topics Limits, Continuity & Differentiability which form the
building blocks of calculus.
Limits lead us to the derivative & the first principle of calculus. We can move on to differential calculus only if the idea of limits is very clear to us.
The basic concept of limits is explained in the video below. I have gone slow with this so that you can
actually get the concept of limits & understand it clearly. This is the concept which is the real essence of
differential calculus.
In the session, we will take that idea of limits & solve various forms of limits & discuss the methods of
solving limits.
The session will deal also with the topics of continuity & differentiability. Now, before actually getting into
differential calculus, the students need to know the nature of functions, at what points the function is
discontinuous or non-differentiable. The best way to find that out if with the help of graphs.
Although, we would discuss all the graphs that appear in the questions during the class itself, i would advice
the students to see the recording of one of my public classes on graphs of functions.
Graphs & Graphical
Transformations By Dev Von De
In the session, we would discuss how these graphs can be used to check the continuity & differentiability of
any function. They make it easy for us to determine the nature of even the transcendental & piecewise functions
apart from the basic algebraic, trigonometric & inverse functions.
Here are a few things you should know to determine continuity & differentiability:
Differentiability Implies Continuity but Continuity Doesn't Imply Differentiability
This simply means that a differentiable function has to continuous but a continuous may or may not be
differentiable at a certain point.
A function which has a cusp at a point is not differentiable at that point
This means that a smooth curve will be differentiable & a curve with a cusp will not be differentiable at a
certain point.
The difference between the two cases is described below.
However, most students are not able to differentiate between a cusp & a smooth curve. it looks clear in the
media above but when we deal with functions, sometimes its difficult to make out the difference. This is
exactly what the presenter will explain in the session.
So, the curve in green is differentiable at x=a while the curve in orange is not.
The above also proves the first statement. The curve in orange is continuous as it does not break at any point
but it is not differentiable.
We will discuss all the cases where the cusps could appear & also discuss the types of
discontinuities (removable/jump/etc)
We can also use our limits knowledge to find out the continuity & differentiability. However, the method is
less intuitive, but we would do some examples of such cases.
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