You will not be asked to prove any of the propositions from the notes.

However show, demonstrate, etc. are all asking for proof. Essentially everything in maths is proof but in the sense of can you prepare certain propositions/theorems? The answer is no.

Regards,

]]>Suppose you are given a function defined piecewise like

Usually (i.e. if this question is on your paper), restricted to their own domains, both and will be both continuous and differentiable. Two questions now pop up:

1. Do the graphs 'line up' at — is continuous?

2. If they do, do they line up smoothly with a well defined tangent at — is differentiable at ?

To answer the first question we must show/ask that

.

We proved that a limit exists if and only if the left- and right-hand limits exist and are equal. Hence we calculate

and

.

If they are equal then the function is continuous… assuming (as will be the case), that they are in turn equal to : in this case . Otherwise is not continuous.

Now the second question. We have two main ideas here:

(A): If the function is not continuous then it is not differentiable.

On the other hand, if is continuous, we write down for :

We proved in class (the proposition on p.82) that if is continuous and in addition

exists then is differentiable. Otherwise is not differentiable. We calculate in the same way by looking at left- and right-hand limits.

(B) Alternatively we calculate

via

and

.

Again, if these left- and right-hand derivatives exist and are equal, then the function is differentiable. Otherwise it is not differentiable.

Regards,

]]>Does testing the limit of f ‘ (x) prove differentiability? Then what does testing the limit of f(x) mean?please. ]]>

Question 1 is compulsory.

All questions are worth 25 marks.

Regards,

J.P.

i.e. does question 1 carry the same marks as the other four questions? ]]>