Approximating roots / function continuity (2 Viewers)

[Dumsum]

In a question where it asks you to show that a root lies between two values, do we really need to make the statement that the function is continuous between these values? If so, can we just assume that it is continuous, or should we actually show it?

 

[Slidey]

Proving it is beyond 3u scope.

 

[mynameisgone]

i don't know if i've missed something but to prove a root lies between two values don't u sub each value into the function and it's proved if they are of opposite values, i.e. one positive one negative.
please correct me if im wrong, i don't wanna stuff this up

 

[Slidey]

You're correct, presuming the function is continuous.

As an example, simply through plugging in values with your method, does a root exist between x=-1 and x=1 on the curve y=1/x?

 

[FinalFantasy]

i just say it's continuous anyway in all those questions even if i dun really know if it is
lol

 

[acmilan]

If its something like a polynomial then you can be guaranteed its continuous. If its a rational algebraic function (ie. polynomial divided by polynomial) its guaranteed to be continuous everywhere except where the denominator is 0 (eg. Slide Rule's 1/x function). Continuity is preserved through arithmetic, so, for example, adding two continuous functions makes a continuous function, eg. y = x + sinx is continuous everywhere since both x and sinx are continuous. If you remember this then you can always know that most of the functions they will give in the HSC exams are continous

 

[AreYouAlright?]

It's quite simple to prove a curve is continuous, make sure it adheres to these three conditions.

At the point in consideration (a)
Lim x->a from above: = b
Lim x-->a from below: = b
and f(a) should = b

Hence curve is continuous at a.