Discontinuity and inequalities.  TOPIC_SOLVED

Limits, differentiation, related rates, integration, trig integrals, etc.

Discontinuity and inequalities.

Postby sepoto on Thu Oct 31, 2013 4:34 am

Question 1D-5:
Define
a) Find all values of and such that is continuous.
b) Find all values of and such that is continuous. (Be careful!)

Solution 1D-5:
a) For continuity, we want when . Ans.: all such that .
b) when . We have also . Therefore, to make continuous, we want . Combining this condition with the condition from part (a), we get finally .

I find myself lost on the concept that is being taught here. I'm thinking about and I see an inequality with what looks like a linear function and a parabolic function with a limit on the domain of each equation. The domain of the linear equation has to be greater than or equal to one and the domain of the parabolic has to be less than one.

The solutions manual seems to imply that the linear equation should be solved for all values of a and b where x = 1. My question is what about all the other values of the domain of the linear equation greater than one? I see that part b) involves the user of f'(x) but beyond that I don't really understand yet. I am familiar with f'(x) and take it to be the slope of the tangent line. Could I get some help in understanding this question a little better?

Thanks in advance...
sepoto
 
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Re: Discontinuity and inequalities.  TOPIC_SOLVED

Postby FWT on Thu Oct 31, 2013 12:18 pm

sepoto wrote:I find myself lost on the concept that is being taught here.

They're teaching that piecewise functions, to be continuous, have to match up at the breaks where the rules change.

sepoto wrote:The domain of the linear equation has to be greater than or equal to one and the domain of the parabolic has to be less than one.

It's not that it "has to be". It's just that it's defined that way; those are the rules on how this is set up.

sepoto wrote:My question is what about all the other values of the domain of the linear equation greater than one?

Polynomial functions are always continuous. The only place that this one might be discontinuous is where the rules change at x=1.
FWT
 
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