Understanding Parametric Equations: Conflicting Answers Explained

[phalanx123]

two conflicting answers?

I was doing a question on differentiate parametric equations which has this result $$\frac{dy}{dx}=\frac{4sin(4\theta)}{sin\theta}$$. it then asks what the value of $$\frac{dy}{dx}$$would be if $$\theta=0$$. if I substitute $$\theta=0$$ into $$\frac{4sin(4\theta)}{sin\theta}$$ than I get $$\frac{0}{0}$$ which I persume would be infinity, i.e. the grdient of the graph at that point is undefinined. but if I transform $$\frac{4sin(4\theta)}{sin\theta}$$ into $$16cos\theta cos(4\theta)$$ and substitute$$\theta=0$$ in than I got 16 which is the correct answer. How can this be possible?

 

[CRGreathouse]

I guess the simplest answer would be that 0/0 doesn't mean infinity. In your case, it essentially means 'try harder' -- use L'Hopital's rule, or a trig substitution, to solve.

 

[Gib Z]

Many trig Identities arise from assumptions that the value ie $\sin \theta$ does not equal zero. If you look at the proofs, many involve divisions which are not possible when it equals zero. Basically when you transformed it you worked out the limit as it approaches zero, as you let theta become an infinitesimal instead of exactly zero, which some picky mathematicians may say are equivalent but you get what i mean.

 

[HallsofIvy]

No, 0/0 is not "infinity"- it is "undetermined". If you are actually given that dy/dx= sin(4x)/sin(x) then the simple answer is that dy/dx is not defined at x= 0. If however, you are given that the dy/dx= sin(4x)/sin(x) for x not equal to 0 and that dy/dx is defined at x= 0, take the limit of dy/dx as x goes to 0. (Derivatives are not necessarily continuous but the do satisfy the "intermediate value property" and so if the limit exists the derivative must be equal to that limit.) An easy application of L'Hopital's rule gives 16 as the value just as you say.

Gib Z, I don't know any mathematicians, picky or not, who say "infinitesmal is the same as zero". I do know that defining "infinitesmal" is a very picky problem and I would not recommend it to undergraduates.