what is the most simplest way of computing the domain of a vector-valued function?

Question

const double pi = 3.14

=)

ex. r(t) = (sin((pi)t) / (t + 7))i + ((lnt^2)/(t^2 + 4))j +((t-3)/t^2-8t+15))k

so, for f(t) it isn't defined at -7, for g(t) it is defined everywhere, and for h(t) it isn't defined at 5. i'm getting a weird domain, can someone help me???

Answer 1

find the intersection of each component.

The first has reals except -7

The second has all reals except 0.

The third has all reals except where t^2 -8t +15 = 0 so solving this we get -3 and -5.

Thus your domain is all reals except, -7,-5,-3,0

Answer 2

I agree that the domain

Isn't defined at -7 for f(t)
Is defined everywhere for g(t)

I disagree on h(t).

h(t) = (t - 3)/(t² - 8t + 15) = (t - 3)/[(t - 3)(t - 5)]

Even though you can canel the t - 3 terms, the function is still undefined at t = 3. However that is just a point punched out of the curve rather than an asymptote as it is at t = 5.

Answer 3

I opt to think of of the vector as a three-tuple, purely for computational sake. so which you have r(t)=(5t,-4t,-a million/t). So for this to be defined each and every ingredient could be defined, and consequently you're maximum suitable in that it is the intersection of the three ingredient domain names. the 1st 2 aspects are defined for all genuine numbers (i'm assuming you're working interior the reals, or some field) . The final is defined for all reals (or all field components, no longer 0), different than 0. So the intersection is all reals, no longer 0.