I'm not sure what concept this is

Question

Which of the following functions grow the fastest?

g(t)=(3t-t)

i(t)= ln(t^100)

a(t) = t^5/2

c(t)=√(t^2-5t)

e(t)=e

Answer 1

Terra,

Your question to Linda sort of answers your own question. I believe it is about limits and derivatives. What does the first derivative represent? It represents rate of change, in other words growth. If we stick with Linda's labeling of the 5 equations as a-e, we get the following derivatives for each:

a) 2

b) 100/t

c) 2.5t^1.5

d) (2t-5)/[2(t^2-5t)^.5]

e) 0

Your answer as to which grows the fastest depends on what time you are evaluating it. At t=1, the function with the greatest growth is actually the natural log function. But if we take the limit of each of these functions as t approaches infinity, c will be your answer. In fact, I think the growth of c overtakes the growth of b sometime between t=4 and t=5.

Answer 2

This question looks to me like you are studying functions, asymptotes, and limits.

I looked over each function and without graphing them I came to the following conclusions:

g(t) looks like a straight line with a slope of 2 since 3t - t is the same as 2t.

I(t) is a natural log. Those graphs tend to rise more and more slowly as t gets bigger and bigger.

So the graph will be a very slow rising curve.

a(t) has an exponent that is greater than 2, so it will rise very quickly as t gets large

c(t) is basically the square root of t² . The other terms influence the graph closer to the origin but less so as to gets larger and larger.

e(t) is a constant and so will be just a straight horizontal line.

So I would choose a(t).

I will be interested to see what other tutors say about this problem.

I hope this helps.