Expert: Paul Klarreich - 3/13/2008

Question
I have had previous experience in Calculus but have not done any maths for a few years now and can't quite remember how to answer this question. Your help is appreciated.

T = 0.001*t^3 + 0.008*t^2 + 0.016*t + 0.5

0 < t < 6
a) What is the domain of T?
b) For what values of t is T increasing and for what values decreasing?
c) What is the smallest value of T in its domain, and when does it occur?

Thx

Answer
Questioner:   Gaylene
Category:  Calculus
Private:  No
 
Subject:  AP calculus
Question:  I have had previous experience in Calculus but have not done any maths for a few years now and can't quite remember how to answer this question. Your help is appreciated.
..................................
Hi, Gaylene,

For your function:


T = 0.001*t^3 + 0.008*t^2 + 0.016*t + 0.5

0 < t < 6
a) What is the domain of T?

You just wrote it -- 0 < t < 6

b) For what values of t is T increasing and for what values decreasing?

That's basic differentiation:

T = 0.001*t^3 + 0.008*t^2 + 0.016*t + 0.5

T' = 0.003t^2 + 0.016t + 0.016.

Now set that to zero to find your STATIONARY POINTS:

0.003t^2 + 0.016t + 0.016 = 0

Clear fractions:

3t^2 + 16t + 16 = 0

Factor it:

(3t + 4)(t + 4) = 0

t = -4/3,  t = -4.

Those are both outside your interval, so you don't have any S.P.'s. Now, since T' is clearly positive for all positive t, your function is increasing throughout its domain.

c) What is the smallest value of T in its domain, and when does it occur?

A Relative Minimum would occur at either:

1. A stationary point, but you don't have any.

2. An endpoint, but you don't have any of those, either.  

That's because you wrote 0 < t < 6, NOT  0 <= t <= 6 as your domain.  So 0,6 are the endpoints, and 0 would give a smallest value but it is not in your domain.

So THERE IS NO SMALLEST VALUE.