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Rate of change problem

  • Thread starter Incog
  • Start date
17
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1. Homework Statement

Suppose that t hours after a piece of food is put in the fridge its temperature (in Celsius) is

T(t) = 15 - 3t + [tex]\frac{4}{t - 1}[/tex]

where 0 [tex]\leq[/tex] t [tex]\leq[/tex] 5.

Find the rate of change of temperature after one hour.


3. The Attempt at a Solution

Since it's asking for rate of change, I'm guessing I have to find the derivative of the equation with respect to t.


T(t) = 15 - 3t + [tex]\frac{4}{t - 1}[/tex]

T`(t) = 0 - 3 + [tex]\frac{0(t - 1) - 1(4)}{(t-1)^{2}}[/tex] (Quotient Rule)

T`(t) = -3 + [tex]\frac{0 - 4}{(t-1)^{2}}[/tex]

T`(t) = -3 + [tex]\frac{-4}{(t-1)^{2}}[/tex]

T`(t) = -3 - [tex]\frac{4}{(t-1)^{2}}[/tex]


Would I just plug in 1 after this?
 

Answers and Replies

HallsofIvy
Science Advisor
Homework Helper
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1. Homework Statement

Suppose that t hours after a piece of food is put in the fridge its temperature (in Celsius) is

T(t) = 15 - 3t + [tex]\frac{4}{t - 1}[/tex]

where 0 [tex]\leq[/tex] t [tex]\leq[/tex] 5.

Find the rate of change of temperature after one hour.


3. The Attempt at a Solution

Since it's asking for rate of change, I'm guessing I have to find the derivative of the equation with respect to t.
Don't guess! The derivative of a function is its rate of change!


T(t) = 15 - 3t + [tex]\frac{4}{t - 1}[/tex]

T`(t) = 0 - 3 + [tex]\frac{0(t - 1) - 1(4)}{(t-1)^{2}}[/tex] (Quotient Rule)

T`(t) = -3 + [tex]\frac{0 - 4}{(t-1)^{2}}[/tex]

T`(t) = -3 + [tex]\frac{-4}{(t-1)^{2}}[/tex]

T`(t) = -3 - [tex]\frac{4}{(t-1)^{2}}[/tex]


Would I just plug in 1 after this?
That's what you would like to do- but this function has serious problem at t= 1. Do you remember that, in order to have a derivative at a point, the function must be continuous there? Are you sure you have copied the problem correctly? That's a very strange temperature function! Isn't it peculiar that the temperature of the food goes up when it is put in the refridgerator?
 
Last edited by a moderator:
17
0
Yes, I checked and checked again and that is the equation.

What if I were to plug in a value slightly greater than 1? Would that give me the rate of change after one hour?
 
Last edited:
HallsofIvy
Science Advisor
Homework Helper
41,738
897
Well, I just don't know what to say about a refrigerator where the temperature goes to infinity in one hour!
 

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