Rate of change problem

[Incog]

1. The problem statement, all variables and given/known data

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

T(t) = 15 - 3t + \frac{4}{t - 1}

where 0 \leq t \leq 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 + \frac{4}{t - 1}

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

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

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

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


Would I just plug in 1 after this?

[HallsofIvy]

1. The problem statement, all variables and given/known data

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

T(t) = 15 - 3t + \frac{4}{t - 1}

where 0 \leq t \leq 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 + \frac{4}{t - 1}

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

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

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

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


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?

[Incog]

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?

[HallsofIvy]

Well, I just don't know what to say about a refrigerator where the temperature goes to infinity in one hour!