## Rate of change problem

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?

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 Quote by 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.
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?

 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?

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## Rate of change problem

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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