Rina takes a casserole out of the oven and sets it on a table. She uses a thermo

Question

Rina takes a casserole out of the oven and sets it on a table. She uses a thermometer to keep track of the temperature of the casserole after it is removed from the oven. She lets g(t) to represent the temperature of the casserole, in degrees Fahrenheit, I minutes after she sets it on the table. Suppose that g(10) = 180 and g'(10) = -12. What do these statements mean about the casserole? (The casserole is not sitting on a hill, so the word "slope" should not be found anywhere in your answer.) What does g(10.5) represent? Based on the information above, guess an approximate value for g(10.5) (and explain how you came up with your estimate). Without knowing any formula for g - just understanding the context and knowing what happens to food after you take it out of the oven - how do you expect g and g' to behave as t increases? Justify your answer in terms of the context.

Explanation / Answer

g(t) is the temp t min after she sets it on the table

g(10) = 180
This means the casserole is at 180 F at a time of 10 minutes after removing it from the oven.

g'(10) = -12
This means the rate of DECREASE of temperature of the casserole
10 minutes after removing it from the oven is 12 degrees fahrenheit
per minute

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b)
g(10.5) :
This represents the temp of the casserole 10.5 minutes after it is
removed from the oven.

Now, we guess as to what it could be...

Using linear approximation, we have :
g(t) = g(a) + g'(a)*(t - a)

g(t) = g(10) + g'(10) * (t - 10)

g(t) = 180 + (-12)(t - 10)

Plug in t = 10.5 :

180 - 12(10.5 - 10)

174 Fahrenheit ----> ANSWER

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c)
g is the temperature t minutes after removing it from the oven.
g would decrease with time. Initially, it would decrease at a very fast rate and then it decreases at a slower and slower rate with passage of time

g' is the rate of change of temperature t minutes after removing it from
the oven.
We would expect g' to be very very negative early on because initially, the temperature is decreasing very fast and then later on it almost reaches 0 with passage of time as later on, the temperature changes very slowly with time

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