The temperature, H, in degrees Celsius, of a cup of coffee placed on the kitchen counter is given by H = f(t), where t is in minutes since the coffee was put on the counter.
a) Is f'(t) positive or negative? Why?
b) What are the units of f'(35)?
c) Suppose that |f'(35)| = 1.5 and f(35) = 68. Fill in the blanks (including units where needed) and select the appropriate terms to complete the following statement about the temperature of the coffee in this case.
At ____ minutes after the coffee was put on the counter, its temperature is ____ and will ____ by about ____ in the next 30 seconds.

Answer:

a

    f'(t) is  negative

b

 The  unit is degC/min

c

At 35 minutes after the coffee was put on the counter, its temperature is 68 and will decrease by about 0.75 in the next 30 seconds.          

Step-by-step explanation:

From the question we are told we are told that  

   The  equation for the temperature of a cup of coffee place on a counter is

         [tex]H= f(t)[/tex]

Here  t is the time in minutes the  coffee was put on the counter

Generally  f(t)'  is the derivative of  f(t) at it represents the change of the temperature of the cup of coffee with respect to time

 Generally for the coffee placed on the counter after a couple of minutes the temperature will decrease hence  f(t)' is  negative

Generally the unit of the temperature is in degrees Celsius while that of time is  in minutes

  So the change in temperature with time f(35)' will be in degrees Celsius/minutes (i.e  degC/min)

  From the question we are told that

        [tex]|f(35)'| =  1.5[/tex]

i.e  the rate of change of temperature after 35 minutes is  1.5

and

         [tex]|f(35)| =  68[/tex]

i.e the temperature of the cup of coffee after 35 minutes is  67 degC

Now after another   t=  30  seconds =  [tex]\frac{30}{60}= 0.5 \ minutes[/tex] the  rate  of change of the temperature of the cup of  coffee is  

          [tex]R =  |f(35)'| *  t[/tex]

=>        [tex]R = 1.5  *  0.5 [/tex]

=>        [tex]R = 0.75  [/tex]    

So

At 35 minutes after the coffee was put on the counter, its temperature is 68 and will decrease by about 0.75 in the next 30 seconds.  

The value of f'(t)  is negative because as the cup of coffee is placed on the counter there is a loss of heat from the cup of coffee to the surroundings as t increases hence f'(t) is negative.

The units of  f'(35) = degC/min.

At 35 minutes after the coffee was put on the counter, its temperature is 68 and will decrease by about 0.75 in the next 30 seconds.    

The temperature, H, in degrees Celsius, of a cup of coffee placed on the kitchen counter is given by H = f(t), where t is in minutes since the coffee was put on the counter.

1. The value of f'(t)  is negative because as the cup of coffee is placed on the counter there is a loss of heat from the cup of coffee to the surroundings as t increases hence f'(t) is negative.

2. What are the units of f'(35)?

The units of  f'(35) = degC/min.

3. At 35 minutes after the coffee was put on the counter, its temperature is 68 and will decrease by about 0.75 in the next 30 seconds.    

At ║f'(35)║ = 1.5 and f'(35) = 68

|f'(35)| = 1.5 represents the rate of change of temperature every 35 minutes = 1.5.

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