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?
(b) What are the units of f′(35)?
(c) Suppose that |f′(35)|=1.5 and f(35)=76. 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 35 minutes after the coffee was put on the counter, its temperature is
76 degC and will decrease by about __________ in the next 15 seconds.
To determine whether f′(t) (the derivative of f(t)) is positive or negative, we need information about the behavior of the temperature function f(t) over time.
(a) If f′(t) is positive, it means that the temperature is increasing as time passes. If f′(t) is negative, it means that the temperature is decreasing as time passes.
To find out whether f′(t) is positive or negative, you will need the function f(t) or additional details about the behavior of the temperature over time.
(b) The units of f′(35) will depend on the units used for the time variable t and the temperature variable H. Since H is given in degrees Celsius, and t is in minutes, f′(t) will have units of degrees Celsius per minute. Therefore, f′(35) will have units of degrees Celsius per minute.
(c) Given |f′(35)|=1.5 and f(35)=76, we can infer that the absolute value of the derivative at t=35 is 1.5 degrees Celsius per minute. The temperature of the coffee at t=35 minutes is 76 degrees Celsius.
To find out how much the temperature will decrease in the next 15 seconds, we can use the approximate rate of change provided by |f′(35)|=1.5.
Rate of change = |f′(35)| = 1.5 degrees Celsius per minute
Since we are interested in the change over 15 seconds, we need to convert 15 seconds to minutes:
15 seconds = (15/60) minutes = 0.25 minutes
Now we can calculate the approximate decrease in temperature using the rate of change:
Decrease in temperature = Rate of change * Time = 1.5 degrees Celsius per minute * 0.25 minutes
Therefore, the temperature of the coffee will decrease by about 0.375 degrees Celsius in the next 15 seconds.