Question

let ( f(t) ) be the temperature of a cup of coffee ( t ) minutes after it has been poured. interpret ( f(9)=130 ) and ( f^{prime}(9)=-7 ). estimate the temperature of the coffee 9.6 minutes.
what does ( f(9)=130 ) imply?
a. 130 minutes after the coffee has been poured, the temperature of the cup of coffee is 9 degrees.
b. 9 minutes after the coffee has been poured, the temperature of the cup of coffee is rising at a rate of 130 degrees per minute
c. 130 minutes after the coffee has been poured, the temperature of the cup of coffee is rising at a rate of 9 degrees per minute.
d. 9 minutes after the coffee has been poured, the temperature of the cup of coffee is 130 degrees.
what does ( f^{prime}(9)=-7 ) imply?
a. 9 minutes after the coffee has been poured, the temperature of the cup of coffee is -7 degrees.
b. 7 minutes after the coffee has been poured, the temperature of the cup of coffee is 9 degrees.
c. 9 minutes after the coffee has been poured, the temperature of the cup of coffee is falling at a rate of 7 degrees per minute.
d. 9 minutes after the coffee has been poured, the temperature of the cup of coffee is rising at a rate of 7 degrees per minute.
after 9 minutes and 36 seconds, the coffee will be ( square ) degrees.
(simplify your answer. type an exact answer.)

Explanation:

Step1: Interpret \( f(9) = 130 \)

Since \( f(t) \) is the temperature \( t \) minutes after pouring, when \( t = 9 \), \( f(9)=130 \) means 9 minutes after pouring, the temperature is 130 degrees.

Step2: Interpret \( f^{\prime}(9)=-7 \)

The derivative \( f^{\prime}(t) \) represents the rate of change of temperature. A negative value means the temperature is decreasing. So \( f^{\prime}(9)=-7 \) means 9 minutes after pouring, the temperature is falling at a rate of 7 degrees per minute.

Step3: Estimate the temperature at \( t = 9.6 \)

We use the linear approximation formula \( f(a + h)\approx f(a)+f^{\prime}(a)h \). Here \( a = 9 \), \( h=9.6 - 9=0.6 \), \( f(9) = 130 \), \( f^{\prime}(9)=-7 \).

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Answer:

For the first question: D. 9 minutes after the coffee has been poured, the temperature of the cup of coffee is 130 degrees.
For the second question: C. 9 minutes after the coffee has been poured, the temperature of the cup of coffee is falling at a rate of 7 degrees per minute.
For the temperature at 9.6 minutes: \( 125.8 \)