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 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.

Explanation:

• For \(f(9) = 130\): The function \(f(t)\) gives the temperature of the coffee at time \(t\). So when \(t = 9\) (9 minutes after pouring), \(f(9)\) is the temperature value.
• For \(f^{\prime}(9)=-7\): The derivative \(f^{\prime}(t)\) represents the rate of change of the temperature function \(f(t)\). A negative value indicates a decrease.

Answer:

• For \(f(9) = 130\): D. 9 minutes after the coffee has been poured, the temperature of the cup of coffee is 130 degrees.
• For \(f^{\prime}(9)=-7\): 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.