Question

let f(t) be the temperature of a cup of coffee t minutes after it has been poured. interpret f(6) = 130 and f’(6) = -7. estimate the temperature of the coffee after 6 minutes and 12 seconds, that is, after 6.2 minutes. what does f(6) = 130 imply? a. 130 minutes after the coffee has been poured, the temperature of the cup of coffee is rising at a rate of 6 degrees per minute. b. 6 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. 6 minutes after the coffee has been poured, the temperature of the cup of coffee is 130 degrees. d. 130 minutes after the coffee has been poured, the temperature of the cup of coffee is 6 degrees. what does f’(6) = -7 imply? a. 6 minutes after the coffee has been poured, the temperature of the cup of coffee is rising at a rate of 7 degrees per minute b. 6 minutes after the coffee has been poured, the temperature of the cup of coffee is -7 degrees. c. 7 minutes after the coffee has been poured, the temperature of the cup of coffee is 6 degrees. d. 6 minutes after the coffee has been poured, the temperature of the cup of coffee is falling at a rate of 7 degrees per minute.

Explanation:

First Question (Interpret \( f(6) = 130 \)):

The function \( f(t) \) represents the temperature of coffee \( t \) minutes after pouring. So \( f(6) = 130 \) means at \( t = 6 \) minutes (6 minutes after pouring), the temperature \( f(6) \) is 130 degrees. Option C says "6 minutes after the coffee has been poured, the temperature of the cup of coffee is 130 degrees," which matches this interpretation. Options A and B incorrectly relate to the rate (which is for the derivative), and D swaps the time and temperature values.

The derivative \( f'(t) \) represents the rate of change of temperature with respect to time. A negative value means the temperature is decreasing (falling). \( f'(6) = -7 \) means at \( t = 6 \) minutes, the temperature is changing at a rate of -7 degrees per minute (falling at 7 degrees per minute). Option D says "6 minutes after the coffee has been poured, the temperature of the cup of coffee is falling at a rate of 7 degrees per minute," which matches. Option A is incorrect (positive rate would be rising), B misinterprets the derivative as temperature, and C is irrelevant to the rate.

Answer:

C. 6 minutes after the coffee has been poured, the temperature of the cup of coffee is 130 degrees.

Second Question (Interpret \( f'(6) = -7 \)):