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. after 6 minutes and 12 seconds, the coffee will be degrees. (simplify your answer. type an exact answer.)

Explanation:

First Question: What does \( f(6) = 130 \) imply?

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 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 talk about the rate (which is related to the derivative, not the function value), and D misstates the temperature.

The derivative \( f'(t) \) represents the rate of change of temperature with respect to time. A negative value means the temperature is decreasing (falling). So \( f'(6) = -7 \) 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 wrong (rising, but derivative is negative), B is wrong (temperature value, not rate), C is wrong (temperature value, not rate).

Step 1: Recall the linear approximation formula

The linear approximation of \( f(t) \) at \( t = a \) is \( f(a + h) \approx f(a) + f'(a) \cdot h \), where \( a = 6 \), \( h = 0.2 \) (since 12 seconds is \( \frac{12}{60}=0.2 \) minutes), \( f(6) = 130 \), and \( f'(6) = -7 \).

Step 2: Substitute values into the formula

\( f(6.2) \approx f(6) + f'(6) \cdot (0.2) \)
\( = 130 + (-7) \cdot 0.2 \)

Step 3: Calculate the result

First, calculate \( (-7) \cdot 0.2 = -1.4 \)
Then, \( 130 - 1.4 = 128.6 \)

Answer:

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

Second Question: What does \( f'(6) = -7 \) imply?