Problem 3 - Calculus

Question

The temperature, \(T\), in degrees Fahrenheit, of a cold yam placed in a hot oven is given by \(T=f(t),\) where \(t\) is the time in minutes since the yam was put in the oven. (a) What is the sign of \(f^{\prime}(t) ?\) Why? (b) What are the units of \(f^{\prime}(20) ?\) What is the practical meaning of the statement \(f^{\prime}(20)=2 ?\)

Step-by-Step Solution

Answer

(a) Positive, because the temperature increases. (b) °F/min; the temperature increases by 2°F per minute at t=20.

Step 1: Understand the Context

The temperature of the yam increases as it is placed in a hot oven. This means the function \( f(t) \), representing the temperature at time \( t \), is increasing.

Step 2: Determine the Sign of the Derivative

The derivative \( f'(t) \) represents the rate of change of temperature with respect to time. Since the temperature increases as the yam heats up, \( f'(t) > 0 \). Thus, the sign of \( f'(t) \) is positive.

Step 3: Determine the Units of the Derivative

The function \( f(t) \) gives temperature in degrees Fahrenheit and \( t \) is in minutes. The derivative \( f'(t) \) is the rate of change of temperature per unit of time, so its units are degrees Fahrenheit per minute \( \text{°F/min} \).

Step 4: Interpret the Meaning of \( f'(20) = 2 \)

The statement \( f'(20) = 2 \) tells us that at \( t = 20 \) minutes, the temperature of the yam is increasing at a rate of 2 degrees Fahrenheit per minute.

Key Concepts

Rate of Change

The concept of rate of change is crucial in understanding how one quantity varies in relation to another. In the context of the exercise, the rate of change is used to describe how the temperature of the yam evolves over time.
The function provided, \( f(t) \), represents the temperature of the yam at any given time \( t \), measured in minutes. When we talk about the rate at which the yam's temperature changes, we are essentially looking at how quickly or slowly this temperature increases.
The rate of change is positive when the temperature increases as time progresses, as is the case in this scenario where the yam is being heated in the oven. This means that as more time passes, the temperature keeps going up at a certain rate.

Derivatives

In calculus, derivatives play a central role in describing the rate of change of functions. For the yam in the oven, the derivative \( f'(t) \) gives us the rate at which the temperature changes with respect to time.
This derivative is essentially a mathematical tool that measures how much \( f(t) \) changes as \( t \) changes.

• When the derivative \( f'(t) \) is positive, as in this case, it indicates that the function is increasing — meaning the temperature is rising.
• If the derivative were zero, it would mean that the temperature is constant, and not actively changing at that moment.
• A negative derivative would signify a decrease in temperature.

Understanding the direction and magnitude of change through derivatives helps us predict and analyze behavior in various contexts.

Interpreting Derivatives

Interpreting derivatives gives us practical insights into the problems we analyze. In the yam example, the derivative \( f'(20) = 2 \) has a specific interpretation that tells us meaningful information about the scenario at hand.
This particular statement means that at 20 minutes into the heating process, the temperature of the yam is increasing at a steady rate of 2 °F per minute.
The units of the derivative, \( \text{°F/min} \), are crucial as they indicate the change in temperature per unit of time.It's important to translate derivative values into real-world contexts:

• We learn about the behavior of the system, in this case, that the yam is getting hotter.
• It allows us to quantify how quickly processes occur, providing practical information for decision-making.

Through the interpretation of \( f'(t) \), we not only solve mathematical equations but also gain insight into how different rates affect real-world situations.