Problem 11 - Applied Calculus

Question

A yam has just been taken out of the oven and is cooling off before being eaten. The temperature, \(T,\) of the yam (measured in degrees Fahrenheit) is a function of how long it has been out of the oven, \(t\) (measured in minutes). Thus, we have \(T=f(t)\) (a) Is \(f^{\prime}(t)\) positive or negative? Why? (b) What are the units for \(f^{\prime}(t) ?\)

Step-by-Step Solution

Answer

(a) \(f^{\prime}(t)\) is negative because the yam is cooling. (b) Units are degrees Fahrenheit per minute.

Step 1: Understanding the Scenario

The problem describes a cooling yam, where the temperature decreases over time. The function, \(f(t)\), gives the temperature, \(T\), at a given time, \(t\). We're asked to analyze the derivative \(f^{\prime}(t)\) and determine its sign and units.

Step 1: Analyze the Derivative Sign

The derivative \(f^{\prime}(t)\) represents the rate of change of temperature with respect to time. Since the yam is cooling, its temperature decreases as time increases. Therefore, \(f^{\prime}(t)\) would be negative, indicating a decrease in temperature over time.

Step 2: Determine Units for the Derivative

The derivative \(f^{\prime}(t)\) has units derived from the original function \(f(t)\). The temperature \(T\) is measured in degrees Fahrenheit and \(t\) is in minutes. Thus, the units for \(f^{\prime}(t)\) are degrees Fahrenheit per minute, denoting the change in temperature per unit of time.

Key Concepts

Derivative

Derivatives are a cornerstone concept in calculus, particularly useful for understanding how a function changes. In simple terms, the derivative of a function gives us the rate at which the function's value changes as its variable changes. The derivative is often symbolized as \( f'(t) \) or \( \frac{d}{dt}[f(t)] \). It represents an instantaneous rate of change at a given point. To visualize, think of driving a car. The speedometer shows how fast the car is moving at any moment. This speed is analogous to a derivative, showing the rate of change of your position over time. When we talk about derivatives in the context of temperature and time, it helps us understand the cooling or heating rate of an object like a yam. It tells us how fast the temperature is falling or rising at any specific moment. Understanding derivatives is key to studying various real-world phenomena where change over time is critical.

Rate of Change

The rate of change is a measure of how a quantity varies with respect to another. Often, in calculus, it's with respect to time. For example, if you have a temperature function \( T = f(t) \), the rate of change \( f'(t) \) tells you how the temperature changes over time. - If the rate of change is positive, it means the temperature is rising. - If it’s negative, the temperature is falling. The rate of change can be visualized as the slope of a tangent line to the curve of the function at any given point. When you're considering a cooling yam, the negative rate of change reflects a cooling process. This concept is crucial for determining trends and patterns within scientific, economic, and engineering contexts.

Temperature Function

A temperature function like \( T = f(t) \) describes how temperature varies over time. In the given problem, the function tells us how hot or cold the yam is at any time after being removed from the oven. - \( T \) represents the temperature in degrees Fahrenheit.- \( t \) stands for time in minutes. Such functions are practical as they allow us to predict the future temperature at any specific time. By using calculus, particularly derivatives, we can also determine the rate at which the temperature changes as time progresses. Temperature functions are used in multiple fields including meteorology, physics, and culinary science to analyze and predict temperature variations.

Cooling Process

The cooling process is a specific type of rate change where the temperature decreases over time. In the context of the exercise, as the yam cools down, the temperature function \( T = f(t) \) shows a declining trend. This process is typically modeled by Newton’s Law of Cooling, which describes how the rate of cooling of an object is proportional to the difference between its temperature and the ambient temperature. For practical purposes, knowing the cooling rate helps in understanding how long it might take for the yam to reach a temperature suitable for eating. When analyzing problems involving cooling, it's important to note factors like initial temperature, surrounding temperature, and material properties, all of which can affect the rate of cooling.