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The rate of change in a function describes how quickly a value, such as temperature, is shifting over time. It's found by taking the derivative of the function because a derivative expresses the instantaneous rate of change. In our temperature function,\[H=5+95 e^{-0.054 t},\]we derive with respect to time \(t\) to find how fast the temperature is changing at any given moment. The derivative we've calculated is\[\frac{dH}{dt} = -5.13 e^{-0.054 t}.\]
This result shows us that the temperature of the soup decreases over time. The rate is \(-5.13 e^{-0.054 t}\) degrees Celsius per minute, portraying a cooling process that slows down as time goes on. The negative sign is crucial as it indicates the direction of change, confirming the decrease.

A temperature function models how temperature changes over time. In this exercise, the function \(H=5+95 e^{-0.054 t}\) represents how the temperature of a pot of soup cools as time passes. The number you see in front of the exponential term, 95, is the initial change in temperature from some baseline. As time increases, the exponential term \(e^{-0.054 t}\) decreases, which tells us the temperature gradually drops.
Consider the components:

• The constant 5 indicates a baseline temperature as time approaches infinity when the exponential term reduces to near zero.
• The exponential component, using the base \(e\), rapidly decreases, meaning it models the rate that gets slower over time.

This combination shows how the soup cools from a high initial temperature towards a stable state of 5 degrees Celsius.

Differential calculus is a fundamental branch of mathematics focused on how things change. In our exercise, it's used to analyze the temperature function and to develop an understanding of how the temperature shifts over time. We employ differential calculus to find the derivative of the function \(H(t)\), yielding the rate of temperature change \(\frac{dH}{dt}\).
Steps to perform this process:

• Identify the function that models the scenario—in this case,\[H=5+95 e^{-0.054 t}.\]
• Use the rules of differentiation to find the rate of change. Specifically, with exponential functions, recognize that the derivative of \(e^{-kt}\) includes multiplying by \(-k\).

This derivative provides insights into the behavior of temperature over time, showcasing the power of differential calculus to illuminate dynamic processes.