91Ó°ÊÓ

Understanding the rate of change is essential in calculus, especially when examining dynamic systems like the weight of a laboratory mouse over time. The rate of change, in this context, refers to how quickly the weight of the mouse increases or decreases each week. Given by the function \( w(t) = \frac{13.785}{t} \), this expression provides a model for how mouse weight changes week by week. As the denominator \( t \) increases, meaning as more time passes, the rate of change reduces. This inversely proportional relationship is typical of many natural processes where growth or change slows over time. In simpler terms, during the earlier weeks, the mouse's weight changes rapidly, but as weeks progress, the changes become subtler. Understanding this concept helps in predicting and quantifying real-world phenomena, using tools provided by calculus.

The definite integral is a powerful tool in calculus for calculating the total accumulation of quantities. In our scenario, to find out how much the mouse's weight changes from week 3 to week 9, we calculate the definite integral \( \int_{3}^{9} w(t) \, dt = \int_{3}^{9} \frac{13.785}{t} \, dt \). This process involves finding the function's antiderivative and applying it across the set range, from 3 to 9 weeks.

Calculating a definite integral gives us the net change over a specified interval. It is distinct from an indefinite integral, which finds the general antiderivative without providing bounds. The fundamental theorem of calculus links the concept of differentiation, the finding of instantaneous rate of change, and integration, the computation of the total change over an interval. Thus, evaluating \( \int_{3}^{9} \) precisely quantifies the mouse weight change from week 3 to week 9, providing a numerical value that is crucial in experimental contexts.

The concept of the antiderivative is crucial for understanding how the definite integral is computed. An antiderivative is a function whose derivative gives back the original function. For the function \( w(t) = \frac{13.785}{t} \), its antiderivative is \( 13.785 \ln |t| \).

Think of it as the reverse of differentiation: if differentiation tells you how fast something changes, finding an antiderivative asks, "What function changed at this rate?" Once the antiderivative is found, it can be used to evaluate the definite integral by applying it to specific limits of integration, in our case, from week 3 to week 9.

To compute these values practically:

• First, find the antiderivative of the function.
• Apply the limits of integration to express the total change.
• Subtract the lower limit result from the upper limit result to determine the net change over the interval.

This process allows us to transition from a rate of change back to a cumulative value, encapsulating the weight changes of the mouse observed in the experiment.