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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.1.53b
Chapter 3, Problem 3.1.53b

Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.<IMAGE>
b. Estimate the value of L'(a) for a ≥ 4 . What does this tell you about the talon lengths on these birds? (Source: ZooKeys, 132, 2011)

Verified step by step guidance
1
Step 1: Understand that L'(a) represents the derivative of the function L(t) at t = a, which gives the rate of change of the talon length with respect to time at that specific week.
Step 2: To estimate L'(a) for a ≥ 4, identify two points on the graph of L(t) that are close to t = a. These points will help you calculate the average rate of change, which approximates the derivative.
Step 3: Use the formula for the average rate of change between two points (t1, L(t1)) and (t2, L(t2)) on the graph: \( \frac{L(t2) - L(t1)}{t2 - t1} \). Choose t1 and t2 such that t1 < a < t2.
Step 4: Substitute the values of L(t1) and L(t2) from the graph into the formula to calculate the average rate of change, which serves as an estimate for L'(a).
Step 5: Interpret the result: A positive L'(a) indicates that the talon length is increasing at week a, while a negative L'(a) would indicate a decrease. The magnitude of L'(a) tells you how quickly the talon length is changing.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. In this context, L'(a) represents the instantaneous rate of change of the average length of the talon with respect to time (weeks). A positive derivative indicates that the talon length is increasing, while a negative derivative suggests it is decreasing.
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Average Rate of Change

The average rate of change of a function over an interval gives a general idea of how the function behaves between two points. For the function L(t), the average rate of change from t = 4 to t = a can provide insights into how the average talon length evolves as the owlet ages, helping to contextualize the derivative's value.
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Biological Growth Patterns

Understanding biological growth patterns is essential for interpreting the results of L'(a). In many species, growth rates can vary with age due to factors like maturity and environmental influences. Analyzing L'(a) in this context can reveal whether the talon lengths are stabilizing, accelerating, or decelerating as the owlets mature.
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