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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 3.1.53a
Chapter 3, Problem 3.1.53a

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>
a. Estimate L' (1.5) and state the physical meaning of this quantity.

Verified step by step guidance
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Step 1: Understand the function L(t) which represents the average length of the middle talon of an Indian spotted owlet at t weeks old. The derivative L'(t) represents the rate of change of the talon length with respect to time.
Step 2: To estimate L'(1.5), we need to find the slope of the tangent line to the curve L(t) at t = 1.5. This can be done by finding two points on the curve close to t = 1.5 and calculating the average rate of change between them.
Step 3: Choose two points on the curve, say (t1, L(t1)) and (t2, L(t2)), where t1 is slightly less than 1.5 and t2 is slightly more than 1.5. For example, you might choose t1 = 1 and t2 = 2.
Step 4: Calculate the average rate of change between these two points using the formula: \( \frac{L(t2) - L(t1)}{t2 - t1} \). This will give an approximation of L'(1.5).
Step 5: Interpret the physical meaning of L'(1.5). It represents the rate at which the length of the middle talon is growing at 1.5 weeks. A positive value indicates growth, while a negative value would indicate a decrease in length.

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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'(t) represents the instantaneous rate of change of the average length of the owlet's talon with respect to time, indicating how quickly the talon length is growing at a specific age, t.
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Average Length Function

The function L(t) describes the average length of the middle talon of the Indian spotted owlet as a function of its age in weeks. Understanding this function is crucial for estimating L'(1.5), as it provides the necessary data to analyze how the talon length varies over time.
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Physical Interpretation

The physical meaning of L'(1.5) relates to the biological growth of the owlet. Specifically, it indicates how much the average length of the talon is expected to increase per week when the owlet is 1.5 weeks old, providing insight into the growth patterns of this species.
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Related Practice
Textbook Question

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = -7x; P(-1,7)

Textbook Question

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = 1/x; P (1,1)

Textbook Question

Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.

73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.

a. f(x) = (x-2)^1/3

Textbook Question

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = √3x; a= 12

Textbook Question

Deriving trigonometric identities

a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.

Textbook Question

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(t) = 3t⁴; a= -2, 2

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