Textbook Question
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.
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.7.106a
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.
Verified step by step guidance1
Step 1: Differentiate the left side of the identity with respect to t. The left side is \( \cos(2t) \). Using the chain rule, the derivative is \(-2\sin(2t)\).
Step 2: Differentiate the right side of the identity with respect to t. The right side is \( \cos^2(t) - \sin^2(t) \). Use the chain rule and the power rule to differentiate each term separately.
Step 3: For \( \cos^2(t) \), use the chain rule: the derivative is \( 2\cos(t)(-\sin(t)) = -2\cos(t)\sin(t) \).
Step 4: For \( \sin^2(t) \), use the chain rule: the derivative is \( 2\sin(t)\cos(t) \).
Step 5: Combine the derivatives from steps 3 and 4: \(-2\cos(t)\sin(t) - 2\sin(t)\cos(t) = -2\sin(t)\cos(t) - 2\sin(t)\cos(t) = -4\sin(t)\cos(t)\). Set this equal to the derivative from step 1, \(-2\sin(2t)\), and simplify to show \( \sin(2t) = 2\sin(t)\cos(t) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They serve as fundamental tools in calculus and can simplify complex expressions. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle identities, which are essential for manipulating and proving relationships between sine and cosine functions.
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Differentiation
Differentiation is a fundamental concept in calculus that involves finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In the context of trigonometric functions, differentiation applies specific rules, such as the derivatives of sine and cosine, to derive new relationships. This process is crucial for proving identities by showing that two expressions have the same derivative.
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Double Angle Formulas
Double angle formulas are specific trigonometric identities that express trigonometric functions of double angles in terms of single angles. For example, the formula sin(2t) = 2sin(t)cos(t) is derived from the sine and cosine functions. Understanding these formulas is essential for simplifying expressions and proving identities, as they provide a direct relationship between angles and their trigonometric values.
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Related Practice
Textbook Question
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
cos y = x; (0, π/2)
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
{Use of Tech} Tree growth Let b represent the base diameter of a conifer tree and let h represent the height of the tree, where b is measured in centimeters and h is measured in meters. Assume the height is related to the base diameter by the function h = 5.67+0.70b+0.0067b².
a. Graph the height function.
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
1
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