Textbook Question
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = sin 2x + 3 on [-π , π]
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Ch. 4 - Applications of the Derivative
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 4, Problem 4.R.63
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 (3 sin² 2Θ) / Θ²
Verified step by step guidance1
First, identify the form of the limit as Θ approaches 0. The expression (3 sin² 2Θ) / Θ² is in the indeterminate form 0/0, which suggests that l'Hôpital's Rule can be applied.
Apply l'Hôpital's Rule, which states that if the limit of f(Θ)/g(Θ) as Θ approaches a value results in an indeterminate form, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately.
Differentiate the numerator: The derivative of 3 sin² 2Θ with respect to Θ involves using the chain rule. First, differentiate sin² 2Θ to get 2 sin 2Θ * cos 2Θ, and then multiply by the derivative of 2Θ, which is 2. Therefore, the derivative of the numerator is 12 sin 2Θ * cos 2Θ.
Differentiate the denominator: The derivative of Θ² with respect to Θ is straightforward, which is 2Θ.
Now, evaluate the limit of the new expression (12 sin 2Θ * cos 2Θ) / (2Θ) as Θ approaches 0. Simplify the expression and check if further application of l'Hôpital's Rule is needed or if the limit can be directly evaluated.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.
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One-Sided Limits
L'Hôpital's Rule
L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule simplifies the process of finding limits in complex functions.
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5:50Power Rules
Trigonometric Functions
Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In calculus, they are often involved in limits and derivatives due to their unique properties, such as periodicity and boundedness. Understanding their behavior, especially near critical points like 0, is essential for evaluating limits involving trigonometric expressions.
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Related Practice
Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_y→0⁺ (ln¹⁰ y) / √y
Textbook Question
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
ln x and log₁₀ x
Textbook Question
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
a . Give the approximate coordinates of the local maxima and minima of ƒ
Textbook Question
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
f. On what intervals (approximately) is f concave down?
Textbook Question
24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = ln( x² + 3) / (x -1)