Textbook Question
Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.
f(x) = 6x² - x³
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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 4.7.34
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (sin² 3x) / x²
Verified step by step guidance1
First, identify the form of the limit as x approaches 0. Substitute x = 0 into the expression (sin²(3x)) / x² to see if it results in an indeterminate form like 0/0.
Since substituting x = 0 gives 0/0, l'Hôpital's Rule is applicable. l'Hôpital's Rule states that if the limit of f(x)/g(x) as x approaches a value results in 0/0 or ∞/∞, then the limit can be found by differentiating the numerator and the denominator separately.
Differentiate the numerator sin²(3x) with respect to x. Use the chain rule: the derivative of sin²(3x) is 2sin(3x)cos(3x) multiplied by the derivative of 3x, which is 3. This results in 6sin(3x)cos(3x).
Differentiate the denominator x² with respect to x. The derivative of x² is 2x.
Apply l'Hôpital's Rule by taking the limit of the new expression: lim_x→0 (6sin(3x)cos(3x)) / (2x). Simplify the expression if possible and evaluate the limit as x approaches 0.
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7mKey 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 used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. The rule 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 process can be repeated if the result remains indeterminate.
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Trigonometric Limits
Trigonometric limits involve evaluating limits that include trigonometric functions, such as sine and cosine. A common limit is lim_x→0 (sin x)/x = 1, which is essential for simplifying expressions involving sine functions. Understanding these limits is crucial when working with functions that oscillate or have periodic behavior, especially near points where they may not be defined.
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6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.
f(x) = x³ - 13x² - 9x
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Textbook Question
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = √(9 - x²) + sin⁻¹ (x/3)
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Textbook Question
Crankshaft A crank of radius r rotates with an angular frequency w It is connected to a piston by a connecting rod of length L (see figure). The acceleration of the piston varies with the position of the crank according to the function <IMAGE>
a (Θ) = w²r (cos Θ + (r cos2Θ) / L) .
For fixed w , L, and r find the values of Θ, with 0 ≤ Θ ≤ 2π , for which the acceleration of the piston is a maximum and minimum.
Textbook Question
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = (x - 1)²
Textbook Question
Sketch the graph of a continuous function ƒ on [0, 4] satisfying the given properties.
ƒ' (x) and ƒ'3 are undefined; ƒ'(2) = 0; has a local maximum at x= 1; ƒ has local minimum at x = 2; and ƒ has an absolute maximum at x= 3; and ƒ has an absolute minimum at x = 4 .