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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.7.34
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

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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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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 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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Related Practice
Textbook Question

5–7. For each function ƒ and interval [a, b], a graph of ƒ is given along with the secant line that passes though the graph of ƒ at x = a and x = b.



a. Use the graph to make a conjecture about the value(s) of c satisfying the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) .


b. Verify your answer to part (a) by solving the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) for c.



ƒ(x) = x² / 4 + 1 ; [ -2, 4] <IMAGE>

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 .