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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 4.7.35
Chapter 4, Problem 4.7.35

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→π (cos x +1 ) / (x - π )²

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First, identify the form of the limit as x approaches π. Substitute x = π into the expression (cos x + 1) / (x - π)². You will find that both the numerator and the denominator approach 0, indicating a 0/0 indeterminate form.
Since the limit is in an indeterminate form, l'Hôpital's Rule can be applied. This rule states that if the limit of f(x)/g(x) as x approaches a point is indeterminate, then the limit is the same as the limit of f'(x)/g'(x) as x approaches that point, provided the derivatives exist.
Differentiate the numerator, cos(x) + 1, with respect to x. The derivative of cos(x) is -sin(x), and the derivative of the constant 1 is 0. Therefore, the derivative of the numerator is -sin(x).
Differentiate the denominator, (x - π)², with respect to x. Using the power rule, the derivative of (x - π)² is 2(x - π).
Apply l'Hôpital's Rule by taking the limit of the new expression: lim_x→π (-sin(x)) / (2(x - π)). Substitute x = π into this expression to evaluate the limit. Since sin(π) = 0, the numerator becomes 0, and the denominator becomes 2(π - π) = 0, indicating another 0/0 form. You may need to apply l'Hôpital's Rule again or simplify further to find the limit.

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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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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, and then re-evaluating the limit. This technique simplifies the process of finding limits in complex scenarios.
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Power Rules

Continuity

Continuity refers to a property of functions where they do not have any abrupt changes, jumps, or breaks at a given point. A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. Understanding continuity is essential for evaluating limits, as discontinuities can lead to different limit behaviors.
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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>

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Closest point on a curve What point on the parabola y = 1 - x² is closest to the point (1, 1)?

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Shortest ladder A 10-ft-tall fence runs parallel to the wall of a house at a distance of 4 ft. Find the length of the shortest ladder that extends from the ground to the house without touching the fence. Assume the vertical wall of the house and the horizontal ground have infinite extent.

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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

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.

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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

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