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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.3.57a
Chapter 4, Problem 4.3.57a

Identifying Extrema


In Exercises 53–60:


a. Find the local extrema of each function on the given interval, and say where they occur.


f(x) = x / 2 − 2sin (x/2), 0 ≤ x ≤ 2π

Verified step by step guidance
1
To find the local extrema of the function \( f(x) = \frac{x}{2} - 2\sin\left(\frac{x}{2}\right) \) on the interval \( 0 \leq x \leq 2\pi \), we first need to find the derivative \( f'(x) \).
Differentiate the function: \( f'(x) = \frac{1}{2} - \cos\left(\frac{x}{2}\right) \). This derivative will help us find the critical points where the slope of the tangent is zero or undefined.
Set the derivative equal to zero to find critical points: \( \frac{1}{2} - \cos\left(\frac{x}{2}\right) = 0 \). Solve for \( x \) to find the critical points within the interval.
Evaluate the function \( f(x) \) at the critical points and at the endpoints of the interval \( x = 0 \) and \( x = 2\pi \) to determine the local extrema.
Compare the values of \( f(x) \) at these points to identify the local maximum and minimum values, and specify where they occur within the interval.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Local Extrema

Local extrema refer to the points in a function where it reaches a local maximum or minimum within a given interval. To find these points, one typically examines the derivative of the function to identify critical points where the derivative is zero or undefined, and then uses the second derivative test or evaluates the function at these points to determine the nature of the extrema.
Recommended video:
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Finding Extrema Graphically

Derivative and Critical Points

The derivative of a function provides the rate of change of the function with respect to its variable. Critical points occur where the derivative is zero or undefined, indicating potential locations for local extrema. By solving f'(x) = 0, we can find these critical points, which are essential for determining where the function's slope changes direction, potentially indicating maxima or minima.
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04:50
Critical Points

Interval Analysis

Interval analysis involves examining the behavior of a function within a specified range of values. For the function f(x) = x/2 − 2sin(x/2) on the interval [0, 2π], it is crucial to evaluate the function at the endpoints and any critical points within this interval to identify local extrema. This ensures that all potential maxima and minima are considered within the given domain.
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Finding Area Between Curves that Cross on the Interval
Related Practice
Textbook Question

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = (x − 2)²ᐟ³.


a. Does f′(2) exist?

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Textbook Question

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = |x³ − 9x|.


a. Does f'(0) exist?

167
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Textbook Question

52. Two masses hanging side by side from springs have positions s_1 = 2 sin t and s_2 = sin 2t,

respectively.

a. At what times in the interval 0 < t do the masses pass each other? (Hint: sin 2t = 2 sint cost.)

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Textbook Question

Identifying Extrema


In Exercises 53–60:


a. Find the local extrema of each function on the given interval, and say where they occur.


f(x) = sin 2x, 0 ≤ x ≤ π

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Textbook Question

Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


k(x) = x³ + 3x² + 3x + 1, −∞ < x ≤ 0

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Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

1 / x²

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