Domanda del libro di testo
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) = sec²x − 2tan x, −π/2 < x < π/2
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 4, Problema 4.3.54a
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 x − cos x,0 ≤ x ≤ 2π
Guida verificata passo dopo passo1
First, find the derivative of the function f(x) = sin(x) - cos(x). The derivative, f'(x), is obtained by differentiating each term: f'(x) = cos(x) + sin(x).
Next, set the derivative equal to zero to find the critical points: cos(x) + sin(x) = 0. Solve this equation for x within the interval [0, 2π].
To solve cos(x) + sin(x) = 0, you can rewrite it as sin(x) = -cos(x). This can be expressed as tan(x) = -1, which gives the solutions x = 3π/4 and x = 7π/4 within the interval [0, 2π].
Evaluate the function f(x) at the critical points and at the endpoints of the interval. Calculate f(0), f(3π/4), f(7π/4), and f(2π).
Compare the values obtained in the previous step to determine the local extrema. The largest value is the local maximum, and the smallest value is the local minimum within the interval [0, 2π].
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Local Extrema
Local extrema refer to the points in a function where it reaches a local maximum or minimum within a specific 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.
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Finding Extrema Graphically
Derivative
The derivative of a function represents the rate at which the function's value changes with respect to changes in its input. It is a fundamental tool in calculus for analyzing the behavior of functions, particularly in finding critical points where the function's slope is zero, indicating potential local maxima or minima. For the function f(x) = sin x − cos x, the derivative helps identify where the function's slope changes direction.
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Percorso guidato
05:44Derivatives
Interval Analysis
Interval analysis involves examining the behavior of a function within a specified range of input values. In the context of finding extrema, it is crucial to consider the endpoints of the interval as well as any critical points within it. For the function f(x) = sin x − cos x on the interval 0 ≤ x ≤ 2π, this means evaluating the function at x = 0, x = 2π, and any critical points found within this range to determine where local extrema occur.
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02:59Finding Area Between Curves that Cross on the Interval
Pratica correlata
Domanda del libro di testo
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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Domanda del libro di testo
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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Domanda del libro di testo
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) = √3cos x + sin x, 0 ≤ x ≤ 2π
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Domanda del libro di testo
20.The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. a.What dimensions will give a box with a square end the largest possible volume?
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Domanda del libro di testo
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
f(x) = (x + 1)², −∞ < x ≤ 0
156
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