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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.3.53a
Capitolo 4, Problema 4.3.53a

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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To find the local extrema of the function f(x) = sin(2x) on the interval [0, π], we first need to find the critical points. This involves taking the derivative of the function and setting it equal to zero.
The derivative of f(x) = sin(2x) is f'(x) = 2cos(2x). Set this equal to zero to find the critical points: 2cos(2x) = 0.
Solve the equation cos(2x) = 0. The solutions to this equation are 2x = π/2 + kπ, where k is an integer. Divide by 2 to solve for x: x = π/4 + kπ/2.
Determine which of these critical points lie within the interval [0, π]. For k = 0, x = π/4; for k = 1, x = 3π/4. Both are within the interval.
Evaluate the function f(x) at the critical points and the endpoints of the interval, x = 0 and x = π, to determine the local extrema. Compare these values to identify the local maxima and minima.

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Local Extrema

Local extrema refer to the points in a function where it reaches a local maximum or minimum value. These points occur where the derivative of the function is zero or undefined, indicating a change in the direction of the function's slope. Identifying local extrema involves finding these critical points and evaluating the function's behavior around them.
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Finding Extrema Graphically

Derivative and Critical Points

The derivative of a function provides the rate of change or slope of the function at any given point. Critical points occur where the derivative is zero or undefined, which are potential locations for local extrema. To find these points, differentiate the function and solve for the values of x that satisfy these conditions within the given interval.
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Critical Points

Trigonometric Functions and Their Derivatives

Understanding the behavior of trigonometric functions like sine is crucial for solving problems involving them. The derivative of sin(2x) is 2cos(2x), which helps in finding critical points. Analyzing the sine and cosine functions over the interval [0, π] allows us to determine where the function increases or decreases, aiding in identifying local extrema.
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Introduction to Trigonometric Functions
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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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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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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

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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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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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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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