Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problema 4.1.29

Capitolo 4, Problema 4.1.29

Absolute Extrema on Finite Closed Intervals

In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.

g(x) = √(4 − x²), −2 ≤ x ≤ 1

Guida verificata passo dopo passo

First, understand that the function g(x) = √(4 − x²) is defined for the interval −2 ≤ x ≤ 1. This is because the expression under the square root, 4 − x², must be non-negative.

Next, identify the critical points of the function within the interval. To do this, find the derivative of g(x) with respect to x. The derivative is g'(x) = -x / √(4 − x²). Set g'(x) = 0 to find critical points, which gives x = 0.

Evaluate the function g(x) at the critical point and at the endpoints of the interval. Calculate g(−2), g(0), and g(1) to determine the function values at these points.

Compare the values obtained in the previous step to identify the absolute maximum and minimum values of the function on the interval. The largest value is the absolute maximum, and the smallest value is the absolute minimum.

Finally, graph the function g(x) = √(4 − x²) over the interval −2 ≤ x ≤ 1. Mark the points where the absolute extrema occur on the graph, and label these points with their coordinates.

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Absolute Extrema

Absolute extrema refer to the highest and lowest values a function attains on a given interval. To find these, evaluate the function at critical points and endpoints of the interval. The largest value is the absolute maximum, and the smallest is the absolute minimum.

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Finding Extrema Graphically

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are potential locations for local extrema. For the function g(x) = √(4 − x²), find the derivative, set it to zero, and solve for x to identify critical points within the interval.

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Critical Points

Graphing Functions

Graphing a function involves plotting its values over a specified interval to visually identify features like extrema. For g(x) = √(4 − x²), plot points within the interval [-2, 1] and mark the coordinates of the absolute maximum and minimum to illustrate where these extrema occur.

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Percorso guidato

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Graph of Sine and Cosine Function

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Checking the Mean Value Theorem

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f(x) = {x² − x, −2 ≤ x ≤−1

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Checking Antiderivative Formulas

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Finding Indefinite Integrals

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