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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 4.3.15
Capitolo 4, Problema 4.3.15

Identifying Extrema


In Exercises 15–18:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local and absolute extreme values, if any, saying where they occur.


Graph of a function showing local minima and maxima, with labeled axes and a curve illustrating increasing and decreasing intervals.

Guida verificata passo dopo passo
1
Step 1: Analyze the graph to determine where the function is increasing or decreasing. Look for intervals where the slope of the graph is positive (increasing) or negative (decreasing).
Step 2: Identify the intervals of increase. From the graph, observe that the function is increasing on the intervals (-3, -2) and (0, 1).
Step 3: Identify the intervals of decrease. From the graph, observe that the function is decreasing on the intervals (-2, 0) and (1, 3).
Step 4: Determine the local extrema by finding the points where the function changes from increasing to decreasing or vice versa. These are typically at the peaks and troughs of the graph.
Step 5: Identify the local and absolute extrema. From the graph, the local maximum occurs at x = -2 and x = 1, and the local minimum occurs at x = 0 and x = 3. The absolute minimum is at x = 3, and the absolute maximum is at x = -2.

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Increasing and Decreasing Functions

A function is said to be increasing on an interval if, for any two points within that interval, the function's value at the second point is greater than at the first. Conversely, a function is decreasing if the value at the second point is less than at the first. Identifying these intervals involves analyzing the function's derivative, where positive values indicate increasing behavior and negative values indicate decreasing behavior.
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Determining Where a Function is Increasing & Decreasing

Local and Absolute Extrema

Local extrema refer to points where a function reaches a maximum or minimum value within a specific neighborhood, while absolute extrema are the highest or lowest values of the function over its entire domain. To find these extrema, one typically uses the first and second derivative tests, evaluating critical points where the derivative is zero or undefined, and determining their nature through further analysis.
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Finding Extrema Graphically

Critical Points

Critical points of a function occur where its derivative is either zero or undefined. These points are essential for identifying local extrema, as they represent potential locations where the function changes from increasing to decreasing or vice versa. Analyzing critical points helps in determining the behavior of the function and locating its maxima and minima.
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Critical Points
Pratica correlata
Domanda del libro di testo

Identifying Extrema


In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.


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Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

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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.


f(t) = 2 − |t|, −1 ≤ t ≤ 3

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Initial Value Problems


Find the curve y = f(x) in the xy-plane that passes through the point (9,4) and whose slope at each point is 3√x.

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Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

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∫ ( 3√ t + 4/t² ) dt

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Domanda del libro di testo

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec θ/3 tan θ/3 dθ

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