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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 4.3.46a
Capitolo 4, Problema 4.3.46a

Identifying Extrema


In Exercises 41–52:


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


f(t) = t³ − 3t², −∞ < t ≤ 3

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First, find the derivative of the function f(t) = t³ − 3t². The derivative, f'(t), will help us identify critical points where local extrema might occur.
Calculate f'(t) by differentiating f(t) = t³ − 3t². Using the power rule, the derivative is f'(t) = 3t² - 6t.
Set the derivative f'(t) equal to zero to find the critical points: 3t² - 6t = 0. Factor the equation to solve for t.
Factor the equation: 3t(t - 2) = 0. This gives the critical points t = 0 and t = 2.
Evaluate the function f(t) at the critical points and at the endpoint t = 3 to determine the local extrema. Compare the values of f(t) at t = 0, t = 2, and t = 3 to identify the local maximum and minimum values within the domain −∞ < t ≤ 3.

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Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are potential locations for local extrema. For the function f(t) = t³ − 3t², finding the derivative and setting it to zero helps identify critical points, which are essential for determining where local maxima or minima might occur.
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Critical Points

First Derivative Test

The First Derivative Test is used to determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign changes of the derivative around the critical points, we can infer the behavior of the function. This test is crucial for identifying the nature of extrema in the function f(t) = t³ − 3t².
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The First Derivative Test: Finding Local Extrema

Endpoints in a Domain

When identifying extrema in a given domain, it's important to evaluate the function at the endpoints of the domain as well. Since the domain for f(t) = t³ − 3t² is −∞ < t ≤ 3, we must consider the behavior of the function as t approaches 3, as this endpoint could potentially be a location for an extreme value.
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Percorso guidato
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Finding the Domain and Range of a Graph
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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.

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Theory and Examples


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


Let f(x) = |x³ − 9x|.


a. Does f'(0) exist?

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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.


f(t) = 12t − t³, −3 ≤ t < ∞

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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.

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Identifying Extrema


In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).


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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.

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Domanda del libro di testo

Finding Antiderivatives

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

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