Domanda del libro di testo
Applications
Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).
Find:
∫f(x) dx
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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 4.3.45a
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 < ∞
Guida verificata passo dopo passo1
To find the local extrema of the function \( f(t) = 12t - t^3 \), we first need to find its critical points. This involves taking the derivative of the function and setting it equal to zero.
Calculate the derivative: \( f'(t) = \frac{d}{dt}(12t - t^3) = 12 - 3t^2 \).
Set the derivative equal to zero to find critical points: \( 12 - 3t^2 = 0 \). Solve for \( t \) to find the critical points.
Solve the equation \( 12 - 3t^2 = 0 \) to get \( t^2 = 4 \), which gives \( t = 2 \) and \( t = -2 \). These are the critical points within the domain \( -3 \leq t < \infty \).
Evaluate the function \( f(t) \) at the critical points and endpoints of the domain to determine the local extrema. Compare the values of \( f(t) \) at \( t = -3, -2, \) and \( 2 \) to identify the local maximum and minimum values.
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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, as they indicate where the function's slope changes direction. To find critical points, compute the derivative of the function and solve for values of the variable where the derivative equals zero or does not exist.
Video consigliato:
04:50
Critical Points
First Derivative Test
The First Derivative Test helps determine whether a critical point is a local maximum or minimum. By analyzing the sign of the derivative before and after the critical point, one can infer the behavior of the function. If the derivative changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it is a local minimum.
Video consigliato:
07:09The First Derivative Test: Finding Local Extrema
Domain Considerations
Understanding the domain of a function is crucial when identifying extrema, as it defines the range of input values to consider. In this problem, the domain is given as −3 ≤ t < ∞, meaning the function is evaluated from t = -3 onwards. This affects where extrema can occur, especially at the boundary points, which must be checked separately.
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Percorso guidato
5:10Finding the Domain and Range of a Graph
Pratica correlata
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.
−3x⁻⁴
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Domanda del libro di testo
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫√(2x + 1) dx = √(x² + x +C)
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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.
2x
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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(t) = t³ − 3t², −∞ < t ≤ 3
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Domanda del libro di testo
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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