Domanda del libro di testo
35. Determine the dimensions of the rectangle of largest area that can be inscribed in the right triangle shown in the accompanying figure.
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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 4.4.110
110. Suppose the derivative of the function y = f(x) is
y'=(x-1)^22(x-2)(x-4).
At what points, if any, does the graph of f have a local minimum, local maximum, or
point of inflection?
Guida verificata passo dopo passo1
Step 1: Identify critical points by setting the derivative y' = (x-1)^2 * 2(x-2)(x-4) equal to zero. Solve for x to find the values where the slope of the tangent line is zero or undefined.
Step 2: Analyze the multiplicity of the factors in y'. For example, (x-1)^2 has even multiplicity, which means it does not change sign at x = 1. Factors with odd multiplicity, such as (x-2) and (x-4), indicate sign changes at x = 2 and x = 4.
Step 3: Use the first derivative test to determine whether each critical point corresponds to a local minimum, local maximum, or neither. Examine the sign of y' on intervals around each critical point (e.g., test values in intervals like (−∞, 1), (1, 2), (2, 4), and (4, ∞)).
Step 4: To find points of inflection, compute the second derivative y'' by differentiating y' = (x-1)^2 * 2(x-2)(x-4). Set y'' equal to zero and solve for x to identify where the concavity changes.
Step 5: Verify the nature of each point of inflection by checking the sign of y'' on intervals around the solutions obtained in Step 4. Points where y'' changes sign indicate points of inflection.
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Derivative and Critical Points
The derivative of a function, denoted as y' or f'(x), represents the rate of change of the function at any point. Critical points occur where the derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection. To find these points, we set the derivative equal to zero and solve for x.
Video consigliato:
04:50
Critical Points
Second Derivative Test
The second derivative test is a method used to determine the nature of critical points. If the second derivative, f''(x), is positive at a critical point, the function has a local minimum; if negative, it has a local maximum. If the second derivative is zero, the test is inconclusive, and further analysis is needed.
Video consigliato:
06:02The Second Derivative Test: Finding Local Extrema
Points of Inflection
Points of inflection occur where the concavity of the function changes, which can be identified by analyzing the second derivative. Specifically, a point of inflection exists where f''(x) changes sign, indicating a transition from concave up to concave down or vice versa. These points are important for understanding the overall shape of the graph.
Video consigliato:
04:50Critical Points
Pratica correlata
Domanda del libro di testo
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
f(x) = x(4 − x)³
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Domanda del libro di testo
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
y = 1 / (x² - 1)
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Domanda del libro di testo
Finding Functions from Derivatives
In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.
f'(x) = 2x − 1, P(0,0)
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Domanda del libro di testo
Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.
7. y=sin|x|, -2π≤x≤2π
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Domanda del libro di testo
22. A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only half as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame.
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