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 4.4.93

Capitolo 4, Problema 4.4.93

93. The accompanying figure shows a portion of the graph of a twice-differentiable function y=f(x). At each of the five labeled points, classify y' and \(\y\)'' as positive, negative, or zero.

Guida verificata passo dopo passo

Step 1: Understand the problem. The graph represents a twice-differentiable function y = f(x). At each labeled point (P, Q, R, S, T), we need to classify the first derivative (y') and the second derivative (y'') as positive, negative, or zero based on the behavior of the graph.

Step 2: Analyze point P. At P, the graph is decreasing (sloping downward), so y' < 0. The graph is concave down (curving downward), so y'' < 0.

Step 3: Analyze point Q. At Q, the graph is at a local minimum, meaning the slope is zero (y' = 0). The graph transitions from concave down to concave up, so y'' > 0.

Step 4: Analyze point R. At R, the graph is increasing (sloping upward), so y' > 0. The graph is concave up (curving upward), so y'' > 0.

Step 5: Analyze point S. At S, the graph is at a local maximum, meaning the slope is zero (y' = 0). The graph transitions from concave up to concave down, so y'' < 0.

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First Derivative (y')

The first derivative of a function, y', represents the slope of the tangent line to the graph at any given point. It indicates the rate of change of the function. If y' is positive, the function is increasing; if y' is negative, the function is decreasing; and if y' is zero, the function has a horizontal tangent, indicating a potential local maximum or minimum.

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The First Derivative Test: Finding Local Extrema

Second Derivative (y'')

The second derivative, y'', provides information about the concavity of the function. If y'' is positive, the graph is concave up, resembling a 'U' shape, indicating that the slope of the tangent line is increasing. If y'' is negative, the graph is concave down, resembling an 'n' shape, indicating that the slope of the tangent line is decreasing. A zero value for y'' may indicate an inflection point where the concavity changes.

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The Second Derivative Test: Finding Local Extrema

Critical Points and Inflection Points

Critical points occur where the first derivative y' is zero or undefined, often corresponding to local maxima or minima. Inflection points occur where the second derivative y'' is zero or changes sign, indicating a change in concavity. Analyzing these points helps in understanding the behavior of the function and its graph, such as identifying peaks, troughs, and points of curvature change.

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Critical Points

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Theory and Examples

67. An inequality for positive integers Show that if a, b, c, and d are positive integers, then

[(a^2+1)(b^2+1)(c^2+1)(d^2+1)]/abcd ≥ 16

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Applications

Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of 20 m/sec². How fast will the rocket be going 1 min later?

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Identifying Extrema

In Exercises 19–40:

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

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

f(x) = (x² − 3) / (x − 2), x ≠ 2

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Checking the Mean Value Theorem

Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.

f(x) = √(x(1 − x)), [0, 1]

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32. Answer Exercise 31 if one piece is bent into a square and the other into a circle.

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Checking Antiderivative Formulas

Right, or wrong? Give a brief reason why.

∫−15(x + 3)² / (x − 2)⁴ dx = ((x + 3)/(x − 2))³ + C

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