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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 16
Capitolo 4, Problema 16

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-(x+1)^3

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Identify the function: The given function is \( y = 1 - (x+1)^3 \). This is a cubic function, which typically has an S-shaped curve.
Find the first derivative: To locate local extreme points, calculate the first derivative \( y' \) of the function. Use the power rule to differentiate: \( y' = -3(x+1)^2 \).
Set the first derivative to zero to find critical points: Solve \( -3(x+1)^2 = 0 \). This gives \( x = -1 \) as the critical point.
Determine the nature of the critical point: Use the second derivative test. Find the second derivative \( y'' = -6(x+1) \). Evaluate \( y''(-1) \) to determine if the critical point is a local maximum, minimum, or neither.
Find inflection points: Set the second derivative \( y'' = -6(x+1) \) to zero and solve for \( x \). This gives \( x = -1 \) as a potential inflection point. Check the sign change of \( y'' \) around \( x = -1 \) to confirm.

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Graphing Functions

Graphing functions involves plotting the curve of a function on a coordinate plane to visualize its behavior. This includes identifying key features such as intercepts, asymptotes, and the overall shape of the graph. For the function y = 1 - (x+1)^3, understanding how transformations affect the graph is crucial, such as shifts and reflections.
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Graph of Sine and Cosine Function

Local Extreme Points

Local extreme points are points where a function reaches a local maximum or minimum within a certain interval. These are found by analyzing the derivative of the function, setting it to zero, and solving for x. For y = 1 - (x+1)^3, finding the derivative will help identify where the slope changes, indicating potential local maxima or minima.
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Critical Points

Inflection Points

Inflection points occur where the curvature of the graph changes, which is determined by the second derivative of the function. An inflection point is where the second derivative equals zero and changes sign. For y = 1 - (x+1)^3, calculating the second derivative will reveal points where the graph transitions from concave up to concave down or vice versa.
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Critical Points
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Theory and Examples


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Applications


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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.

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