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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 16
Chapter 4, Problem 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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1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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
Related Practice
Textbook Question

As a result of a heavy rain, the volume of water in a reservoir increased by 1400 acre-ft in 24 hours. Show that at some instant during that period the reservoir’s volume was increasing at a rate in excess of 225,000 gal/min. (An acre-foot is 43,560 ft³, the volume that would cover 1 acre to the depth of 1 ft. A cubic foot holds 7.48 gal.) 

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Textbook Question

Theory and Examples


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


Let f(x) = |x³ − 9x|.


d. Determine all extrema of f.

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Textbook Question

Applications


Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).


Find:


∫[f(x) + g(x)] dx

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Textbook Question

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 = (x² - 49) / (x² + 5x - 14)

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Textbook Question

The Mean Value Theorem                                                                                                                                                                  

                                                                                                                                                                                        

 a. Show that the equation 𝓍⁴ + 2𝓍² ― 2 = 0 has exactly one solution on [0,1] .

         

[Technology Exercises] b.Find the solution to as many decimal places as you can.  

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Textbook Question

106. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (d) When is the acceleration positive? Negative?

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