Problem 19
Question
In Exercises 5–24, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ y=2-x-x^{3} $$
Step-by-Step Solution
Verified Answer
The function \(y=2-x-x^3\) has x-intercepts at approximately \(x = -1.26, 0, 1.26\), a y-intercept at \(y = 2\), relative extrema at approximately \(x = -0.577, 0.577\) with \(y \approx 2.15\), and a point of inflection at \(x = 0, y = 2\)
1Step 1: Find the Intercepts
Set \( y \) equal to zero and solve for \( x \). This gives the x-intercepts of the function: \(0 = 2 - x - x^3 \). Solving this cubic equation gives \( x \approx -1.26,0,1.26 \). Setting \( x = 0 \) and solve for \( y \) gives \( y = 2 \), which is the y-intercept.
2Step 2: Find the Relative Extrema
Find the first derivative of the function, \( y' = -1 - 3x^2 \), and set it to zero, solve for \( x \). This gives \( -1 - 3x^2 = 0 \), which implies \( x \approx -0.577,0.577 \). These are the x-coordinates of the relative extrema. Substitute \( x \) into the original function gives the y-coordinates of the extrema: \( y \approx 2.15 \).
3Step 3: Find the Points of Inflection
Find the second derivative of the function, \( y'' = -6x \), and set it to zero, solve for \( x \). This gives \( -6x = 0 \), which implies \( x = 0 \). This is the x-coordinate of the point of inflection. Substitute \( x = 0 \) into the original function gives the y-coordinate of the inflection point: \( y = 2 \).
4Step 4: Sketch the Graph
Using the information from steps 1-3, sketch the graph. Mark the intercepts, extrema, points of inflection, and asymptotes when plotting.
5Step 5: Verify with a Graphing Utility
Use a graphing calculator or computer software to plot the function, and confirm that it matches the sketch from step 4.
Key Concepts
Finding X-InterceptsRelative ExtremaPoints of InflectionUsing Graphing Utilities
Finding X-Intercepts
Understanding where a function crosses the x-axis is crucial for analyzing its behavior. To find the x-intercepts of the function
\( y=2-x-x^{3} \),
we set the output (y-value) to zero and solve for x. In this case, we're solving the equation
\(0 = 2 - x - x^3 \).
This is a cubic equation, and graphing utilities or numerical methods may be used to approximate the solutions, which are
\(x \approx -1.26,~0,~1.26\).
These solutions represent the points where the curve of the function intersects the x-axis, and they are essential for sketching the graph accurately.
\( y=2-x-x^{3} \),
we set the output (y-value) to zero and solve for x. In this case, we're solving the equation
\(0 = 2 - x - x^3 \).
This is a cubic equation, and graphing utilities or numerical methods may be used to approximate the solutions, which are
\(x \approx -1.26,~0,~1.26\).
These solutions represent the points where the curve of the function intersects the x-axis, and they are essential for sketching the graph accurately.
Relative Extrema
Relative extrema are the high and low points on a graph within a particular viewing window. To find these points on the curve of the function
\( y=2-x-x^{3} \),
we first calculate the derivative to be
\( y' = -1 - 3x^2 \).
Setting the derivative to zero, we solve the equation
\( -1 - 3x^2 = 0 \).
The solutions
\( x \approx -0.577,~0.577 \)
are the x-coordinates where the slope of the tangent to the function is zero, indicating either a maximum or a minimum. To find the corresponding y-coordinates, we substitute back into the original function to get
\( y \approx 2.15 \).
Charting these points allows us to identify peaks and valleys on the function's graph.
\( y=2-x-x^{3} \),
we first calculate the derivative to be
\( y' = -1 - 3x^2 \).
Setting the derivative to zero, we solve the equation
\( -1 - 3x^2 = 0 \).
The solutions
\( x \approx -0.577,~0.577 \)
are the x-coordinates where the slope of the tangent to the function is zero, indicating either a maximum or a minimum. To find the corresponding y-coordinates, we substitute back into the original function to get
\( y \approx 2.15 \).
Charting these points allows us to identify peaks and valleys on the function's graph.
Points of Inflection
A point of inflection is where a curve changes from being concave (curved upwards) to convex (curved downwards), or vice versa. To locate these points on the graph of
\( y=2-x-x^{3} \),
we determine the second derivative, which is
\( y'' = -6x \).
Setting this derivative to zero gives us
\( -6x = 0 \),
revealing that
\(x = 0\)
is our only point of inflection for this particular function. By substituting x into the original equation, we can find the y-coordinate,
\( y = 2 \),
which helps us understand how the curvature of the graph changes at that specific point.
\( y=2-x-x^{3} \),
we determine the second derivative, which is
\( y'' = -6x \).
Setting this derivative to zero gives us
\( -6x = 0 \),
revealing that
\(x = 0\)
is our only point of inflection for this particular function. By substituting x into the original equation, we can find the y-coordinate,
\( y = 2 \),
which helps us understand how the curvature of the graph changes at that specific point.
Using Graphing Utilities
Graphing utilities such as calculators or software are invaluable tools for visually confirming the algebraic work done in previous steps. After finding the intercepts, extrema, and points of inflection for the function
\( y=2-x-x^{3} \),
we can use these technologies to plot the curve and verify our results. These utilities display an accurate graph and are particularly useful when dealing with complex functions that are difficult to visualize. By comparing our sketch with the graph generated by the utility, we ensure that all significant features of the function's behavior have been properly identified and represented in our analysis.
\( y=2-x-x^{3} \),
we can use these technologies to plot the curve and verify our results. These utilities display an accurate graph and are particularly useful when dealing with complex functions that are difficult to visualize. By comparing our sketch with the graph generated by the utility, we ensure that all significant features of the function's behavior have been properly identified and represented in our analysis.
Other exercises in this chapter
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