Textbook Question
Sketch the graphs of the rational functions in Exercises 53–60.
𝓍²
y = ------------------
𝓍² ― 4
5. Graphical Applications of Derivatives
Curve Sketching
14:55 minutesProblem 30
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.
30. y = (x² - 4) / (x² - 2)
Verified step by step guidance1
Step 1: Analyze the domain of the function y = (x² - 4) / (x² - 2). Identify any restrictions on x by setting the denominator equal to zero. Solve x² - 2 = 0 to find the values of x where the function is undefined.
Step 2: Determine the critical points by finding the derivative of the function y = (x² - 4) / (x² - 2). Use the quotient rule: \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \), where \( u = x^2 - 4 \) and \( v = x^2 - 2 \). Set the derivative equal to zero to solve for x.
Step 3: Identify local extreme points by analyzing the critical points found in Step 2. Use the first derivative test or second derivative test to determine whether each critical point corresponds to a local maximum, local minimum, or neither.
Step 4: Find inflection points by calculating the second derivative of the function. Set the second derivative equal to zero and solve for x. Verify changes in concavity by testing intervals around these points.
Step 5: Determine absolute extreme points by evaluating the function at critical points and endpoints of the domain (if applicable). Compare the values of the function at these points to identify the absolute maximum and minimum.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Graphing Rational Functions
Graphing rational functions involves analyzing the function's behavior by identifying key features such as intercepts, asymptotes, and the overall shape of the graph. For the function y = (x² - 4) / (x² - 2), one must determine where the function is undefined (vertical asymptotes) and where it crosses the axes (x and y intercepts) to sketch an accurate graph.
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Local and Absolute Extrema
Local extrema refer to points where a function reaches a maximum or minimum value within a specific interval, while absolute extrema are the highest or lowest points over the entire domain of the function. To find these points, one typically uses the first derivative test to identify critical points and the second derivative test to determine their nature.
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Inflection Points
Inflection points occur where the concavity of a function changes, which can be identified by analyzing the second derivative. For the function y = (x² - 4) / (x² - 2), finding inflection points involves determining where the second derivative equals zero or is undefined, indicating a change in the curvature of the graph.
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