Textbook Question
Sketch the graphs of the rational functions in Exercises 53–60.
𝓍⁴ ― 1
y = ------------------
𝓍²
5. Graphical Applications of Derivatives
Curve Sketching
15:33 minutesProblem 4.4.27
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 = 1 / (x² - 1)
Verified step by step guidance1
Step 1: Analyze the domain of the function y = 1 / (x² - 1). The denominator x² - 1 cannot be zero, so solve x² - 1 = 0 to find the values of x where the function is undefined. These values are x = ±1. The domain of the function is all real numbers except x = ±1.
Step 2: Find the first derivative y' to determine critical points and analyze local extrema. Use the quotient rule for differentiation: if y = u/v, then y' = (u'v - uv') / v². Here, u = 1 and v = x² - 1. Compute y' and simplify.
Step 3: Set the first derivative y' equal to zero to find critical points. Solve the resulting equation for x. Also, check where y' is undefined, as these points may correspond to vertical asymptotes or other critical behavior.
Step 4: Find the second derivative y'' to analyze concavity and locate inflection points. Differentiate y' again using the quotient rule. Set y'' = 0 and solve for x to find potential inflection points. Check the sign of y'' on intervals to determine concavity.
Step 5: Identify absolute extrema by evaluating the function at critical points and endpoints of the domain (if applicable). Also, analyze the behavior of the function as x approaches the vertical asymptotes (x = ±1) and as x approaches ±∞ to understand the overall graph shape.
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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 understanding their behavior based on the numerator and denominator. For the function y = 1 / (x² - 1), we identify vertical asymptotes where the denominator is zero, which occurs at x = ±1. Additionally, we analyze the horizontal asymptote, which is determined by the degrees of the numerator and denominator.
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Local Extreme Points
Local extreme points are points on the graph where the function reaches a local maximum or minimum. To find these points, we calculate the first derivative of the function and set it to zero to identify critical points. We then use the first or second derivative test to determine whether these points are maxima, minima, or neither.
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Inflection Points
Inflection points occur where the concavity of the function changes, which can be found by analyzing the second derivative. For the function y = 1 / (x² - 1), we compute the second derivative and set it to zero to find potential inflection points. Evaluating the sign of the second derivative around these points helps confirm whether a change in concavity occurs.
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