Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Statements and Logic
Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.
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### Problem 9 #### Task: Sketch the graph of the function \( f(x) = \dfrac{x^2}{x-1} \). Label the asymptotes and local (relative) extrema. You do not have to label concavity or inflection points. #### Detailed Steps: 1. **Find the Asymptotes:** - **Vertical Asymptote:** Set the denominator equal to zero and solve for \( x \). The vertical asymptote is at \( x = 1 \). - **Horizontal Asymptote:** Analyze the degrees of the numerator and the denominator. Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is no horizontal asymptote. Instead, check for an oblique asymptote by performing polynomial long division. 2. **Determine the Local Extrema:** - Find the first derivative \( f'(x) \) to determine the critical points. - Set \( f'(x) = 0 \) and solve for \( x \). - Use the second derivative test or the first derivative test to confirm whether these points are local maxima or minima. 3. **Plot Key Points and Asymptotes:** - Draw the vertical asymptote at \( x = 1 \). - Plot any calculated critical points. - Draw the graph based on the identified behavior, ensuring it approaches the asymptotes appropriately. For detailed calculations and the sketch, students are encouraged to work through the derivations and plotting individually or refer to supplemental graphing tools or software.
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Transcribed Image Text:### Problem 9 #### Task: Sketch the graph of the function \( f(x) = \dfrac{x^2}{x-1} \). Label the asymptotes and local (relative) extrema. You do not have to label concavity or inflection points. #### Detailed Steps: 1. **Find the Asymptotes:** - **Vertical Asymptote:** Set the denominator equal to zero and solve for \( x \). The vertical asymptote is at \( x = 1 \). - **Horizontal Asymptote:** Analyze the degrees of the numerator and the denominator. Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is no horizontal asymptote. Instead, check for an oblique asymptote by performing polynomial long division. 2. **Determine the Local Extrema:** - Find the first derivative \( f'(x) \) to determine the critical points. - Set \( f'(x) = 0 \) and solve for \( x \). - Use the second derivative test or the first derivative test to confirm whether these points are local maxima or minima. 3. **Plot Key Points and Asymptotes:** - Draw the vertical asymptote at \( x = 1 \). - Plot any calculated critical points. - Draw the graph based on the identified behavior, ensuring it approaches the asymptotes appropriately. For detailed calculations and the sketch, students are encouraged to work through the derivations and plotting individually or refer to supplemental graphing tools or software.
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