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A detailed guideline for sketching functions by discussing the importance of function type, finding the domain, asymptotes, limits, derivatives, critical points, inflection points, and determining increasing/decreasing and concave up/down properties. Three examples are given to illustrate the process.

Typology: Study notes

2009/2010

1 / 4

Download Curve Sketching: Comprehensive Guide to Domains, Asymptotes, Critical & Inflection Points and more Study notes Mathematics in PDF only on Docsity! Chapter 4. Section 4 Page 1 of 4 C. Bellomo, revised 14-Apr-09 Section 4.4 โCurve Sketching A Quick Guideline: โข When looking to sketch a function, it is a good idea to start with the function type. โข Find the domain. โข Find any asymptotes. โข Find any limits (if needed). โข Find the first and second derivative. โข Find any critical points, inflection points. โข Determine when it is increasing/decreasing, concave up/down. โข See the supplemental to sketching for a more detailed guideline. Chapter 4. Section 4 Page 2 of 4 C. Bellomo, revised 14-Apr-09 โข Example. Sketch 2( 1) xy x = โ This is a rational expression. We expect a restricted domain and vertical/horizontal asymptotes. Domain: 2( 1) 0 when 1x xโ = = Since x = 1 does not make the top go to zero, it is a vertical asymptote. Roots: x = 0 is a root. Note 2 0 ( 1) 0x โ โ Horizontal Asymptote: 2 2 1lim as gets large, it behaves like ( 1)x x xx x x xโโ = โ . So there is a H.A. at 0. Evaluating the first derivative: 2 1 4 3 3 ( 1) 2 ( 1) ( 1) 1 ( 1) 1 ( 1) x x xy x x x x x โ โ โโฒ = โ โ โ = โ + = โ โ 0 when 1y xโฒ = = โ , undefined when x = 1. Evaluating the second derivative: 3 2 6 4 ( 1) ( 1)3( 1) ( 1) 2 4 ( 1) x x xy x x x โ โ โ โ โ โโฒโฒ = โ + = โ . There is an inflection point at x = โ2 So doing a sign diagram we see the basic shape Evaluating 2 1( 1) 0.25 ( 1 1) y โโ = = โ โ โ and 2 2( 2) 0.22 ( 2 1) y โโ = = โ โ โ We can also evaluate the limit as x tends to 1 from the right and left to verify.

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