# Lect26 - Curve Sketching 1 2 3 4 5 6 7 8 Domain Intercepts...

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92.131 Lecture 26 1 of 14 Ronald Brent © 2009 All rights reserved. Curve Sketching 1. Domain 2. Intercepts 3. Symmetry 4. Asymptotes 5. Intervals of Increase or Decrease 6. Local Maximum and Minimum Values 7. Concavity and Points of Inflection 8. Sketch the curve

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92.131 Lecture 26 2 of 14 Ronald Brent © 2009 All rights reserved. Using these steps let’s analyze the function 1 2 4 1 ) ( 2 4 + = x x x f . 1) Domain: Since f is a polynomial, ) , ( −∞ is the domain. 2) Intercepts: y -intercept is (0, 1) x intercepts come from solving 0 1 2 4 1 2 4 = + x x , Using 2 x w = gives 0 1 2 4 1 2 = + w w 3 2 4 ± = w , and so 3 2 4 ± ± = x . 3) Symmetry: Since ) ( ) ( x f x f = , the function is symmetric w.r.t. y - axis. 4) Asymptotes: None (Since it’s a polynomial.)
92.131 Lecture 26 3 of 14 Ronald Brent © 2009 All rights reserved. x -4 -3 -2 -1 0 1 2 3 4 5) Intervals of Increase and Decrease. Here x x x f 4 ) ( 3 = , and 4 3 ) ( 2 = x x f . We can factor the first derivative as ) 2 )( 2 ( ) ( + = x x x x f and diagram its sign below. − − − − − − − + + + + + + − − − − − − + + + + + + + Sign of ) ( x f ) ( x f is increasing on the intervals ) 0 , 2 ( and ) , 2 ( and decreasing on ) 2 , ( and ) 2 , 0 (. 2 , 0 , 2 = x are critical points. 6) Relative Extrema: From the diagram 3 ) 2 ( = ± f are relative minimum values and 1 ) 0 ( = f is a relative maximum value.

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92.131 Lecture 26 4 of 14 Ronald Brent © 2009 All rights reserved. x -4 -3 -2 -1 0 1 2 3 4 7) Concavity: Diagramming the second derivative 4 3 ) ( 2 = x x f : + + + + + + + + + − − − − − − − − + + + + + + + + + + + Sign of ) ( x f Concave up on the intervals 3 2 , and , 3 2
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Lect26 - Curve Sketching 1 2 3 4 5 6 7 8 Domain Intercepts...

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