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11 x1 t12 05 curve sketching (2013)

N
Nigel Simmons

The document provides guidance on sketching curves based on information derived from the derivative of the curve function. Key points include looking for x-intercepts, asymptotes, points where the derivative is zero (stationary points), points where the second derivative is zero (inflection points), and using the first and second derivatives to determine concavity and points of maximum/minimum. Examples are given demonstrating these concepts, such as finding the x-value of a stationary point and determining if a point is a local max/min based on the sign of the second derivative.

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Curve Sketching
Curve Sketching Look for;
Curve Sketching 0,0, 1 e.g.itydiscontinuofpoints  yx x y Look for;
Curve Sketching 0,0, 1 e.g.itydiscontinuofpoints  yx x y Look for; xyx x xy  ,0, 1 e.g.asymptotes
Curve Sketching 0,0, 1 e.g.itydiscontinuofpoints  yx x y  xfy curveOn the Look for; xyx x xy  ,0, 1 e.g.asymptotes
Curve Sketching 0,0, 1 e.g.itydiscontinuofpoints  yx x y  xfy curveOn the     0occur whenpointsstationary1  xf Look for; xyx x xy  ,0, 1 e.g.asymptotes
Curve Sketching 0,0, 1 e.g.itydiscontinuofpoints  yx x y  xfy curveOn the     0occur whenpointsstationary1  xf Look for; xyx x xy  ,0, 1 e.g.asymptotes       0,0ifpointturningmaximum2  xfxf
Curve Sketching 0,0, 1 e.g.itydiscontinuofpoints  yx x y  xfy curveOn the     0occur whenpointsstationary1  xf Look for; xyx x xy  ,0, 1 e.g.asymptotes       0,0ifpointturningmaximum2  xfxf       0,0ifpointturningminimum3  xfxf
Curve Sketching 0,0, 1 e.g.itydiscontinuofpoints  yx x y  xfy curveOn the     0occur whenpointsstationary1  xf Look for; xyx x xy  ,0, 1 e.g.asymptotes       0,0ifpointturningmaximum2  xfxf       0,0ifpointturningminimum3  xfxf       0concavity inchangeaisthereand0ifinflectionofpoint4   xf xf
Curve Sketching 0,0, 1 e.g.itydiscontinuofpoints  yx x y  xfy curveOn the     0occur whenpointsstationary1  xf Look for; xyx x xy  ,0, 1 e.g.asymptotes       0,0ifpointturningmaximum2  xfxf       0,0ifpointturningminimum3  xfxf       0concavity inchangeaisthereand0ifinflectionofpoint4   xf xf is;curveA
Curve Sketching 0,0, 1 e.g.itydiscontinuofpoints  yx x y  xfy curveOn the     0occur whenpointsstationary1  xf Look for;   0ifincreasing  xf xyx x xy  ,0, 1 e.g.asymptotes       0,0ifpointturningmaximum2  xfxf       0,0ifpointturningminimum3  xfxf       0concavity inchangeaisthereand0ifinflectionofpoint4   xf xf is;curveA
Curve Sketching 0,0, 1 e.g.itydiscontinuofpoints  yx x y  xfy curveOn the     0occur whenpointsstationary1  xf Look for;   0ifincreasing  xf xyx x xy  ,0, 1 e.g.asymptotes       0,0ifpointturningmaximum2  xfxf       0,0ifpointturningminimum3  xfxf       0concavity inchangeaisthereand0ifinflectionofpoint4   xf xf is;curveA   0ifdecreasing  xf
Curve Sketching 0,0, 1 e.g.itydiscontinuofpoints  yx x y  xfy curveOn the     0occur whenpointsstationary1  xf Look for;   0ifincreasing  xf xyx x xy  ,0, 1 e.g.asymptotes       0,0ifpointturningmaximum2  xfxf       0,0ifpointturningminimum3  xfxf       0concavity inchangeaisthereand0ifinflectionofpoint4   xf xf is;curveA   0ifdecreasing  xf   0ifupconcave  xf
Curve Sketching 0,0, 1 e.g.itydiscontinuofpoints  yx x y  xfy curveOn the     0occur whenpointsstationary1  xf Look for;   0ifincreasing  xf xyx x xy  ,0, 1 e.g.asymptotes       0,0ifpointturningmaximum2  xfxf       0,0ifpointturningminimum3  xfxf       0concavity inchangeaisthereand0ifinflectionofpoint4   xf xf is;curveA   0ifdecreasing  xf   0ifupconcave  xf   0ifdownconcave  xf
596curveSketch thee.g. 23  xxxy
596curveSketch thee.g. 23  xxxy 9123 2  xx dx dy
596curveSketch thee.g. 23  xxxy 9123 2  xx dx dy 1262 2  x dx yd
596curveSketch thee.g. 23  xxxy 9123 2  xx dx dy 1262 2  x dx yd 63 3  dx yd
596curveSketch thee.g. 23  xxxy 9123 2  xx dx dy 1262 2  x dx yd 0occur whenpointsStationary  dx dy 63 3  dx yd
596curveSketch thee.g. 23  xxxy 9123 2  xx dx dy 1262 2  x dx yd 0occur whenpointsStationary  dx dy 09123i.e. 2  xx 63 3  dx yd
596curveSketch thee.g. 23  xxxy 9123 2  xx dx dy 1262 2  x dx yd 0occur whenpointsStationary  dx dy 09123i.e. 2  xx    031 0342   xx xx 63 3  dx yd
596curveSketch thee.g. 23  xxxy 9123 2  xx dx dy 1262 2  x dx yd 0occur whenpointsStationary  dx dy 09123i.e. 2  xx    031 0342   xx xx 3or1  xx 63 3  dx yd
596curveSketch thee.g. 23  xxxy 9123 2  xx dx dy 1262 2  x dx yd 0occur whenpointsStationary  dx dy 09123i.e. 2  xx    031 0342   xx xx 3or1  xx 63 3  dx yd   06 1216,1when 2 2   dx yd x
596curveSketch thee.g. 23  xxxy 9123 2  xx dx dy 1262 2  x dx yd 0occur whenpointsStationary  dx dy 09123i.e. 2  xx    031 0342   xx xx 3or1  xx 63 3  dx yd   06 1216,1when 2 2   dx yd x   pointturningmaximumais11  ,
  06 1236,3when 2 2   dx yd x
  06 1236,3when 2 2   dx yd x   pointturningminimumais53  ,
  06 1236,3when 2 2   dx yd x   pointturningminimumais53  , 0occur wheninflectionofpointsPossible 2 2  dx yd
  06 1236,3when 2 2   dx yd x   pointturningminimumais53  , 0occur wheninflectionofpointsPossible 2 2  dx yd 0126.. xei
  06 1236,3when 2 2   dx yd x   pointturningminimumais53  , 0occur wheninflectionofpointsPossible 2 2  dx yd 0126.. xei 2x
  06 1236,3when 2 2   dx yd x   pointturningminimumais53  , 0occur wheninflectionofpointsPossible 2 2  dx yd 0126.. xei 2x 06,2when 3 3  dx yd x
  06 1236,3when 2 2   dx yd x   pointturningminimumais53  , 0occur wheninflectionofpointsPossible 2 2  dx yd 0126.. xei 2x 06,2when 3 3  dx yd x   inflectionofpointais32  ,
x y
x y  11 ,
x y  11 ,  53 , 5
x y  11 ,  53 ,  32 ,
x y  11 ,  53 ,  32 , 5
x y  11 , 596 23  xxxy  53 ,  32 , 5
x y  11 , 596 23  xxxy  53 ,  32 , 5 Exercise 10F; 1bdf, 2, 4b, 5ace etc, 6bd Exercise 10G 1, 2ace, 4b, 5c

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