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11 x1 t10 05 curve sketching (2012)

N
Nigel Simmons

The document provides guidance on sketching curves. It explains that to sketch a curve defined by y = f(x), one should look for discontinuities, asymptotes, stationary points where the derivative f'(x) = 0, maximum/minimum turning points where f''(x) changes sign, and points of inflection where f''(x) = 0 but f'''(x) ≠ 0. It also defines concepts like a curve being increasing, decreasing, concave up, or concave down based on the signs of f'(x) and f''(x). An example of sketching the curve y = x3 - 6x2 + 9x - 5 is provided to demonstrate the process.

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Curve Sketching
Curve Sketching Look for;
Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x
Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x
Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x 
Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0
Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0
Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0
Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0 4 point of inflection if f  x   0 and there is a change in concavity  f  x   0
Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0 4 point of inflection if f  x   0 and there is a change in concavity  f  x   0 A curve is;
Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0 4 point of inflection if f  x   0 and there is a change in concavity  f  x   0 A curve is;  increasing if f  x   0
Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0 4 point of inflection if f  x   0 and there is a change in concavity  f  x   0 A curve is;  increasing if f  x   0  decreasing if f  x   0
Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0 4 point of inflection if f  x   0 and there is a change in concavity  f  x   0 A curve is;  increasing if f  x   0  concave up if f  x   0  decreasing if f  x   0
Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0 4 point of inflection if f  x   0 and there is a change in concavity  f  x   0 A curve is;  increasing if f  x   0  concave up if f  x   0  decreasing if f  x   0  concave down if f  x   0
e.g. Sketch the curve y  x 3  6 x 2  9 x  5
e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx
e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx
e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx
e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx dy Stationary points occur when  0 dx
e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx dy Stationary points occur when  0 dx i.e. 3x 2  12 x  9  0
e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx dy Stationary points occur when  0 dx i.e. 3x 2  12 x  9  0 x2  4x  3  0  x  1 x  3  0
e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx dy Stationary points occur when  0 dx i.e. 3x 2  12 x  9  0 x2  4x  3  0  x  1 x  3  0 x  1 or x  3
e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx dy Stationary points occur when  0 dx i.e. 3x 2  12 x  9  0 x2  4x  3  0  x  1 x  3  0 2 x  1 or x  3 d y when x  1, 2  61  12 dx  6  0
e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx dy Stationary points occur when  0 dx i.e. 3x 2  12 x  9  0 x2  4x  3  0  x  1 x  3  0 2 x  1 or x  3 d y when x  1, 2  61  12 dx  6  0  1,  1 is a maximum turning point
d2y when x  3, 2  63  12 dx 60
d2y when x  3, 2  63  12 dx 60  3,  5 is a minimum turning point
d2y when x  3, 2  63  12 dx 60  3,  5 is a minimum turning point d2y Possible points of inflection occur when 2  0 dx
d2y when x  3, 2  63  12 dx 60  3,  5 is a minimum turning point d2y Possible points of inflection occur when 2  0 dx i.e. 6 x  12  0
d2y when x  3, 2  63  12 dx 60  3,  5 is a minimum turning point d2y Possible points of inflection occur when 2  0 dx i.e. 6 x  12  0 x2
d2y when x  3, 2  63  12 dx 60  3,  5 is a minimum turning point d2y Possible points of inflection occur when 2  0 dx i.e. 6 x  12  0 x2 d3y when x  2, 3  6  0 dx
d2y when x  3, 2  63  12 dx 60  3,  5 is a minimum turning point d2y Possible points of inflection occur when 2  0 dx i.e. 6 x  12  0 x2 d3y when x  2, 3  6  0 dx  2,  3 is a point of inflection
y x
y x 1,  1
y x 1,  1 5 3,  5
y x 1,  1 2,  3 3,  5
y x 1,  1 2,  3 5 3,  5
y x 1,  1 2,  3 y  x3  6 x 2  9 x  5 5 3,  5
y x 1,  1 Exercise 10F; 1bdf, 2, 4b, 2,  3 y  x3  6 x 2  9 x  5 5ace etc, 6bd Exercise 10G 1, 2ace, 4b, 5c 5 3,  5

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11 x1 t10 05 curve sketching (2012)

  • 3. Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x
  • 4. Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x
  • 5. Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x 
  • 6. Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0
  • 7. Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0
  • 8. Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0
  • 9. Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0 4 point of inflection if f  x   0 and there is a change in concavity  f  x   0
  • 10. Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0 4 point of inflection if f  x   0 and there is a change in concavity  f  x   0 A curve is;
  • 11. Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0 4 point of inflection if f  x   0 and there is a change in concavity  f  x   0 A curve is;  increasing if f  x   0
  • 12. Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0 4 point of inflection if f  x   0 and there is a change in concavity  f  x   0 A curve is;  increasing if f  x   0  decreasing if f  x   0
  • 13. Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0 4 point of inflection if f  x   0 and there is a change in concavity  f  x   0 A curve is;  increasing if f  x   0  concave up if f  x   0  decreasing if f  x   0
  • 14. Curve Sketching Look for; 1  points of discontinu ity e.g. y  , x  0, y  0 x 1  asymptotes e.g. y  x  , x  0, y  x x On the curve y  f  x  1 stationary points occur when f  x   0 2 maximum turning point if f  x   0, f  x   0 3 minimum turning point if f  x   0, f  x   0 4 point of inflection if f  x   0 and there is a change in concavity  f  x   0 A curve is;  increasing if f  x   0  concave up if f  x   0  decreasing if f  x   0  concave down if f  x   0
  • 15. e.g. Sketch the curve y  x 3  6 x 2  9 x  5
  • 16. e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx
  • 17. e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx
  • 18. e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx
  • 19. e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx dy Stationary points occur when  0 dx
  • 20. e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx dy Stationary points occur when  0 dx i.e. 3x 2  12 x  9  0
  • 21. e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx dy Stationary points occur when  0 dx i.e. 3x 2  12 x  9  0 x2  4x  3  0  x  1 x  3  0
  • 22. e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx dy Stationary points occur when  0 dx i.e. 3x 2  12 x  9  0 x2  4x  3  0  x  1 x  3  0 x  1 or x  3
  • 23. e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx dy Stationary points occur when  0 dx i.e. 3x 2  12 x  9  0 x2  4x  3  0  x  1 x  3  0 2 x  1 or x  3 d y when x  1, 2  61  12 dx  6  0
  • 24. e.g. Sketch the curve y  x 3  6 x 2  9 x  5 dy  3x 2  12 x  9 dx d2y 2  6 x  12 dx d3y 3 6 dx dy Stationary points occur when  0 dx i.e. 3x 2  12 x  9  0 x2  4x  3  0  x  1 x  3  0 2 x  1 or x  3 d y when x  1, 2  61  12 dx  6  0  1,  1 is a maximum turning point
  • 25. d2y when x  3, 2  63  12 dx 60
  • 26. d2y when x  3, 2  63  12 dx 60  3,  5 is a minimum turning point
  • 27. d2y when x  3, 2  63  12 dx 60  3,  5 is a minimum turning point d2y Possible points of inflection occur when 2  0 dx
  • 28. d2y when x  3, 2  63  12 dx 60  3,  5 is a minimum turning point d2y Possible points of inflection occur when 2  0 dx i.e. 6 x  12  0
  • 29. d2y when x  3, 2  63  12 dx 60  3,  5 is a minimum turning point d2y Possible points of inflection occur when 2  0 dx i.e. 6 x  12  0 x2
  • 30. d2y when x  3, 2  63  12 dx 60  3,  5 is a minimum turning point d2y Possible points of inflection occur when 2  0 dx i.e. 6 x  12  0 x2 d3y when x  2, 3  6  0 dx
  • 31. d2y when x  3, 2  63  12 dx 60  3,  5 is a minimum turning point d2y Possible points of inflection occur when 2  0 dx i.e. 6 x  12  0 x2 d3y when x  2, 3  6  0 dx  2,  3 is a point of inflection
  • 32. y x
  • 33. y x 1,  1
  • 34. y x 1,  1 5 3,  5
  • 35. y x 1,  1 2,  3 3,  5
  • 36. y x 1,  1 2,  3 5 3,  5
  • 37. y x 1,  1 2,  3 y  x3  6 x 2  9 x  5 5 3,  5
  • 38. y x 1,  1 Exercise 10F; 1bdf, 2, 4b, 2,  3 y  x3  6 x 2  9 x  5 5ace etc, 6bd Exercise 10G 1, 2ace, 4b, 5c 5 3,  5