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X2 T04 01 curve sketching - basic features/ calculus

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X2 T04 01 curve sketching - basic features/ calculus

1. 1. (A) Features You Should Notice About A Graph
2. 2. (A) Features You Should Notice About A Graph (1) Basic Curves
3. 3. (A) Features You Should Notice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation;
4. 4. (A) Features You Should Notice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation; a) Straight lines: y  x (both pronumerals are to the power of one)
5. 5. (A) Features You Should Notice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation; a) Straight lines: y  x (both pronumerals are to the power of one) b) Parabolas: y  x 2 (one pronumeral is to the power of one, the other the power of two)
6. 6. (A) Features You Should Notice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation; a) Straight lines: y  x (both pronumerals are to the power of one) b) Parabolas: y  x 2 (one pronumeral is to the power of one, the other the power of two) NOTE: general parabola is y  ax 2  bx  c
7. 7. (A) Features You Should Notice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation; a) Straight lines: y  x (both pronumerals are to the power of one) b) Parabolas: y  x 2 (one pronumeral is to the power of one, the other the power of two) NOTE: general parabola is y  ax 2  bx  c c) Cubics: y  x 3 (one pronumeral is to the power of one, the other the power of three)
8. 8. (A) Features You Should Notice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation; a) Straight lines: y  x (both pronumerals are to the power of one) b) Parabolas: y  x 2 (one pronumeral is to the power of one, the other the power of two) NOTE: general parabola is y  ax 2  bx  c c) Cubics: y  x 3 (one pronumeral is to the power of one, the other the power of three) NOTE: general cubic is y  ax 3  bx 2  cx  d
9. 9. (A) Features You Should Notice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation; a) Straight lines: y  x (both pronumerals are to the power of one) b) Parabolas: y  x 2 (one pronumeral is to the power of one, the other the power of two) NOTE: general parabola is y  ax 2  bx  c c) Cubics: y  x 3 (one pronumeral is to the power of one, the other the power of three) NOTE: general cubic is y  ax 3  bx 2  cx  d d) Polynomials in general
10. 10. 1 e) Hyperbolas: y  OR xy  1 x (one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together)
11. 11. 1 e) Hyperbolas: y  OR xy  1 x (one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together) f) Exponentials: y  a x (one pronumeral is in the power)
12. 12. 1 e) Hyperbolas: y  OR xy  1 x (one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together) f) Exponentials: y  a x (one pronumeral is in the power) g) Circles: x 2  y 2  r 2 (both pronumerals are to the power of two, coefficients are the same)
13. 13. 1 e) Hyperbolas: y  OR xy  1 x (one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together) f) Exponentials: y  a x (one pronumeral is in the power) g) Circles: x 2  y 2  r 2 (both pronumerals are to the power of two, coefficients are the same) h) Ellipses: ax 2  by 2  k (both pronumerals are to the power of two, coefficients are NOT the same)
14. 14. 1 e) Hyperbolas: y  OR xy  1 x (one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together) f) Exponentials: y  a x (one pronumeral is in the power) g) Circles: x 2  y 2  r 2 (both pronumerals are to the power of two, coefficients are the same) h) Ellipses: ax 2  by 2  k (both pronumerals are to the power of two, coefficients are NOT the same) (NOTE: if signs are different then hyperbola)
15. 15. 1 e) Hyperbolas: y  OR xy  1 x (one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together) f) Exponentials: y  a x (one pronumeral is in the power) g) Circles: x 2  y 2  r 2 (both pronumerals are to the power of two, coefficients are the same) h) Ellipses: ax 2  by 2  k (both pronumerals are to the power of two, coefficients are NOT the same) (NOTE: if signs are different then hyperbola) i) Logarithmics: y  log a x
16. 16. 1 e) Hyperbolas: y  OR xy  1 x (one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together) f) Exponentials: y  a x (one pronumeral is in the power) g) Circles: x 2  y 2  r 2 (both pronumerals are to the power of two, coefficients are the same) h) Ellipses: ax 2  by 2  k (both pronumerals are to the power of two, coefficients are NOT the same) (NOTE: if signs are different then hyperbola) i) Logarithmics: y  log a x j) Trigonometric: y  sin x, y  cos x, y  tan x
17. 17. 1 e) Hyperbolas: y  OR xy  1 x (one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together) f) Exponentials: y  a x (one pronumeral is in the power) g) Circles: x 2  y 2  r 2 (both pronumerals are to the power of two, coefficients are the same) h) Ellipses: ax 2  by 2  k (both pronumerals are to the power of two, coefficients are NOT the same) (NOTE: if signs are different then hyperbola) i) Logarithmics: y  log a x j) Trigonometric: y  sin x, y  cos x, y  tan x 1 1 1 k) Inverse Trigonmetric: y  sin x, y  cos x, y  tan x
18. 18. (2) Odd & Even Functions
19. 19. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch
20. 20. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree rotational symmetry)
21. 21. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f  x   f  x  (symmetric about the y axis)
22. 22. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f  x   f  x  (symmetric about the y axis) (3) Symmetry in the line y = x
23. 23. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f  x   f  x  (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = x e.g. x 3  y 3  1 (in other words, the curve is its own inverse)
24. 24. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f  x   f  x  (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = x e.g. x 3  y 3  1 (in other words, the curve is its own inverse) (4) Dominance
25. 25. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f  x   f  x  (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = x e.g. x 3  y 3  1 (in other words, the curve is its own inverse) (4) Dominance As x gets large, does a particular term dominate?
26. 26. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f  x   f  x  (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = x e.g. x 3  y 3  1 (in other words, the curve is its own inverse) (4) Dominance As x gets large, does a particular term dominate? a) Polynomials: the leading term dominates e.g. y  x 4  3 x 3  2 x  2, x 4 dominates
27. 27. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f  x   f  x  (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = x e.g. x 3  y 3  1 (in other words, the curve is its own inverse) (4) Dominance As x gets large, does a particular term dominate? a) Polynomials: the leading term dominates e.g. y  x 4  3 x 3  2 x  2, x 4 dominates b) Exponentials: e x tends to dominate as it increases so rapidly
28. 28. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f  x   f  x  (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = x e.g. x 3  y 3  1 (in other words, the curve is its own inverse) (4) Dominance As x gets large, does a particular term dominate? a) Polynomials: the leading term dominates e.g. y  x 4  3 x 3  2 x  2, x 4 dominates b) Exponentials: e x tends to dominate as it increases so rapidly c) In General: look for the term that increases the most rapidly i.e. which is the steepest
29. 29. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree rotational symmetry) b) EVEN : f  x   f  x  (symmetric about the y axis) (3) Symmetry in the line y = x If x and y can be interchanged without changing the function, the curve is relected in the line y = x e.g. x 3  y 3  1 (in other words, the curve is its own inverse) (4) Dominance As x gets large, does a particular term dominate? a) Polynomials: the leading term dominates e.g. y  x 4  3 x 3  2 x  2, x 4 dominates b) Exponentials: e x tends to dominate as it increases so rapidly c) In General: look for the term that increases the most rapidly i.e. which is the steepest NOTE: check by substituting large numbers e.g. 1000000
30. 30. (5) Asymptotes
31. 31. (5) Asymptotes a) Vertical Asymptotes: the bottom of a fraction cannot equal zero
32. 32. (5) Asymptotes a) Vertical Asymptotes: the bottom of a fraction cannot equal zero b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the fraction cannot equal zero
33. 33. (5) Asymptotes a) Vertical Asymptotes: the bottom of a fraction cannot equal zero b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the fraction cannot equal zero NOTE: if order of numerator  order of denominator, perform a polynomial division. (curves can cross horizontal/oblique asymptotes, good idea to check)
34. 34. (5) Asymptotes a) Vertical Asymptotes: the bottom of a fraction cannot equal zero b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the fraction cannot equal zero NOTE: if order of numerator  order of denominator, perform a polynomial division. (curves can cross horizontal/oblique asymptotes, good idea to check) (6) The Special Limit
35. 35. (5) Asymptotes a) Vertical Asymptotes: the bottom of a fraction cannot equal zero b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the fraction cannot equal zero NOTE: if order of numerator  order of denominator, perform a polynomial division. (curves can cross horizontal/oblique asymptotes, good idea to check) (6) The Special Limit sin x Remember the special limit seen in 2 Unit i.e. lim 1 x0 x , it could come in handy when solving harder graphs.
36. 36. (B) Using Calculus
37. 37. (B) Using Calculus Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections.
38. 38. (B) Using Calculus Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections. (1) Critical Points
39. 39. (B) Using Calculus Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections. (1) Critical Points dy When dx is undefined the curve has a vertical tangent, these points are called critical points.
40. 40. (B) Using Calculus Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections. (1) Critical Points dy When dx is undefined the curve has a vertical tangent, these points are called critical points. (2) Stationary Points
41. 41. (B) Using Calculus Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections. (1) Critical Points dy When dx is undefined the curve has a vertical tangent, these points are called critical points. (2) Stationary Points dy When dx  0 the curve is said to be stationary, these points may be minimum turning points, maximum turning points or points of inflection.
42. 42. (3) Minimum/Maximum Turning Points
43. 43. (3) Minimum/Maximum Turning Points dy d2y a) When  0 and 2  0, the point is called a minimum turning point dx dx
44. 44. (3) Minimum/Maximum Turning Points dy d2y a) When  0 and 2  0, the point is called a minimum turning point dx dx dy d2y b) When  0 and 2  0, the point is called a maximum turning point dx dx
45. 45. (3) Minimum/Maximum Turning Points dy d2y a) When  0 and 2  0, the point is called a minimum turning point dx dx dy d2y b) When  0 and 2  0, the point is called a maximum turning point dx dx dy NOTE: testing either side of for change can be quicker for harder dx functions
46. 46. (3) Minimum/Maximum Turning Points dy d2y a) When  0 and 2  0, the point is called a minimum turning point dx dx dy d2y b) When  0 and 2  0, the point is called a maximum turning point dx dx dy NOTE: testing either side of for change can be quicker for harder dx functions (4) Inflection Points
47. 47. (3) Minimum/Maximum Turning Points dy d2y a) When  0 and 2  0, the point is called a minimum turning point dx dx dy d2y b) When  0 and 2  0, the point is called a maximum turning point dx dx dy NOTE: testing either side of for change can be quicker for harder dx functions (4) Inflection Points d2y d3y a) When 2  0 and 3  0, the point is called an inflection point dx dx
48. 48. (3) Minimum/Maximum Turning Points dy d2y a) When  0 and 2  0, the point is called a minimum turning point dx dx dy d2y b) When  0 and 2  0, the point is called a maximum turning point dx dx dy NOTE: testing either side of for change can be quicker for harder dx functions (4) Inflection Points d2y d3y a) When 2  0 and 3  0, the point is called an inflection point dx dx d2y NOTE: testing either side of 2 for change can be quicker for harder dx functions
49. 49. (3) Minimum/Maximum Turning Points dy d2y a) When  0 and 2  0, the point is called a minimum turning point dx dx dy d2y b) When  0 and 2  0, the point is called a maximum turning point dx dx dy NOTE: testing either side of for change can be quicker for harder dx functions (4) Inflection Points d2y d3y a) When 2  0 and 3  0, the point is called an inflection point dx dx d2y NOTE: testing either side of 2 for change can be quicker for harder dx functions dy d2y d3y b) When  0, 2  0 and 3  0, the point is called a horizontal dx dx dx point of inflection
50. 50. (5) Increasing/Decreasing Curves
51. 51. (5) Increasing/Decreasing Curves dy a) When  0, the curve has a positive sloped tangent and is dx thus increasing
52. 52. (5) Increasing/Decreasing Curves dy a) When  0, the curve has a positive sloped tangent and is dx thus increasing dy b) When  0, the curve has a negative sloped tangent and is dx thus decreasing
53. 53. (5) Increasing/Decreasing Curves dy a) When  0, the curve has a positive sloped tangent and is dx thus increasing dy b) When  0, the curve has a negative sloped tangent and is dx thus decreasing (6) Implicit Differentiation
54. 54. (5) Increasing/Decreasing Curves dy a) When  0, the curve has a positive sloped tangent and is dx thus increasing dy b) When  0, the curve has a negative sloped tangent and is dx thus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions
55. 55. (5) Increasing/Decreasing Curves dy a) When  0, the curve has a positive sloped tangent and is dx thus increasing dy b) When  0, the curve has a negative sloped tangent and is dx thus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3  y 3  1
56. 56. (5) Increasing/Decreasing Curves dy a) When  0, the curve has a positive sloped tangent and is dx thus increasing dy b) When  0, the curve has a negative sloped tangent and is dx thus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3  y 3  1 Note:• the curve has symmetry in y = x
57. 57. y y=x x
58. 58. (5) Increasing/Decreasing Curves dy a) When  0, the curve has a positive sloped tangent and is dx thus increasing dy b) When  0, the curve has a negative sloped tangent and is dx thus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3  y 3  1 Note:• the curve has symmetry in y = x • it passes through (1,0) and (0,1)
59. 59. y y=x 1 1 x
60. 60. (5) Increasing/Decreasing Curves dy a) When  0, the curve has a positive sloped tangent and is dx thus increasing dy b) When  0, the curve has a negative sloped tangent and is dx thus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3  y 3  1 Note:• the curve has symmetry in y = x • it passes through (1,0) and (0,1) • it is asymptotic to the line y = -x
61. 61. (5) Increasing/Decreasing Curves dy a) When  0, the curve has a positive sloped tangent and is dx thus increasing dy b) When  0, the curve has a negative sloped tangent and is dx thus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3  y 3  1 Note:• the curve has symmetry in y = x • it passes through (1,0) and (0,1) • it is asymptotic to the line y = -x  y 3  1  x3 i.e. y 3   x 3 y  x
62. 62. y y=x 1 1 x y=-x
63. 63. (5) Increasing/Decreasing Curves dy a) When  0, the curve has a positive sloped tangent and is dx thus increasing dy b) When  0, the curve has a negative sloped tangent and is dx thus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3  y 3  1 On differentiating implicitly; Note:• the curve has symmetry in y = x • it passes through (1,0) and (0,1) • it is asymptotic to the line y = -x  y 3  1  x3 i.e. y 3   x 3 y  x
64. 64. (5) Increasing/Decreasing Curves dy a) When  0, the curve has a positive sloped tangent and is dx thus increasing dy b) When  0, the curve has a negative sloped tangent and is dx thus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3  y 3  1 On differentiating implicitly; 2 dy Note:• the curve has symmetry in y = x 3x  3 y 2 0 • it passes through (1,0) and (0,1) dx • it is asymptotic to the line y = -x dy  x 2  2  y  1 x 3 3 dx y i.e. y 3   x 3 y  x
65. 65. (5) Increasing/Decreasing Curves dy a) When  0, the curve has a positive sloped tangent and is dx thus increasing dy b) When  0, the curve has a negative sloped tangent and is dx thus decreasing (6) Implicit Differentiation This technique allows you to differentiate complicated functions e.g. Sketch x 3  y 3  1 On differentiating implicitly; 2 dy Note:• the curve has symmetry in y = x 3x  3 y 2 0 • it passes through (1,0) and (0,1) dx • it is asymptotic to the line y = -x dy  x 2  2  y  1 x 3 3 dx y dy i.e. y 3   x 3 This means that  0 for all x dx y  x Except at (1,0) : critical point & (0,1): horizontal point of inflection
66. 66. y y=x 1 x3  y 3  1 1 x y=-x