X2 T07 01 features calculus (2011)
- 1. (A) Features You Should Notice About A Graph
- 2. (A) Features You Should Notice About A Graph (1) Basic Curves
- 3. (A) Features You Should Notice About A Graph (1) Basic Curves The following basic curve shapes should be recognisable from the equation;
- 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. (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. (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. (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. (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. (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. 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. 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. 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. 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. 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. 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. 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. 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. (2) Odd & Even Functions
- 19. (2) Odd & Even Functions These curves have symmetry and are thus easier to sketch
- 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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (5) Asymptotes
- 31. (5) Asymptotes a) Vertical Asymptotes: the bottom of a fraction cannot equal zero
- 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. (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. (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. (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 x0 x , it could come in handy when solving harder graphs.
- 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. (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. (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. (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. (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. (3) Minimum/Maximum Turning Points
- 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. (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. (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. (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. (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. (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. (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
- 51. (5) Increasing/Decreasing Curves dy a) When 0, the curve has a positive sloped tangent and is dx thus increasing
- 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. (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. (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. (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. (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. y y=x x
- 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. y y=x 1 1 x
- 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. (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. y y=x 1 1 x y=-x
- 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. (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. (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. y y=x 1 x3 y 3 1 1 x y=-x