# X2 T07 01 features calculus (2011)

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<ul><li> 1. (A) Features You Should Notice About A Graph</li></ul><p> 2. (A) Features You Should Notice About A Graph(1) Basic Curves 3. (A) Features You Should Notice About A Graph(1) Basic CurvesThe following basic curve shapes should be recognisable from theequation; 4. (A) Features You Should Notice About A Graph(1) Basic CurvesThe following basic curve shapes should be recognisable from theequation;a) Straight lines: y x (both pronumerals are to the power of one) 5. (A) Features You Should Notice About A Graph(1) Basic CurvesThe following basic curve shapes should be recognisable from theequation;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 otherthe power of two) 6. (A) Features You Should Notice About A Graph(1) Basic CurvesThe following basic curve shapes should be recognisable from theequation;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 otherthe power of two) NOTE: general parabola is y ax 2 bx c 7. (A) Features You Should Notice About A Graph(1) Basic CurvesThe following basic curve shapes should be recognisable from theequation;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 otherthe power of two) NOTE: general parabola is y ax 2 bx cc) Cubics: y x 3 (one pronumeral is to the power of one, the other thepower of three) 8. (A) Features You Should Notice About A Graph(1) Basic CurvesThe following basic curve shapes should be recognisable from theequation;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 otherthe power of two) NOTE: general parabola is y ax 2 bx cc) Cubics: y x 3 (one pronumeral is to the power of one, the other thepower of three) NOTE: general cubic is y ax 3 bx 2 cx d 9. (A) Features You Should Notice About A Graph(1) Basic CurvesThe following basic curve shapes should be recognisable from theequation;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 otherthe power of two) NOTE: general parabola is y ax 2 bx cc) Cubics: y x 3 (one pronumeral is to the power of one, the other thepower of three) NOTE: general cubic is y ax 3 bx 2 cx dd) Polynomials in general 10. 1e) 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. 1e) 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. 1e) 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. 1e) 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. 1e) 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. 1e) 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. 1e) 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 xj) Trigonometric: y sin x, y cos x, y tan x 17. 1e) 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 xj) Trigonometric: y sin x, y cos x, y tan x1 1 1k) Inverse Trigonmetric: y sin x, y cos x, y tan x 18. (2) Odd &amp; Even Functions 19. (2) Odd &amp; Even FunctionsThese curves have symmetry and are thus easier to sketch 20. (2) Odd &amp; Even FunctionsThese curves have symmetry and are thus easier to sketcha) ODD : f x f x (symmetric about the origin, i.e. 180 degreerotational symmetry) 21. (2) Odd &amp; Even FunctionsThese curves have symmetry and are thus easier to sketcha) ODD : f x f x (symmetric about the origin, i.e. 180 degreerotational symmetry)b) EVEN : f x f x (symmetric about the y axis) 22. (2) Odd &amp; Even FunctionsThese curves have symmetry and are thus easier to sketcha) ODD : f x f x (symmetric about the origin, i.e. 180 degreerotational symmetry)b) EVEN : f x f x (symmetric about the y axis)(3) Symmetry in the line y = x 23. (2) Odd &amp; Even FunctionsThese curves have symmetry and are thus easier to sketcha) ODD : f x f x (symmetric about the origin, i.e. 180 degreerotational symmetry)b) EVEN : f x f x (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve isrelected in the line y = x e.g. x 3 y 3 1 (in other words, the curve is its own inverse) 24. (2) Odd &amp; Even FunctionsThese curves have symmetry and are thus easier to sketcha) ODD : f x f x (symmetric about the origin, i.e. 180 degreerotational symmetry)b) EVEN : f x f x (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve isrelected 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 &amp; Even FunctionsThese curves have symmetry and are thus easier to sketcha) ODD : f x f x (symmetric about the origin, i.e. 180 degreerotational symmetry)b) EVEN : f x f x (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve isrelected 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 &amp; Even FunctionsThese curves have symmetry and are thus easier to sketcha) ODD : f x f x (symmetric about the origin, i.e. 180 degreerotational symmetry)b) EVEN : f x f x (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve isrelected 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 dominatese.g. y x 4 3 x 3 2 x 2, x 4 dominates 27. (2) Odd &amp; Even FunctionsThese curves have symmetry and are thus easier to sketcha) ODD : f x f x (symmetric about the origin, i.e. 180 degreerotational symmetry)b) EVEN : f x f x (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve isrelected 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 dominatese.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 &amp; Even FunctionsThese curves have symmetry and are thus easier to sketcha) ODD : f x f x (symmetric about the origin, i.e. 180 degreerotational symmetry)b) EVEN : f x f x (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve isrelected 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 dominatese.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 &amp; Even FunctionsThese curves have symmetry and are thus easier to sketcha) ODD : f x f x (symmetric about the origin, i.e. 180 degreerotational symmetry)b) EVEN : f x f x (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve isrelected 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 dominatese.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) Asymptotesa) Vertical Asymptotes: the bottom of a fraction cannot equal zero 32. (5) Asymptotesa) Vertical Asymptotes: the bottom of a fraction cannot equal zerob) Horizontal/Oblique Asymptotes: Top of a fraction is constant, thefraction cannot equal zero 33. (5) Asymptotesa) Vertical Asymptotes: the bottom of a fraction cannot equal zerob) Horizontal/Oblique Asymptotes: Top of a fraction is constant, thefraction cannot equal zeroNOTE: if order of numerator order of denominator, perform apolynomial division. (curves can cross horizontal/oblique asymptotes,good idea to check) 34. (5) Asymptotesa) Vertical Asymptotes: the bottom of a fraction cannot equal zerob) Horizontal/Oblique Asymptotes: Top of a fraction is constant, thefraction cannot equal zeroNOTE: if order of numerator order of denominator, perform apolynomial division. (curves can cross horizontal/oblique asymptotes,good idea to check)(6) The Special Limit 35. (5) Asymptotesa) Vertical Asymptotes: the bottom of a fraction cannot equal zerob) Horizontal/Oblique Asymptotes: Top of a fraction is constant, thefraction cannot equal zeroNOTE: if order of numerator order of denominator, perform apolynomial division. (curves can cross horizontal/oblique asymptotes,good idea to check)(6) The Special Limit sin xRemember the special limit seen in 2 Unit i.e. lim 1 x0 x, it could come in handy when solving harder graphs. 36. (B) Using Calculus 37. (B) Using CalculusCalculus is still a tremendous tool that should not be disregarded whencurve sketching. However, often it is used as a final tool to determinecritical points, stationary points, inflections. 38. (B) Using CalculusCalculus is still a tremendous tool that should not be disregarded whencurve sketching. However, often it is used as a final tool to determinecritical points, stationary points, inflections.(1) Critical Points 39. (B) Using CalculusCalculus is still a tremendous tool that should not be disregarded whencurve sketching. However, often it is used as a final tool to determinecritical points, stationary points, inflections.(1) Critical PointsdyWhen dx is undefined the curve has a vertical tangent, these pointsare called critical points. 40. (B) Using CalculusCalculus is still a tremendous tool that should not be disregarded whencurve sketching. However, often it is used as a final tool to determinecritical points, stationary points, inflections.(1) Critical PointsdyWhen dx is undefined the curve has a vertical tangent, these pointsare called critical points.(2) Stationary Points 41. (B) Using CalculusCalculus is still a tremendous tool that should not be disregarded whencurve sketching. However, often it is used as a final tool to determinecritical points, stationary points, inflections.(1) Critical PointsdyWhen dx is undefined the curve has a vertical tangent, these pointsare 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 Pointsdyd2ya) When 0 and 2 0, the point is called a minimum turning pointdxdx 44. (3) Minimum/Maximum Turning Pointsdyd2ya) When 0 and 2 0, the point is called a minimum turning pointdxdxdyd2yb) When 0 and 2 0, the point is called a maximum turning pointdxdx 45. (3) Minimum/Maximum Turning Pointsdyd2ya) When 0 and 2 0, the point is called a minimum turning pointdxdxdyd2yb) When 0 and 2 0, the point is called a maximum turning pointdxdx dyNOTE: testing either side offor change can be quicker for harder dxfunctions 46. (3) Minimum/Maximum Turning Pointsdyd2ya) When 0 and 2 0, the point is called a minimum turning pointdxdxdy d2yb) When 0 and 2 0, the point is called a maximum turning pointdx dx dyNOTE: testing either side offor change can be quicker for harder dxfunctions (4) Inflection Points 47. (3) Minimum/Maximum Turning Pointsdyd2ya) When 0 and 2 0, the point is called a minimum turning pointdxdxdy d2yb) When 0 and 2 0, the point is called a maximum turning pointdx dx dyNOTE: testing either side offor change can be quicker for harder dxfunctions (4) Inflection Points d2y d3ya) When 2 0 and 3 0, the point is called an inflection point dxdx 48. (3) Minimum/Maximum Turning Pointsdyd2ya) When 0 and 2 0, the point is called a minimum turning pointdxdxdy d2yb) When 0 and 2 0, the point is called a maximum turning pointdx dx dyNOTE: testing either side offor change can be quicker for harder dxfunctions (4) Inflection Pointsd2yd3ya) When 2 0 and 3 0, the point is called an inflection pointdxdxd2y NOTE: testing either side of2 for change can be quicker for harderdx f...</p>