The document discusses key features to notice when analyzing graphs of functions. It identifies basic curve shapes defined by common equations, such as straight lines, parabolas, cubics, and circles. It also covers concepts like odd and even functions, symmetry, and dominance - how certain terms in an equation become more prominent at higher values of x. Basic curve shapes and these analytical concepts can help recognize and sketch the shape of graphs defined by functions.
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
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Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
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2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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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
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
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.
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
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
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)
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
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