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Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
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This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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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.
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27. As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)
x
28. As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)
x
29. As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)
c x
A(c) is the area from 0 to c
30. As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)
c x x
A(c) is the area from 0 to c
A(x) is the area from 0 to x
31. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
32. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
33. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
A x Ac x c f x
34. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
A x Ac x c f x
A x Ac
f x
xc
35. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
A x Ac x c f x
A x Ac
f x
xc
Ac h Ac
h = width of rectangle
h
36. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
A x Ac x c f x
A x Ac
f x
xc
Ac h Ac
h = width of rectangle
h
As the width of the rectangle decreases, the estimate becomes more
accurate.
38. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
39. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
40. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
41. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
the equation of the curve is the derivative of the Area function.
42. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
the equation of the curve is the derivative of the Area function.
The area under the curve y f x between x a and x b is;
43. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
the equation of the curve is the derivative of the Area function.
The area under the curve y f x between x a and x b is;
b
A f x dx
a
44. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
the equation of the curve is the derivative of the Area function.
The area under the curve y f x between x a and x b is;
b
A f x dx
a
F b F a
45. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
the equation of the curve is the derivative of the Area function.
The area under the curve y f x between x a and x b is;
b
A f x dx
a
F b F a
where F x is the primitive function of f x
46. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
x= 2
47. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
48. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
49. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
2 04
1 4
4
50. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
2 04
1 4
4
4 units 2
51. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
2 04
1 4
4
4 units 2
3
ii x 2 1dx
2
52. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
2 04
1 4
4
4 units 2
3 3
ii x 1dx
2 1 x 3 x
2 3 2
53. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
2 04
1 4
4
4 units 2
3 3
ii x 1dx
2 1 x 3 x
2 3 2
1 33 3 1 2 3 2
3 3
54. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
2
x= 2
A x 3 dx
0
2
x4
1
4 0
2 04
1 4
4
4 units 2
3 3
ii x 1dx
2 1 x 3 x
2 3 2
1 33 3 1 2 3 2
3 3
22
3