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5. Calculus
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Subjects
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Maths AA
/
Topics
/
5. Calculus
/
Sub-topics
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If
f
(
x
)
f(x)
f
(
x
)
is as close as we like to some real number
A
A
A
for all
x
x
x
sufficiently close to (but not equal to)
a
a
a
, then we say that
f
(
x
)
f(x)
f
(
x
)
has a
limit
of
A
A
A
as
x
x
x
approaches
a
a
a
, and we write
lim
x
→
a
f
(
x
)
=
A
\lim\limits_{x \to a} f(x) = A
x
→
a
lim
f
(
x
)
=
A
.
In this case,
f
(
x
)
f(x)
f
(
x
)
is said to
converge to
A
A
A
as
x
x
x
approaches
a
a
a
. Otherwise, we say a limit is
divergent
.
Figure 5.1.1
Convergence of limits for continuous (left) and discontinuous function (right)
We define continuity using limits. Consider a real function
f
f
f
defined on an open interval containing the value
a
a
a
. We say
f
f
f
is continuous at
x
=
a
x = a
x
=
a
if
lim
x
→
a
f
(
x
)
=
f
(
a
)
\lim\limits_{x \to a} f(x) = f(a)
x
→
a
lim
f
(
x
)
=
f
(
a
)
.
If
f
f
f
is continuous at
x
=
a
x = a
x
=
a
for all
a
ϵ
R
a \epsilon ℝ
a
ϵ
R
, we say
f
f
f
is continuous on
R
ℝ
R
.
5.1 Introduction to Differential Calculus
Derivatives can describe a function in multiple ways.
Figure 5.2.1
Stationary points of a curve
Suppose
S
S
S
is an interval in the domain of
f
(
x
)
f(x)
f
(
x
)
, so
f
(
x
)
f(x)
f
(
x
)
is defined for all
x
x
x
in
S
S
S
. Then, we have:
1.
f
(
x
)
f(x)
f
(
x
)
is increasing on
S
S
S
if
f
′
(
x
)
>
0
f'(x) > 0
f
′
(
x
)
>
0
for all
x
x
x
in
S
S
S
2.
f
(
x
)
f(x)
f
(
x
)
is decreasing on
S
S
S
if
f
′
(
x
)
<
0
f'(x) < 0
f
′
(
x
)
<
0
for all
x
x
x
in
S
S
S
3.
f
(
x
)
f(x)
f
(
x
)
has a stationary point when
f
′
(
x
)
=
0
f'(x) = 0
f
′
(
x
)
=
0
:
5.2 Properties of Curves
There are different applications you need to be aware of.
5.3 Application of Differential Calculus
Integration is an inverse operation of differentiation.
Thus, we call
F
(
x
)
=
f
(
x
)
d
x
F(x) = f(x) dx
F
(
x
)
=
f
(
x
)
d
x
as the
antiderivative
of
f
(
x
)
f(x)
f
(
x
)
, and
d
d
x
F
(
x
)
=
f
(
x
)
\frac d{dx}F(x) = f(x)
d
x
d
F
(
x
)
=
f
(
x
)
.
Indefinite integral
denotes an integration without the bounds, and thus has a constant of integration
C
C
C
.
Figure 5.5.1
Definite integral represented as the area under the curve for a specific range
Definite integral
denotes an integration with the bounds, and represents the area below
f
f
f
from
a
a
a
to
b
b
b
.
5.4 Integration
There are 3 different applications you need to be aware of.
5.5 Application of Integration
Another application is
differential equation
, an equation involving a derivative of a function. (HL)
There are two ways to solve a differential equation,
numerical method
, and
analytic method
. A general solution of a differential equation is a function with a constant
c
c
c
that satisfies the differential equation.
Below regards the analytic method.
5.6 Differential Equations
