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5. Calculus
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5. Calculus
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5. Calculus
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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
.
There are three different methods to solve limits.
We can identify asymptotes using limits.
Always investigate
→
∞
\rightarrow ∞
→
∞
,
→
−
∞
\rightarrow -∞
→
−
∞
and possible points of singularity
x
=
a
x = a
x
=
a
.
Point of singularity, in its simplest term denotes the point of discontinuity.
There is a
squeeze theorem
, used in Riemann sum later in integration.
If
g
(
x
)
≤
f
(
x
)
≤
h
(
x
)
g(x) ≤ f(x) ≤ h(x)
g
(
x
)
≤
f
(
x
)
≤
h
(
x
)
and
lim
x
→
a
g
(
x
)
=
lim
x
→
a
h
(
x
)
=
L
\lim\limits_{x\rightarrow a}g(x) = \lim\limits_{x\rightarrow a}h(x) = L
x
→
a
lim
g
(
x
)
=
x
→
a
lim
h
(
x
)
=
L
, then
lim
x
→
a
f
(
x
)
=
L
\lim\limits_{x\rightarrow a} f(x) = L
x
→
a
lim
f
(
x
)
=
L
.
You can also use L’Hopital’s rule if you have the form:
0
0
\frac00
0
0
,
∞
∞
\frac∞∞
∞
∞
, or
0
∞
\frac 0∞
∞
0
, and
lim
x
→
a
f
′
(
x
)
g
′
(
x
)
\lim\limits_{x\rightarrow a}\frac{f'(x)}{g'(x)}
x
→
a
lim
g
′
(
x
)
f
′
(
x
)
exists.
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
:
4.
f
(
x
)
f(x)
f
(
x
)
is concave up on
S
S
S
if
f
′
′
(
x
)
>
0
f''(x) > 0
f
′′
(
x
)
>
0
for all
x
x
x
in
S
S
S
5.
f
(
x
)
f(x)
f
(
x
)
is concave down on
S
S
S
if
f
′
′
(
x
)
<
0
f''(x) <0
f
′′
(
x
)
<
0
for all
x
x
x
in
S
S
S
6.
f
(
x
)
f(x)
f
(
x
)
has an (non-stationary) inflection point where
f
′
′
(
x
)
=
0
f''(x) =0
f
′′
(
x
)
=
0
Figure 5.2.2
Example of a cubic function with a sign diagram on the right
Represent the sign changes in a sign diagram to identify each stationary point.
5.3 Properties of curves
done
5.2 Properties of Curves
There are different applications you need to be aware of.
Kinematics
is the study of motion; modeling and analyzing the motion of objects in a straight line.
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
.
Given
F
F
F
as the antiderivative of
f
f
f
,
∫
a
b
f
(
x
)
d
x
=
F
(
a
)
−
F
(
b
)
\int_a^bf(x) dx = F(a) - F(b)
∫
a
b
f
(
x
)
d
x
=
F
(
a
)
−
F
(
b
)
. (
Fundamental theorem of calculus.
)
Integrals have different properties:
We have different techniques to solve the integration.
We can estimate the area under the curve (i.e.
∫
a
b
f
(
x
)
d
x
\int _a^bf(x) dx
∫
a
b
f
(
x
)
d
x
) numerically.
Both use different numbers of
partitions
, a subdivision of a given interval into non-empty sets.
For instance, if we divide
a
≤
x
≤
b
a ≤ x ≤ b
a
≤
x
≤
b
into
n
n
n
subintervals, we obtain equal width of
∆
x
=
b
−
a
n
∆x = \frac{b-a}n
∆
x
=
n
b
−
a
per partition. Then, we have
x
i
=
a
+
i
∆
x
x_i=a+i∆x
x
i
=
a
+
i
∆
x
, and the
i
i
i
th subinterval would be:
x
i
−
1
≤
x
≤
x
i
x_{i-1} ≤ x ≤ x_i
x
i
−
1
≤
x
≤
x
i
.
5.4 Integration
There are 3 different applications you need to be aware of.
5.6 Applications of integration
done
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
