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| Differential
calculus |
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| |
Properties
of continuous functions |
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Continuous function definition |
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Intermediate value theorem - Bolzano's theorem |
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Existence of roots |
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Extreme value theorem |
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Monotone function |
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Differential
Calculus
- Derivatives and differentials |
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The derivative of a
function |
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Definition of the derivative of a function |
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Tangent to
a curve |
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The
equation of the line tangent to the given curve at the given point |
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Determining the derivative of a function as the limit of the
difference quotient |
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The
equation of the line tangent to the given curve at the given point,
example |
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Determining
the lines tangent to the graph of a function from a point outside the
function |
| Determining
the lines tangent to the graph of a function from a point outside the
function, example |
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| Properties
of continuous functions |
| Continuous
function definition |
| A
real function
y = f(x)
is continuous at a point a
if it is defined at x =
a
and |
 |
| that
is, if for every e
> 0 there is a d(e)
> 0 such that |
f(x)
-
f(a)
| < e
whenever | x
-
a | < d(e). |
| Therefore,
if a function changes gradually as independent variable changes,
so that at every value a,
of the |
| independent
variable, the difference between f(x)
and f(a)
approaches zero as x
approaches a. |
| Thus,
a function is continuous at a point if both one-sided limits (the
left-handed and right-handed limits) are |
| equal, |
 |
|
|
| that
is, if it is continuous both on the left and on the right at that point. |
| A
point at which the function value is not equal to its limit, as x
approaches that point, is called point
of |
| discontinuity.
A
function having points of discontinuity is discontinuous. |
| A
function is said to be continuous if it is continuous at all
points |
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| 1.
Intermediate value theorem - Bolzano's theorem |
| If
a function is continuous on the closed interval [a,
b] then it takes every value
between f(a)
and f(b)
for at |
| least
one argument between a
and b. |
| That
is, for every y
between f(a)
and f(b)
there exists at least
one argument x
between a
and b whose |
| function's
value f(x)
= y, as
shows the figure below. |
|
|
|
| 2.
Existence of roots |
| If
a function is continuous on the closed interval [a,
b] then, if
f(a)
and f(b)
have opposite signs, or |
| f(a)
·
f(b)
< 0, then there exists x
Î
[a, b] such that f
(x) = 0. |
|
That
is, the function has at least one real root. For example,
inside the closed interval [a, b],
the function
|
| shown
in the figure below, has three roots, x1,
x2
and x3, |
|
that is, f
(x1) = 0, f
(x2) = 0
and f
(x3) = 0 since
f(a)
·
f(b) < 0. |
 |
|
| 3.
Extreme value theorem |
| If
a function is continuous on the closed interval [a,
b] then, there
exist xmin
and xMax
Î
[a, b] such that |
|
for all x
Î
[a, b],
values of the function f
(xmin)
< f
(x) < f
(xMax). |
|
|
| 4.
Monotone function |
|
A continuous
function is monotone on an interval if it is consistently
increasing or decreasing in value,
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|
so that
either
f
(x1) < f
(x2)
for
all
x1 < x2,
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or
f
(x1) > f
(x2)
for
all
x1 < x2.
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These
may be called strictly monotone
functions to distinguish them from those satisfying either
|
|
f
(x1) < f
(x2)
for
all
x1 < x2,
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|
or
f
(x1) > f
(x2)
for
all
x1 < x2,
|
| that
are called weakly monotone. |
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| The
derivative of a function |
|
The
slope or the gradient of the secant line joining points (x,
f(x)) and (x
+ Dx,
f(x + Dx))
given by |
|
| is
called the difference quotient, and
represents the average rate of change of the function y
= f(x). |
|
|
Note
that in the above equation x
is the given fixed point and the only variable quantity is the
distance
|
| increment,
Dx. |
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Definition of the derivative of a function |
|
As
the distance
Dx
gets smaller, the average rate of change
becomes more accurate, that is the secant line
|
| becomes
more and more like the tangent line of the curve at the point (x,
f(x)),
as shows the figure above. |
| Therefore,
the limit of the difference quotient as Dx
®
0,
that
equals
the slope of the line tangent to the |
|
curve at
the
point (x,
f(x)),
|
 |
|
represents the instantaneous rate of change of the function
with respect to x,
or when written as
|
|
|
|
where
Dx
is substituted with h,
is
called the derivative of a function
f
at a given point x.
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Geometrically,
the derivative is the slope m
(gradient) of the tangent at the point (x,
y) to the curve y
= f(x).
|
| There
are other common notations for the derivative like, |
 |
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| The
equation of the line tangent to the given curve at the given point |
| The
equation of the tangent line at the given point (x0,
y0) of a function y
= f(x) we write using the
point |
|
slope
equation of a line,
|
|
y
-
y0
= f
'
(x0) · (x
-
x0)
|
| where
m = f '
(x0)
is the slope of the tangent line at
the point
(x0,
y0).
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|
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Determining the derivative of a function as the limit of the
difference quotient |
| Example: Find
the equation of the tangent line at the point (2,
4)
of the function f(x)
= x2.
|
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| Solution: The
slope of the tangent line at
the given point
we calculate using
the formula |
|
|
| then, |
| m
= f '
(x0)
= 2x0,
m
= f '
(2)
= 2 · 2
= 4. |
| By
substituting m
= 4 and the point (2,
4) into |
| y
-
y0
= f
'
(x0) · (x -
x0), |
| we
get |
| y
-
4
= 4 · (x -
2) or
y
= 4x -
4 |
| the
equation of the tangent line at the given point
of |
| the function
f(x)
= x2, as shows the
right figure.
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|
 |
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Determining the lines tangent to the graph of a function from a point outside the
function
|
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Lines
tangent to the graph of a function y
= f(x)
from a given point (x1,
y1)
outside the function are defined
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by two
points they pass through, the given point (x1,
y1)
and the point of tangency (x0,
y0).
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To
find the coordinates x0
and y0
of the points of tangency we should solve the following two
equations,
|
|
(1) y0
=
f(x0) |
|
(2) y1
-
y0
=
f
'
(x0) · (x1
-
x0)
|
|
as
the system of the two equations with the two unknowns x0
and y0.
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The
first equation states that the point of tangency must lie on the
given function while, at the same time, its
|
|
coordinates
x0
and y0
must satisfy the equation of the tangent line (2),
since the point of tangency is the
|
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common point of the
tangent
and the curve.
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Thus
for example, if given is the quadratic function
f(x)
= ax2
+ bx
+ c
and
the point P1(x1,
y1)
then,
|
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applying
the above system of equations we can calculate
the coordinates of the points of tangency
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P0(x0,
y0)
directly from the formulas,
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| Example: |
Find equations of the lines tangent
to the function f(x)
= -
x2
+ 2x
+ 4
drawn
from the point |
|
|
(3,
2).
|
| Solution: To
find the coordinates x0
and y0
of the points of tangency we should solve the system of equations.
Substitute given quantities, the point (3,
2)
and
f(x)
= -
x2
+ 2x
+ 4
into equations, |
|
|
|
Let calculate the slope
f
'
(x0)
of the tangent line, the equation
(2).
|
|
| Then
substitute the slope and the y0
into second equation to get the points of tangency, |
|
| Therefore, the points of tangency
are (2,
4)
and (4, -4). |
| To
write the equations of tangents t1
and t2
we calculate |
| their
slopes m1
and m2
from |
| f
'
(x0)
= -
2x0
+ 2. |
| Thus,
m1
=
f
'
(2)
= -
2
· 2
+ 2
= -
2,
m1
= -
2, |
|
and m2
=
f
'
(4)
= -
2
· 4
+ 2
= -
6,
m2
= -
6. |
| The
tangent
t1
through the point (2,
4)
and m1
= -
2, and |
| the
tangent
t2
through the point (4, -4)
and m2
= -
6, |
| t1 ::
y
= -
2x
+ 8
and
t2 ::
y
= -
6x
+ 20. |
|
 |
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Check
the coordinates of the points of tangency by plugging the
corresponding parameters into the above
|
|
formulas.
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