Transcript Document

Lecture II
The elements of
higher mathematics
.
The derivative of
function
Lecture questions
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Function
Representation of a function
Function derivative
Geometric interpretation of function derivative
Some differentiation rules
Physical interpretation of function derivative
Second-order derivative, higher derivatives
Extremum of function
• Definition
of
function.
Independent and dependent
variables. The domain and the
range
A relationship between two variables, typically
x and y, is called a function if there is a rule
that assigns to each value of x one and only
one value of y.
When that is the
case, we say that
y is a function of
argument of x.
• The values that x may assume are called
the domain of the function. We say that
those are the values for which the function
is defined.
• Once the domain has been defined, then
the values of y that correspond to the
values of x are called the range.
Representation of a function
There are many ways to represent or
visualize functions: a function may be
described by a formula, by a plot or graph,
by an algorithm that computes it, by arrows
between objects, or by a description of its
properties. Sometimes, a function is
described through its relationship to other
functions (for example, inverse functions). In
applied disciplines, functions are frequently
specified by tables of values or by formulas.
The equation y = ƒ(x) is viewed as a
functional relationship between dependent
and independent variables.
Differential calculus
• Differential
calculus,
a
field
in
mathematics, is the study of how functions
change when their inputs change. The
primary object of study in differential
calculus is the derivative. A closely related
notion is the differential.
Increments
• The increment of a variable in changing
from one numerical value to another is the
difference found by subtracting the second
value from the first. An incrementof
x x is
denoted by the symbol
• And
y the corresponding increment of the
function is
Function derivative
• The derivative of a function is the limit of
the ratio of the function increment to the
argument increment, when the latter
increment varies and approaches zero. If
the limit exists, then f(x) is differentiable
at x.
dy
y
dx
 lim
x0
x
Derivative notations
dy
• Leibniz's notation
dx
• Lagrange's notation
• Euler's notation Dy
• Newton's notation y
y
General Rule for differentiation
• First Step: In a function replacex x 
by
,
x
 function,
y
giving a new value of ythe
• Second Step: Subtract the given value of the
function from the new value in order to find y (the
increment of the function).
• Third Step: Divide the remainder
.
y x by
• Fourth Step: Find the limit of this quotient, when
x varies and approaches the limit zero. This is the
derivative required.
Geometrical interpretation of the
derivative of a function
Geometrical interpretation of the
derivative of a function
• The value of the derivative at any point of
a curve is equal to the slope of the line
drawn tangent to the curve at that point.
Differentiation of
C   0
• Constant
n 
• Power function x   n  x n 1
• Exponential functions a x   a x  ln a

e 
x
• Logarithmic functions
e
x

1
log a x  
x  ln a
 1
ln x  
x
Differentiation of
• Trigonometric functions

sin x 

cos x 
 cos x
  sin x

1
tgx   2
cos x

1
ctgx    2
sin x
Rules of differentiation
Constant factor rule:
• The derivative of the product of a constant
and a function is equal to the product of
the constant and the derivative of the
function.
(c  f x)  c  f x
Sum rule:
• The derivative of the algebraic sum of a
finite number of functions is equal to the
same algebraic sum of their derivatives.

u ( x)  v( x)  w( x) 
 u ( x)  v( x)  w( x)
Product rule:
• The derivative of the product of two
functions is equal to the derivative of the
first function times the second function
plus the derivative of the second function
times the first function.

u  v 
 u   v  v  u
Quotient rule:
• The derivative of a fraction is equal to the
derivative of the numerator times the
denominator, minus the derivative of the
denominator times the numerator, all
divided by the square of the denominator.

 u  u   v  v  u
  
2
v
v
 
Chain rule:
• If f is a function of g and g is a function of
x, then the derivative of f with respect to x
is equal to the derivative of f(g) with
respect to g times the derivative of g(x)
with respect to x.
d
df ( g ) dg( x)
f ( g ( x)) 

dx
dg
dx
Physical interpretation of the
derivative of a function
• a derivative of a coordinate (displacement
or distance from the original position) with
respect to time is a velocity.
dx
v(t ) 
dt
• Generalization: the derivative of a function
at a given point is the instantaneous rate
of change of the function with respect to its
argument at a given point.
Second-order derivative, higher
derivatives
• The derivative of a function of x is in
general also a function of x. This new
function may also be differentiable, in
which case the derivative of the first
derivative is called the second derivative
of the original function.
• Similarly, the derivative of the second
derivative is called the third derivative;
and so on to the nth derivative.
Notations
of the second derivative:
d2y
2
dx
y
2
D y
of a higher order
derivatives:
n
d y
dx
y
(n )
Dn y
y
Applications of derivatives in
investigation of functions
• Domain interior points, in which a derivative of a
function is equal to zero or doesn’t exist, are
critical points of this function.
• Critical points divide a function domain for intervals
that within each of them a derivative saves a
constant
f ( x)  sign.
0
a, b
• If
at every point of an interval
,
then
f(x) increases withina,
this
f ( xa) function
0
b interval.
• If
at every point of an interval
,
then a function f(x) decreases within this interval.
Extreme points
• Sufficient conditions of extreme:
• if a derivative changes its sign from plus to
minus at a critical point, then that is a
point of maximum. If a derivative
changes its sign from minus to plus at a
critical point, then that is a point of
minimum.
At B and R points, where a function is increasing,
the tangent makes an acute angle with the axis of x,
hence the slope is positive. At M and Q points,
where a function is decreasing, the tangent makes
an obtuse angle with the axis of x, therefore the
slope is negative.
C and P points are maxima. N point is a minimum.
To find intervals of function increase and
decrease and the extreme points
•
•
•
•
Find the domain of a function.
Differentiate the function.
Obtain the roots of equation:
.
f ( x)  0
Form open intervals with the zeros (roots) of the
first derivative and the points of discontinuity (if
any).
• Take a value from every interval and find the
sign they have in the first derivative.
• Determine the intervals of function increase and
decrease and extreme points, where the
derivative changes its sign.
Thank you for your attention !