Graphics Reference
In-Depth Information
32 Givean exampleof a function
f
which shows that it is possiblefor
thefunction to havea local maximum at
x
a
without being
differentiable at
a
.
Questions 15 and 17 enable us to find derived functions for
polynomials and rational functions. In chapter 11 we will give a formal
definition of logarithmic and exponential functions and obtain
log
(
x
)
1/
x
, for positive
x
, and exp
(
x
)
exp (
x
), for all real
x
, and we
will give a formal definition of the circular (or trigonometric) functions
sineand cosineand obtain sin
(
x
)
cos (
x
) and cos
(
x
)
sin (
x
) for
all real
x
.
According to the Leibnizian description of derived functions, when,
for example,
y
x
,
dy
dx
3
x
.
For the product rule the Leibnizian expression is
d
(
uv
)
dx
u
·
dv
dx
v
·
du
dx
.
For the quotient rule the Leibnizian expression is
u
v
v
·
du
dx
u
·
dv
d
dx
dx
.
v
The notation of Leibniz is particularly suggestive when describing the
chain rule:
dy
dx
dy
du
·
du
dx
.
33 Apply the chain rule to find the derived function of
f
where
f
(
x
)
sin
x
by considering
y
sin
u
and
u
x
.
34 Give algebraic expressions for the derived function
f
of qn 22.
Is thefunction
f
continuous at
x
0?
Second derivatives
35 A real function
f
is defined by
0
when
x
0,
f
(
x
)
x
when 0
x
1,
x
when 1
x
.
