7. Definition of the Derivative.The derivative of a sum, of a product and of a quotient of two functions.
Answer:
Definition.
A function f is differentiable at an interior point x x0
of its domain, if the difference quotient:
, x
x0
approaches
a limit as x->x0
in which case the limit is called the derivative of f
at x0
, thus f’(x0)
=
(1)
It is sometimes convenient to lot x=x0 + n and write (1) as
f’(x0)=
If f is defined on an open set S, we say that f is differentiable on S if f is differentiable at every point of S.
If so, we denote the derivative of f at x0 by f’’(x0). This is second derivative of f at x0: f(0)=f; f’(x0)=(f(x))’; f’’=(f’)’ ; f(n)=(fn-1)’.
The derivative also has a geometric interpretation. The equation of the line through two points (x0; f(x0)) and (x1 ; f(x1)) on the curve y=f(x) is
y
=f(x0)
+
(x-x0)
Varying xn generates lines through (x0, f(x0)) that rotate in to the line:
y= f(x0) + f’(x0)(x-x0)
as x1 approaches x0. This is the taking to the curve y=f(x) at the point (x0; f(x0))
Theorem.
If f is differentiable at x0
then f is continuous (f
).
7.2 The converse of this theorem is false.
Theorem.If f and g are differentiable at x0 , then so are f+g,f-g, and fg with:
(f+g)’(x0)=f’(x0) + g’(x0)
(f-g)’(x0) =f’(x0) – g’(x0)
(fg)’(x0) = f’(x0)g(x0) + f(x0)g’(x0)
’(x0) =f’(x0)g(x0) - f(x0)g’(x0)/(g(x0))2
Theorem. Suppose that g is differentiable at x0 and f is differentiable at g(x0). Then the composite function h=fg , defined by h(x)=f(g(x)) is differentiable at x0 and h’(x0)=f’(g(x)g’(x0))
One-sided Derivatives.
One-sided limits as (1) and (2) are called one-sided or right-hand and left-hand derivatives.
That is , if f is defined on [x0;b) , the right-hand derivative of f at x0 is defined to be:
f(x0)= (x) – f(x0)/x-x0 ; if the limit exists , while if f is defined on (a;x0]; the left-hand derivative, of f at x0 is defined to be
f’(x0)= (x) – f(x0)/x-x0
Corollary. A function f is differentiable at x0 if:
f’+(x0)=f’-(x0)=f’(x0)
8. Rolle’s Theorem. Lagranzh’s Theorem.Darbu’s Theorem.Cauch’s Theorem.
Answer:
Extreme
values: We say that f(x0)
is a local extreme value of , if
δ>0
such that f(x)-f(x0)
does not change sign on (x0-δ;x0+δ)
f(x0)
is a local maximum of f is f(x)
f(x0)
and f(x0)
is a local minimum of f is f(x)
f(x0)
Rolle’s theorem.
If a function f is differentiable on the closed interval [a;b] and f is continuous on the open interval (a;b) and f(a)=f(b) then c (a;b) f’(c)=0
Proof: Since f C[a;b] , f attains a maximum and a minimum value on [a;b], if these two extreme values are the some the f is constant on (a;b). so f’(x0)=0 for all x in (a;b).
If the extreme values differ , then of least one must be attained at some point c in the open interval (a;b) and f’(c)=0.
Dorbu’s
Theorem.
Suppose that f is differentiable on [a;b] , f’(a)
f’(b)
and
is between f’(a) and f’(b). Then f’(c) =
c
(a;b).
Cauch’s Theorem. If f and g are continuous on the closed interval [a;b] and f and g are differentiable on the open interval (a;b) then (g(b)-g(a))f’(c) = (f(b)-f(a)g’(c)) (4) for some c in (a;b).