Differentiability Flashcards
Differentiability at a point xo
A real function f: (a,b) -> |R is differentiable at a point xo e (a,b) if the limit
lim (as h tend to 0) of (f(xo+h)-(f(xo)/h) exists.
If the limit exists, it is called the derivative of f at xo and denoted f’(xo).
Differentiable => continuous
If a function is differentiable at a point xo e R, then it is continuous at xo.
Not continuous => not differentiable
If f is not continuous at a, then f is not differentiable at a.
Rules for differentiation
Suppose that f and g are both differentiable at a point xo. Then
(1) the sum f+g is differentiable at xo, and
(f+g)’(xo) = f’(xo) + g’(xo)
(2) the difference f - g is differentiable at xo and
(f-g)’(xo) = f’(xo) - g’(xo)
(3) The product fg is differentiable at xo, and
(fg)’(xo) = f’(xo)g(xo) + f(xo)g’(xo)
(4) the quotient f/g is differentiable at xo and
(f/g)’(xo) = ((f’(xo)g(xo) - f(xo)g’(xo))/ (g(xo))^2
Alternative criterion for differentiability at a point
A function f: (a,b) ->|R is differentiable at xo e (a,b) iff there exists a function F: (a,b) -> |R that satisfies
f(x) = f(xo) + (x-xo)F(x) for all x e (a,b)
and is continuous at xo.
The function F
If f is differentiable at xo, then the function F is given by
F(x) = { (f(x) - f(xo)) / ( x -xo ) if x=/ xo
{ f’(xo) if x = xo
Chain rule
Suppose that f:(a,b) -> (c,d) is differentiable at xo e (a,b) and that g : (c,d) -> |R is differentiable at f(xo). Then the composed function g.f is differentiable at xo and
(g.f)’(xo) = g’(f(xo))f’(xo).
Inverse function theorem and differentiability
Suppose that f satisfies the conditions of the inverse function theorem and that xo e (a, b) and yo = f(xo). If in addition, f is differentiable at xo and f’(xo) =/ 0, then f^-1 is differentiable at yo and
(f^-1)’(yo) = 1/f’(xo).
Strictly increasing at a point xo
A function f is strictly increasing at a point xo if there exists d in |R+ such that
f(xo + h) > f(xo) when 0 0 whenever 0< | h | < d
Strictly decreasing at a point xo
A function f is strictly decreasing at a point xo if there exists d in |R + such that
f(xo + h) < f(xo) when 0< h < d and f(xo + h) > f(xo) when -d
Differentiables => monotonicity
Suppose that f is differentiable at xo. If f’(xo) > 0, the f is strictly increasing at xo and if f’(xo) < 0, then f is strictly decreasing at xo.
Differentiable
Suppose that a,b e R and a< b. A function f:(a,b) -> |R iis differentiable it is differentiable at all points in (a,b).
General theorems (AOL)
Suppose that the functions f:(a,b) -> |R and g: (a,b) -> |R are differentiable. Then
(1) f + g is differentiable
(2) f-g is differentiable
(3) fg is differentiable
(4) If g(x)=/0 for all x e (a,b), then f/g is differentiable
Composition of differentiable functions
Suppose that f: (a,b) -> (c, d) and g: (c,d) -> |R are differentiable functions. Then the composed function g.f: (a,b) -> |R is differentiable.
Rolles Theorem
Suppose that the function f is continuous on [a,b] and differentiable on (a,b) and that f(a)=f(b) Then there is a point c e (a,b) such that f’(c) = 0.
Zeros of f => zeros of f’
If f: (a,b) -> |R is differentiable, then between any two zeros of f there is a zero of f’.
No. of zeros of f’
If f: (a,b) -> |R is differentiable and has n distinct zeros, then f’ has n-1 distinct zeros
The mean value theorem
Generalisation of Rolle’s Theorem
Suppose that f is continuous on [a,b] and differentiable on (a,b). Then there exists c e (a,b) such that
( f(b) - f(a) ) / (b - a) = f ‘ (c)
Monotonicity on an interval
Suppose that f is continuous on [a,b] and differentiable on (a,b).
(1) If f’(x) = 0 for all x e (a,b), then f is constant on [a,b]
(2) If f’(x) > 0 for all x e (a,b), then f is strictly increasing on [a,b]
(3) If f’(x) < 0 for all x e (a,b), then f is strictly decreasing on [a,b]
Generalised mean value theorem/ Cauchy’s Mean Value Theorem
Suppose that f and g are continuous on [a,b] and differentiable on (a,b), and that g’(x) =/0 for all x e (a,b). Then there exists c e (a,b) such that
( f(b) - f(a) ) / ( g(b) - g(a) ) = f’(c) / g’(c)
0/0 form of L’Hopital’s Rule
Suppose that f and g are differentiable on (a,b). Suppose further that xo e (a , b) and that g'(x) =/0 for all x e (a , b) \{xo}. If f(xo) = g(xo) = 0 and
f'(x)/g'(x) -> A as x-> xo
then f(x) / g(x) -> A as x-> xoDifferentiable on a open interval
A function f : [a,b] -> |R is differentiable if it is differentiable on the open interval (a,b) and both limits
lim, as h tend to o+, of ( f( a+ h ) - f( a ) ) / h and
lim, as h tend to o- (f ( b + h ) - f(b)) / h exist
nth derivative
Suppose that f is differentiable with derivative f’ on an interval, and that f’ is itself differentiable. Then we denote the derivative f’ by f’’ and it call it the second derivative of f. Continuing in this way we obtain functions
f, f’, f’’, f’’’, … , f^(n)
And f^(n) is the nth derivative of f.
Taylors theorem (theory)
For ‘suitable’ functions the behaviour of the functions at a+t is given (approximately) by a polynomial
(whose coefficients depend on f and the derivatives of f)
Taylors theorem
Suppose that f, f’, f’’, … , f^(n) all exist and are continuous on [a,b] and that f^(n+1) exists on ( a , b ). Then there exists c e (a,b) s.t.
f(b) = SUM, k=0 to n, ( ( f^( k )( a ) / k! (b-a)^k + f^( n+1 )( c ) / (n+1)! ( b-a )^( n+1 )
Taylors theorem equivilant
If f, f’, …, f^(n) exist and are continuous on [a, a+t] and f^(n+1) exists on (a, a+t) then there exist § e (0 , 1), such that
f( a+t ) = f(a) + f’(a) t + f’‘(a) / 2 t^2 + … + f^(n) (a) /n! t^n + f^(n+1) (a + §t) / (n+1)! t^(n+1)
Taylors theorem for polynomials of degree at most n
If f is a polynomial of degree at most n, then f^(n+1) is identically zero. Therefore, Taylors theorem says
f(a+t) = f(a) + f’(a)t + f’‘(a) / 2 t^2 + … + f^(n) (a) /n! t^n
Approximating functions using Taylor’s theorem
Taylor's theorem says that we may approximate certain functions t -> f(a+t) by the polynomial f(a) + f'(a) + ... + f^(n)(a)/n! t^n with an error term Rn(t) = (f^(n+1) (a + §t) / (n+1)! t^n+1 where § depends of f, n, a and t.
Local maximum
Suppose that a real function f is defined on the open interval (a,b) and that xo e (a,b). We say that f has a local maximum at xo if there exists d e |R+ such that
f(x) =< f(xo) whenever x e (x0-d, xo+d)
Global maximum
We say that f has a global maximum at xo if
f(x) =< f(xo) whenever x e (a,b)
Local minimum
We say f has a local minimum at xo if there exists d e |R+ such that
f(x)>=f(xo) whenever x e (xo-d, xo+d)
Global minimum
We say f has a global minimum at xo if
f(x)>=f(xo) whenever x e ( a, b )
Stationary point
Suppose that f is defined on an open interval (a,b). A point xo e (a, b) is a stationary point of f if f is differentiable at xo and f’(xo) = 0.
Classification of stationary points (1)
Suppose that f, f’, f’’, … , f^(n+1) exist and are continuous on (a , b) and that for some point xo e (a , b)
f^(k) (xo) = 0 when 1 =< k =< n while f^(n+1) (xo) e (a,b)
for some positive integer n.
Classification of stationary points (2. Even case)
In the case, n+1 is even,
If f^(n+1)(xo) > 0, then f has a local minimum at xo
If f^(n+1)(xo) < 0, then f has a local maximum at xo
Classification of stationary points (3. Odd Case)
In the case, n+1 is odd,
If f^(n+1)(xo) > 0, then f is strictly increasing at xo
If f^(n+1)(xo)< 0, then f is strictly decreasing at xo
Taylor Series
The taylor series of f centred at xo
f(x) = SUM (from k=0 to inf) f^(k)(xo)/k! (x-xo)^k
Power series
A series in the form SUM (n=0 to inf) (an) (x^n)
Set of convergence for a power series
S:= {x e |R : SUM (n=0 to inf) (an) (x^n) converges
Radius of convergence
For such a power series, the radius of convergence is given by
R = { sup {|x| : x e S} if S is bounded
inf if S is unbounded
Distance
Let X= empty set. A function d: X x X -> |R satisfying the following conditions
(M1) d(x , y) >=0 and d(x, y) =0 iff x=y
(M2) d(x , y) = d (y , x)
(M3) d(x , y) =< d(x , z) + d(z , y)
for all x, y, z in X is called a distance or metric on X.
Metric space
The pair (X , d).
Convergence of a sequence to a point on a metric space
Let (X, d) be a metric space. Let (xn) (from n=1 to inf) be a sequence of points in X, and l e X. We say (xn) converges to l iff for all e>0 there exists N e |N s.t
d(xn, l) < e whenever n>N
Continuity in metric spaces
Let (X, dX) and (Y , dY) be metric spaces, and f: X-> Y be a function. We say that f is continuous at a e X if for all e>0 there exists d>0 such that
dY( f(x) , f(a) ) < e whenever dX (x , a) < d
