Differentiability

Question 1

Differentiability at a point xo

Answer 1

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).

Question 2

Differentiable => continuous

Answer 2

If a function is differentiable at a point xo e R, then it is continuous at xo.

Question 3

Not continuous => not differentiable

Answer 3

If f is not continuous at a, then f is not differentiable at a.

Question 4

Rules for differentiation

Answer 4

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

Question 5

Alternative criterion for differentiability at a point

Answer 5

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.

Question 6

The function F

Answer 6

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

Question 7

Chain rule

Answer 7

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).

Question 8

Inverse function theorem and differentiability

Answer 8

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).

Question 9

Strictly increasing at a point xo

Answer 9

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

Question 10

Strictly decreasing at a point xo

Answer 10

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

Question 11

Differentiables => monotonicity

Answer 11

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.

Question 12

Differentiable

Answer 12

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).

Question 13

General theorems (AOL)

Answer 13

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

Question 14

Composition of differentiable functions

Answer 14

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.

Question 15

Rolles Theorem

Answer 15

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.

Question 16

Zeros of f => zeros of f’

Answer 16

If f: (a,b) -> |R is differentiable, then between any two zeros of f there is a zero of f’.

Question 17

No. of zeros of f’

Answer 17

If f: (a,b) -> |R is differentiable and has n distinct zeros, then f’ has n-1 distinct zeros

Question 18

The mean value theorem

Answer 18

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)

Question 19

Monotonicity on an interval

Answer 19

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]

Question 20

Generalised mean value theorem/ Cauchy’s Mean Value Theorem

Answer 20

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)

Question 21

0/0 form of L’Hopital’s Rule

Answer 21

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-> xo

Question 22

Differentiable on a open interval

Answer 22

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

Question 23

nth derivative

Answer 23

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.

Question 24

Taylors theorem (theory)

Answer 24

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)

Question 25

Taylors theorem

Answer 25

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 )

Question 26

Taylors theorem equivilant

Answer 26

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)

Question 27

Taylors theorem for polynomials of degree at most n

Answer 27

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

Question 28

Approximating functions using Taylor’s theorem

Answer 28

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.

Question 29

Local maximum

Answer 29

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)

Question 30

Global maximum

Answer 30

We say that f has a global maximum at xo if

f(x) =< f(xo) whenever x e (a,b)

Question 31

Local minimum

Answer 31

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)

Question 32

Global minimum

Answer 32

We say f has a global minimum at xo if

f(x)>=f(xo) whenever x e ( a, b )

Question 33

Stationary point

Answer 33

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.

Question 34

Classification of stationary points (1)

Answer 34

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.

Question 35

Classification of stationary points (2. Even case)

Answer 35

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

Question 36

Classification of stationary points (3. Odd Case)

Answer 36

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

Question 37

Taylor Series

Answer 37

The taylor series of f centred at xo

f(x) = SUM (from k=0 to inf) f^(k)(xo)/k! (x-xo)^k

Question 38

Power series

Answer 38

A series in the form SUM (n=0 to inf) (an) (x^n)

Question 39

Set of convergence for a power series

Answer 39

S:= {x e |R : SUM (n=0 to inf) (an) (x^n) converges

Question 40

Radius of convergence

Answer 40

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

Question 41

Distance

Answer 41

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.

Question 42

Metric space

Answer 42

The pair (X , d).

Question 43

Convergence of a sequence to a point on a metric space

Answer 43

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

Question 44

Continuity in metric spaces

Answer 44

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