Chapter 5 differentiation

Question 1

Differentiation definition

Answer 1

Let f be a real-valued function with domain D_f. We say that f is differentiable at a ∈D_f if Lim( x->a) [ (f(x)-f(a))/(x-a)] exists and is finite/real.
Then we have
f’(a) =
Lim( x->a) [ (f(x)-f(a))/(x-a)] = Lim( h->0) [ (f(a+h)-f(a))/(h)]
And we call f’(a) in R the derivative of f at a.

We say that f is differentiable on S subset of D_f is it is differentiable at every point a in S

Question 2

Derivative function

Answer 2

From the limit,
Real valued function f’ with domain
D_f’ = {x∈D_f| f’(x) exists}

Notation for n∈N with n>2 we define the nth derivative of f at a by f^(n) (a) = ( f^(n-1)’)(a)
Whenever the limit on the right hand side exists we sat f us infinitely differentiable or smooth at a if f’(a) exists

Question 3

Theorem 5.2.4 differentiable and continuous?

Answer 3

If a real valued function f is differentiable at a∈D_f then f is continuous at a

Not true that every function that is continuous is differentiable

Proof:
By showing that f(x) -f(a) =( [f(x)-f(a) ]/ [x-a] )•(x-a) and as differentiable by taking limits = f’(a)•0 =0 .

Question 4

Definition: left derivative and right derivative

Answer 4

We say that the REAL VALUED function f has a left derivative at a∈D_f if f’_(a) = lim h↑0 ( f(a+h) -f(a))/h exists and is finite that it has a right derivative at a∈D_f if f’+(a) =lim h↓0 (f(a+h)-f(a))/h exists and is finite

Question 5

Theorem 5.2.7 when is a real valued function is differentiable

Answer 5

A real valued function f is differentiable at a∈D_f
IF AND ONLY IF
Both the left and right derivatives at a exist and are equal. In this case f’(a) = f’_(a) = f’+(a)

(Left and right equal)

Proof:
Exercise

Question 6

Theorem 5.3.1 Rules for differentiation
Sum
Muliplied

Quotient

Of functions

Answer 6

Let f and g be real-valued functions that are differentiable at a∈(D_f ∩ D_g). Then the following hold:

1) for each α,β∈R, the function of αf+βg is differentiable at a and (αf+βg)’(a) = αf’(a) +βg’(a)
2) the product rule: the function fg is differentiable at a and (fg)’(a)=f’(a)g(a)+f(a)g’(a).
3. The quotient rule: if g(a)≠0 then f/g is differentiable at a and (f/g)’(a) = (g(a)f’(a)-f(a)g’(a))/(g(a)²)

Proof:

1) algebra of limits
2) relevant limits
3) …

Question 7

Theorem 5.3.2

Differentiation for composites

Answer 7

The chain rule:
Let f,g be real valued functions such that the range of g is contained in the domain of f. Suppose that g is differentiable at a and that f is differentiable at g(a). Then (f∘g)’(a) = f’(g(a))g’(a)

Question 8

Definition 5.4.1: for functions local minimum and local maximum and turning points

Answer 8

A REAL-valued function f has a local minimum at a∈D_f if there exists δ>0 such that (a-δ,a+δ) ⊂D_f and f(x) ≥ f(a) for all x∈(a-δ,a+δ).

A real-valued function f has a local maximum at a ∈D_f if there exists δ>0 such that (a-δ,a+δ) ⊂D_f and f(x) ≤ f(a) for all x∈(a-δ,a+δ).

A turning point/extreme point for f is a point in its domain that is either a local minimum or local maximum.

Question 9

Difference between local and global maxima minima

Answer 9

Eg if f: [a,b] -> R us continuous then by theorem 4.3.4 it attains its supremum and infimum on [a,b]. So these are the global maximum and minimum

But aren’t necessarily turning points as they might be at the end points of the interval [a,b].

Local need not be global.

Question 10

Theorem 5.4.3 differentiable turning points values

Answer 10

If f is DIFFERENTIABLE at a∈D_f and a is a turning point for f then f’(a) =0.

These are sometimes called stationary points
Turning points are stationary points but not all stationary points are turning points: some are inflection points

Question 11

Theorem 5.4.4 Rolle’s theorem

Answer 11

Let f be a real-values function that is continuous on [a,b] and differentiable on (a,b) with f(a) =f(b) then there exists ∈(a,b) such that f’(c) =0

Diagram

Question 12

Mean value theorem 5.5.1

**

Answer 12

If a real-valued function f is continuous on [a,b] and differentiable on (a,b), then there exists c∈(a,b) such that

f’(c) = (f(b) -f(a))/(b-a)

So when you know it’s differentiable …
Does not say unique c!

Question 13

Example- let a and b in R and consider limit as n tends to infinity of ( |a|^n + |b|^n) ^(1/n)
Does the limit exist? Prove your guess.

Answer 13

By considering cases of a and b eg 0s we guess that converges to max{|a|,|b|}

Proving this:
Let M equal this guess. Then we have |a| and |b| less than or equal to M.

So raising to the power of n.
|a|^n + |b|^n less than or equal to 2M^n
And so
(|a|^n + |b|^n)^(1/n) less than or equal to (2M^n) = 2^(1/n) M
Using the algebra of limits limit as n tends to infinity of this gives M. Hence it follows by the sandwich our original limit converges to M.

Question 14

Example:
f(x)=c where c in R us constant

Checking if differentiable

f(x) = x^n

Answer 14

1) directly from definition of differentiability
2) f(x) = x^n for x in R where n in N fixed. By the binomial theorem

f is differentiable at all x in R:

( f(a+h) - f(a)) /h

= ((a+h)^n - a^n)/h
= ( a^n + na^{n-1} h + 0.5•n•(n-1)•a^(n-2)•h²+…+ nah^{n-1} + h^n -a^n) /h
= na^{n-1} + + 0.5•n•(n-1)•a^(n-2)•h+…+ nah^{n-2} + h^{n-1}
Thus the limit as h tends to 0 is na^{n-1}
So f’(x) =nx^n-1. For all x in R and D_f’ = D_f =R

SUMMARY: find the limit defined in the differentiable definition, for any a in the domain. (This was found using the BINOMIAL THEOREM)

Question 15

Example: consider a function f(x) =|x| with D_f =R which is continuous at every point in R and clearly differentiable at every x not equal to 0.

WE SHOW THAT ITS NOT DIFFERENTIAVLE AT 0 by showing the Left and RIGHT limits are DIFFERENT

Answer 15

Lim_ h↑0 [ (f(0+h - f(0) )/h] = lim_h↑0[ |h| /h] = lim_h↑0[-h/h] =-1

Lim_ h↓0 [ (f(0+h - f(0) )/h] = lim_h↓0[ |h| /h] = lim_h↑0[h/h] =1

Thus the left and right limits are different and so limit as h tends to 0 of f(0+h) -f(0) /h does not exist.

Question 16

Example: consider function f; [-3,2] to R defined by

f(x) =
{ x+2 if x in [-3,-1]
{x^2 if x in [-1,2]

Identify any turning points and global maximum and minimum

Answer 16

The function is continuous: at x=-1 we can check this.

Global max 4 at x=2
Global min -1 x=-3
These aren’t turning points as they’re end points. That is we can’t find a delta st all points within distance delta are contained in the domain.
Local minimum at x=0.

Question 17

Stationary points and turning points

Answer 17

Tps are stationary points
Not all stationary points are tps

f(x)=x^3. The. f’(0) =0 but 0 is neither local max nor local min but is an INFLECTION POINT

Question 18

Corollary 5.5.2
Monotonicity revisited

f’(x) for monotone functions

Answer 18

Suppose that a real valued function f is continuous on [a,b] and differentiable on (a,b)

If for all x in (a,b) we have

f’(x) ≥ 0 : f is monotone increasing on [a,b]
f’(x) > 0 : f is strictly monotone increasing on [a,b]
f’(x) ≤ 0 : f is monotone decreasing on [a,b]
f’(x) < 0 : f is strictly monotone decreasing on [a,b]

By using the MVT showing existence and bigger than or equal to 0 etc
•corollary useful tool to study inverses of functions when used in conjunction with the inverse function theorem

Question 19

Theorem 5.5.3

Inverses revisited.
Continuity of inverses and delta

Answer 19

Suppose that f:R to R is continuous on [a,b] and differentiable on (a,b) and that f’ is continuous at c∈(a,b). If f’(c) ≉ 0 then the following hold:

1) there exists δ>0 so that f is invertible on ]c-δ, c+δ]
And f^- is continuous on ( f(c-δ), f(c+δ)) if f’(c) > 0, and on ( f(c+δ), f(c-δ)) if f’(c) < 0.

2) the mapping f^-1 is differentiable at f(c) and

(f^-1)’(f(c)) = 1/ [f’(c)]
• in calculus du/dx = 1/ (du/dx)

Question 20

Theorem 5.5.4

Cauchys mean value theorem

Answer 20

Let f and g each be continuous on [a,b] and differentiable on (a,b) with
g’(x) ≠ 0 for all x∈(a,b).

Then there exists c∈(a,b) so that

f’(c)/g’(c)
= ( f(b) -f(a)) / ( g(b) - g(a))

Proof:

By rolled theorem…

Question 21

Corollary 5.5.5 l’Hôpitals rule

Answer 21

Suppose that f and g are each differentiable on (a,b) with g’(x) ≠ 0 for all x∈(a,b).

1) if c∈(a,b) with f(c) = g(c)=0 then

Limit as x tends to c of (f(x)/g(x))
= limit as x tends to c of (f’(x)/g’(x))

Whenever the limit on RHS exists and is finite.
2) and 3) similarly if limit as x tends to a-= limit as x tends to b- =0

And if limit as x tends to b+ = limit as x tends to a+ =0

Works for infinity also

Question 22

Example 5.5.6 evaluate

Limit as x↓0 of x^x

Define f(x) =a^x as f(x) = e^{xlnx} for x in the reals.

Answer 22

By l’Hôpitals

Limit as x↓0 of xlnx = Limit as x↓0 of (ln(x)/(1/x))
= -Limit as x↓0 of ((1/x)/(1/x²)) = -Limit as x↓0 of x =0

So by the CONTINUITY of the exponential function

Limit as x↓0 of x^x = Limit as x↓0 of e^xln(x) = e^ ( Limit as x↓0 of xlnx) = e^0 = 1

Question 23

Definition 5.6.1:

For each n in N real vector space for

Answer 23

For each n in N we define the real vector space C^n (a,b) to consist of functions f([a,b] to R for which

• the nth derivative f^(n) of f exists for all points (a,b)
• f^(n) is continuous on (a,b)

We define vector space with n= infinity as that of functions that are infinitely differentiable on (a,b)
Clearly this is a subset of n=n which is a subset of n= n-1 … and of C(a,b) (which is a space of continuous functions on (a,b))

Question 24

Definition 5.6.2

TAYLOR COEFFICIENTS
AND. TAYLOR POLYNOMIAL..

Answer 24

Let f∈C^n (a,b) for some n∈N. fix x₀∈ (a,b). The real numbers f^(k) (x₀)/ k! For k=0,1,..,n are TAYLOR COEFFICIENTS of f at x₀.

We define function T^(n) _f ∈ C^n (a,b) by

T^(n) _f (x) = Σ from k=0 to n
Of [( f^(k) (x₀)) / k!] • (x-x₀)^k

This function is the TAYLOR POLYNOMIAL of f of degree n around x₀

Question 25

Theorem 5.6.3 Taylor’s theorem

Answer 25

Let f∈C^{n+1} (a,b) for some n∈N. fix x₀∈ (a,b).

Then for all x∈ (a,b),

f(x) = Σ from k=0 to n
Of
[( f^(k) (x₀)) / k!] • (x-x₀)^k + (R^{n+1}_f) (x)

Where R^{n+1}_f (x) + (f^{n+1}(c)) /( (n+1)!) • (x- x₀)^{n+1}, for some c depending on x with c ∈ ( x₀,x) if x > x₀ and c ∈ ( x,x₀) if x< x₀

The R term measures the error in approximating f by it’s Taylor polynomial degree n at f (remainder term of degree n+1) = f(x) - T^(n) _f (x)

If 0 in (a,b) can take x_0 as 0 Maclaurins

Question 26

f is represented by its Taylor series

Answer 26

For f in C^ infinity (a,b) and if series terms of Taylor’s converges for all x in (a,b) then we can write f(x) = this limit as n tends to infinity

Question 27

Examples we can check differentiability of functions

Answer 27

By the definition of the limit of differentiation. Showing it exists for all a in the domain of f.