Calculus - DIW fileDifferentiation I Deﬁnition Let f : R !R. Its derivative with respect to a...

Calculus

Anton Velinov

DIW WS 2016

Takeaways

Differentiation: rules, implicit differentiationTaylor and power seriesMaximizationDefinite and indefinite integralsDifferential equationsMultivariate function limits and continuityPartial derivatives, gradient and directional derivativesComplex numbersImplicit function theorem

Anton Velinov Calculus 2 / 57

Differentiation I

DefinitionLet f : R→ R. Its derivative with respect to a point x is definedby the following limit

f ′(x) =dfdx

= lim∆→0

f (x + ∆)− f (x)

∆.

Class ExerciseUse the above definition to calculate the derivative of

f (x) = a + bx

andf (x) =

xx + 1

with respect to x .


Differentiation II

Useful Rulesdxk/dx = kxk−1

d(sin(x))/dx = cos(x) and d(cos(x))/dx = −sin(x)

d(ex )/dx = ex

d(log(x))/dx = 1/x


Differentiation III

TheoremIf a function f (x) is differentiable at some point x , then it iscontinuous at x .

differentiability⇒ continuity.

The converse is not true

continuity ; differentiability.

Class ExerciseShow that f (x) = |x − 2|+ 1 is continuous but not differentiableat x = 2.


Differentiation Rules I

Product RuleIf functions f (x) and g(x) are differentiable then their product isdifferentiable and given as follows:

(f (x)g(x))′ =d(f (x)g(x))

dx= f ′(x)g(x) + f (x)g′(x).

Quotient RuleIf functions f (x) and g(x) are differentiable then their quotient isdifferentiable and given as follows:(

f (x)

g(x)

)′=

d(f (x)/g(x))

dx=

f ′(x)g(x)− f (x)g′(x)

(g(x))2 .


Differentiation Rules II

ExampleProve the product rule. Hint: Add and subtract f (x + ∆)g(x).

Chain RuleFor the composite function g(x) = f (u(x)) or g(x) = (f ◦ u)(x)the derivative is given as follows:

dg(x)

dx=

df (·)du

du(x)

dx.

Class ExerciseUse the chain rule to find the derivative of

f (x) = 2x

with respect to x .


Implicit Differentiation I

It is not always possible to differentiate x explicitly from f (x). Asa direct consequence of the chain rule we can therefore makeuse of implicit differentiation

ExampleFind the derivative of

x2 + y2 = 1

with respect to x and compare this to the explicit solution.

Class ExerciseFind the derivative of

(x2 + y2)3 = 5x2y2

with respect to x .


Implicit Differentiation II

Class Exercise (optional)Find the derivative of

y = xx

with respect to x .


Higher Order Derivatives

DefinitionThe second derivative of f (x) is given by

f ′′(x) =d2f (x)

dx2 =ddx

[df (x)

dx

].

DefinitionIn general the k th-order derivative is the derivative of the(k − 1)th-order derivative.


Taylor Series Approximation

DefinitionAn r-th-order Taylor series approximation of a function f (x) atx = c + ∆ is as follows:

f (x) = f (c) +dfdx

∣∣∣∣∣x=c

·∆ +12!

d2fdx2

∣∣∣∣∣x=c

·∆2 +

+13!

d3fdx3

∣∣∣∣∣x=c

·∆3 + · · ·+ 1r !

d r fdx r

∣∣∣∣∣x=c

·∆r + R(c, x)

provided that all r derivatives of f (x) exist and are continuous ina neighbourhood of c.


First Order Taylor Series Approximation

Figure: First Order Approximation

C


Power Series I

DefinitionA Power series assumes that the remainder term, R(c, x) of theTaylor series goes to 0 as r →∞. It also centres the expansionat c = 0.

Class ExerciseFind the Power series of

f (x) = ex

evaluated at x = 1.


Power Series II

Class Exercise (Optional)Use the previous result to evaluate

limn→∞

(1 +

1n

)n

.


Maxima and Minima

DefinitionThe function f : X → R has a global maximum at x = c iff (x) ≤ f (c) for all x ∈ XThe function f : X → R has a local maximum at x = c iff (x) ≤ f (c) for all x in some open interval around x = c.Analogous definitions hold for global and local minima.

Class ExerciseWhat does the Extreme Value Theorem have to do with theabove definition?


Maximization I

Fermat’s TheoremIf a function f (x) has an extreme point at x = c and f ′(c) exists,then f ′(c) = 0.

This is a necessary condition for an interior maximum.

Class ExerciseFind an example where f ′(x) = 0 does not lead to a maximumor a minimum.


Maximization II

Second Derivative TestIf the second derivative of f (x) exists and if x = c is a criticalpoint of f (x) then:

If f ′′(x) < 0, f (x) has a local maximum at xIf f ′′(x) > 0, f (x) has a local minimum at xIf f ′′(x) = 0, the test is inconclusive.

Class ExerciseUse the second derivative test for the previous exercise.


Indefinite Integral

DefinitionInformally speaking, in case of a continuous function,integration, denoted by

∫·dx , is the inverse of differentiation:∫f ′(x)dx = f (x).

However, we assume that we only know f ′(x) and hence, whenintegrating a function we always add a constant, C to it. This iscalled an indefinite integral. It is over ±∞.

NoteIntegration is not quite the opposite of differentiation asfunctions that are non-differentiable can still be integrated.


Indefinite Integral Rules

Integration Rules∫xkdx = xk+1

k+1 + C for k 6= −1∫x−1dx =log(|x |) + C∫cos(x)dx = sin(x) + C∫sin(x)dx = −cos(x) + C∫eaxdx = (1/a)eax + C∫af (x)dx = a

∫f (x)dx for some constant a not depending

on x∫[af (x) + bg(x)]dx = a

∫f (x)dx + b

∫g(x)dx for some

constants a,b not depending on x .


Definite Integral

DefinitionFrom the fundamental theorem of calculus a definite integral isdefined as:∫ b

af (x)dx =

[∫f (x)dx

]∣∣∣∣∣x=b

−

[∫f (x)dx

]∣∣∣∣∣x=a

= F (b)− F (a).

Intuitively, it is the area under a function over the domain [a,b].

Class ExerciseFind the integral of ∫ b

ax

1b − a

dx .


Integration by Substitution

DefinitionFor the function that can be composed as f (g(x))g′(x), itsintegral over the domain a,b is given as∫ b

af (g(x))g′(x)dx =

∫ g(b)

g(a)f (u)du.

This is known as integration by substitution. This definitionapplies to indefinite integration analogously.

Class Exercise1 What is

∫x/(x2 + 1)dx?

2 What is∫

(x + 1)3dx?


Integration by Parts

DefinitionThrough the product rule we can derive the following formulafor integration by parts∫

f (x)g′(x)dx = f (x)g(x)−∫

f ′(x)g(x)dx .

Class ExerciseFind the integral of ln x .


Differential Equations I

DefinitionA differential equation is an equation that involves an unknownfunction as its solution. It usually involves a general and aparticular solution given some initial conditions.

Intuitively, we are given the nth order derivatives of a function inan equation and we need to solve this in order to find theoriginal function.In economics differential equations are mainly used ingrowth theory

ExampleFind the solution to

dydx

=√

x/y

given the initial condition y = 4, x = 1.


Differential Equations II

Class ExerciseFind the solution to

dydx

= x/y

given the initial condition y = 1, x = 1.

DefinitionThe previous two exercises deal with separable differentialequations, i.e. when the xs and ys can be separated on bothsides of the equation.


Homogeneous Differential Equations (Optional)

DefinitionAn equation of the form

dydx

= f (x , y)

is referred to as a homogeneous differential equation. If the xsand ys are not separable, v = y/x is used to solve theequation.

Class ExerciseFind the solution to

1dydx = x+y

x

2dydx = x2+3y2

2xy (optional).


Multivariate Function Limits I

DefinitionA multivariate or multivariable function, f has n > 1 argumentsmapping onto R, i.e. f : Rn → R. Written as f (x1, x2, . . . , xn) orf (x).

Multivariate function limitIn Euclidean space f has a limit L as x approaches x0 if

∀ ε > 0 ∃ δ(ε) > 0; |f (x)−L| < εwhenever 0 < ‖x−x0‖2 < δ(ε).

We write f (x)→ L as x→ x0 or limx→x0 f (x) = L.

NoteUnlike the univariate case, there are an infinite number ofdirections for x to approach x0.


Multivariate Function Limits II

Class ExerciseDetermine whether the following limits exist

1 lim(x ,y)→(0,0)x2

x2+y2

2 lim(x ,y)→(0,0)xy

x2+y2 .


Multivariate Function Limits III

Class ExerciseShow that the following limit exists

lim(x ,y)→(0,0)

x2yx2 + y2 .


Continuity of Multivariate Functions

Epsilon-Delta Definition

A function f : Rn → R is said to be continuous at x0 ∈ Rn if

∀ ε > 0 ∃ δ(ε,x0) > 0 ; |f (x)− f (x0)| < ε

whenever 0 < ‖x− x0‖2 < δ(ε,x0)

Limit DefinitionA function f : Rn → R is said to be continuous at x0 ∈ Rn if

limx→x0

f (x) = f (x0).

TheoremPolynomial functions are continuous everywhere. Rationalfunctions are continuous everywhere they are defined.


Partial Derivatives I

DefinitionFor the function f : Rn → R, its partial derivative w.r.t. xi isdefined as

fxi =∂f∂xi

= lim∆→0

[f (x1, x2, . . . , xi−1, xi + ∆, xi+1, . . . , xn)

− f (x1, x2, . . . , xi−1, xi , xi+1, . . . , xn)]/∆.

Class ExerciseFind the partial derivatives with respect to x and y for thefollowing function

f (x , y) = x4 + 6√

y + 10.


Partial Derivatives II


Gradient I

DefinitionThe column vector of partial derivatives of the n dimensionalvector valued function f (x) is known as the gradient (denoted∇) of f :

∂f (x)/∂x = ∇ =

∂f (x)/∂x1∂f (x)/∂x2

...∂f (x)/∂xn

=∂f (x)

∂x1e1+

∂f (x)

∂x2e2+· · ·+∂f (x)

∂xnen,

where the eis are orthogonal unit vectors.

Geometric InterpretationThe gradient is always orthogonal to the level set f (x) = k . Itgives the direction of the steepest ascent (in Rn).


Gradient II


Directional Derivatives I

DefinitionPartial derivatives examine the rate of change when allowingone variable to change while the rest are kept constant.Directional derivatives examine the rate of change whenallowing all of a function’s variables to vary. For the functionf : Rn → R the directional derivative with the direction of therate of change denoted by the n dimensional vectoru = [u1 u2 . . . un]′ is given by

Duf (x1, x2, . . . , xn) = lim∆→0

[f (x1 + u1∆, x2 + u2∆, . . . , xn + un∆)

− f (x1, x2, . . . , xn)]/∆


Directional Derivatives II

The limit in the above definition is sometimes difficult tocalculate so an equivalent formula based on the partialderivative is often used

Duf (x1, x2, . . . , xn) = (∂f/∂x1)u1 + (∂f/∂x2)u2 + · · ·+ (∂f/∂xn)un

=n∑

i=1

(∂f/∂xi)ui =n∑

i=1

fxi ui .

Notice that this can be expressed in terms of the gradientvector as

Duf (x) = ∇ · uwhere · denotes the dot product.

ExampleShow that the above formula is equivalent to the one on theprevious slide. Hint: Define the univariate functiong(z) = f (x0

1 + u1z, x02 + u2z, . . . , x0

n + unz) and take itsderivative at z = 0.


Directional Derivatives III

The u VectorNote that the vector u can describe a rate of change in anyinfinite amount of directions. It is normalized to a unit length as

‖uN‖2 =

√√√√ n∑i=1

u2iN = 1,

whereuN =

u‖u‖2

,

and ‖ · ‖2 is the L− 2 vector norm.


The Gradient and the Directional Derivative

NoteUnlike a directional derivative, the gradient is not a scalar but avector.

Class ExerciseFor the function

f (x , y , z) = x2z + y3z2 − xyz

1 Calculate the gradient2 Determine the directional derivative in the direction of

u = [−1 0 3]′

3 Would the directional derivative go in the steepestdirection?


Hessian I

DefinitionThe second derivative of the n dimensional vector valuedfunction f (x) is given by:

∂2f (x)

∂xi∂xj∀i , j = 1, . . . ,n.

These derivatives can be grouped in a matrix known as theHessian matrix, H:

H =∂f (x)

∂x∂x′=

∂2f/∂x1∂x1 ∂2f/∂x1∂x2 · · · ∂2f/∂x1∂xn∂2f/∂x2∂x1 ∂2f/∂x2∂x2 · · · ∂2f/∂x2∂xn

...... · · ·

...∂2f/∂xn∂x1 ∂2f/∂xn∂x2 · · · ∂2f/∂xn∂xn

.Since the order of differentiation does not matter, H is asymmetric matrix.


Hessian II

Second Derivative TestIf x = c ∈ Rn is a critical point of f (x) and if the matrix H exists,then it can be used as a second derivative test in determininglocal optima as follows:

If H is positive definite, then f (·) attains a local minimum atcIf H is negative definite, then f (·) attains a local maximumat cIf the Eigenvalues of H alternate in sign, then f (·) has asaddle point at c.


Jacobian and the Vector-Valued Function

DefinitionSuppose we have the following vector-valued functionf : Rn → Rm, that is we have fi(x1, x2, . . . , xn) = fi(x),x ∈ Rn fori = 1, . . . ,m. Collect all the m functions in an (m × 1) columnvector

f (x) = [f1(x) f2(x) · · · fm(x)]′,

then the derivative of f (x) w.r.t. x is called the Jacobian matrix:

J =∂f (x)

∂x′=

∂f1/∂x1 ∂f1/∂x2 · · · ∂f1/∂xn∂f2/∂x1 ∂f2/∂x2 · · · ∂f2/∂xn

...... · · ·

...∂fm/∂x1 ∂fm/∂x2 · · · ∂fm/∂xn

.

NoteBoth H and J are sometimes defined as their respectivedeterminants.Anton Velinov Calculus 40 / 57

Matrix Differentiation Rules

Matrix Differentiation RulesFor the (n × 1) vector a we have ∂a′x/∂x = ∂x′a/∂x = aFor the (m × n) matrix A we have ∂Ax/∂x′ = AFor the (n × n) matrix C, the quadratic form x′Cx has thederivative ∂x′Cx/∂x = (C′ + C)x, which is 2Cx if C issymmetricFor the (n × n) matrix C, ∂C−1/∂x = −C−1(∂C/∂x)C−1

Class ExerciseShow that the first two points hold.


Multiple Integrals

DefinitionMultiple integrals are of the form∫ ∫

· · ·∫

f (x1, x2, . . . , xn)dx1dx2 · · · dxn.

They can be solved by integrating x1 out, then x2 and so on.Note however that the order of integration does not matter.

Class ExerciseSolve ∫ 2

0

∫ 2

1e−(x+y)dxdy

by integrating out x and y and then y and x . Compare youranswers.


Logarithms in Economics

DefinitionThe natural logarithm is the inverse of e

ln(ex ) = x .

For x > 0 it also follows that

x = eln(x).

In economics texts this is usually denoted as log instead of ln.

Class ExerciseFrom the above definition show that

ln(xa) = a ln(x).


Logarithms as Percentages

Logarithms as PercentagesIn economics the logarithm is often used to describe apercentage. For some small ∆

ln(1 + ∆) ≈ ∆.

Class ExerciseShow that this holds by taking a first-order Taylor expansion ofthe function f (x) = ln(x) around c = 1 evaluated at x = 1.


Complex Numbers C I

DefinitionA complex number, c is written in the form of

c = a + bi

where a and b are real numbers and i is an imaginary numbersuch that i =

√−1.

The Set CThe real number set is a subset of the complex numberset, R ⊂ CC is a field obeying the field axioms (since R is also a field)C however does not satisfy the order axioms, unlike R.


Complex Numbers C II

Figure: Complex Circle

Imaginary axis (i)

Real axis (a)

a+bi

bi

a

R

θ


Complex Numbers C III

DefinitionA complex conjugate, of a complex number a + bi is given by

a− bi .

Both numbers are known as a complex conjugate pair. Additionand multiplication of complex conjugate pairs produce realnumbers.

DefinitionThe modulus or absolute value of a complex number, is itsdistance from the origin as measured by the L− 2 norm

R ≡ ‖a + bi‖2 =√

a2 + b2.


Complex Numbers C IV

Polar CoordinatesA complex number can also be written in Polar coordinate form.Note that

cos(θ) = a/R

andsin(θ) = b/R

hence,a + bi = R[cos(θ) + i · sin(θ)].


Complex Numbers C V

Exponent FormUsing the Power series it is also possible to write a complexnumber as

a + bi = Reiθ.

ExampleVerify the above result by taking the Power series of f (x) =sin(x) and f (x) = cos(x). Hint: Also take the Power series ofex evaluated at x = iθ and plug in all the results in the Polarcoordinate form.


L’ Hôpital’s Rule

DefinitionLet f (·) and g(·) be functions, which are differentiable on theopen interval set X . Then for some point x0 ∈ X if

limx→x0 f (x) = 0 and limx→x0 g(x) = 0 and limx→x0f ′(x)g′(x)

exists or iflimx→x0 f (x) = ±∞ and limx→x0 g(x) = ±∞ andlimx→x0

f ′(x)g′(x) exists

then limx→x0f (x)g(x) = limx→x0

f ′(x)g′(x) .

Class ExerciseUse L’ Hôpital’s Rule to evaluate

limx→0

f (x)

g(x)= lim

x→0

2 sin(x)− sin(2x)

x − sin(x).


Homogenous Functions

DefinitionThe function f (x1, x2, . . . , xn) is homogenous of degree r ∈ R iffor any t > 0

f (tx1, tx2, . . . , txn) = t r f (x1, x2, . . . , xn).

Such functions are commonly used in economics. A functionthat is homogenous of degree 1 can be manipulated as

f (1, x2/x1, . . . , xn/x1) = (1/x1)f (x1, x2, . . . , xn).

Class Exercise1 Give an example of a homogenous function in economics.2 Show that f (x1, x2) =

√x1x2 is homogenous of degree 1.


Implicit Function Theorem I

SettingSuppose we have a system of n functions fi : Rn+m → R, whichdepend on n ’parameters’ x = [x1 x2 · · · xn]′ and m’endogenous’ variables y = [y1 y2 · · · ym]′, i.e. fi(x,y),where x ∈ X ⊂ Rn and y ∈ Y ⊂ Rm. Further, suppose that

fi(x,y) = 0 i = 1, . . . ,n. (1)

Assume the vectors x0 ∈ X and y0 ∈ Y satisfy (1).

GoalWe want to solve for y as a function of x locally around x0 andy0.


Implicit Function Theorem II

DefinitionEquation (1) can be solved locally around (x0,y0) so that y is afunction of x if there are open neighbourhoods of x0 and y0

such that

fi(x, (g1(x),g2(x), . . . ,gn(x))) = 0, i = 1, . . . ,n, (2)

whereyi = gi(x), i = 1, . . . ,n,

provided that x is in its open neighbourhood. gi(x) are know asunique implicit functions.

DefinitionAn open neighbourhood of x ∈ Rn is defined as

{x0 ∈ Rn : ‖x0 − x‖2 < ε}.Anton Velinov Calculus 53 / 57

Implicit Function Theorem III

Implicit Function TheoremSuppose all n functions are continuously partially differentiablew.r.t. their n + m variables. A solution of the type of (2),involving unique implicit functions for the endogenous variables,is possible provided that

|Jy|(x=x0,y=y0) 6= 0.

Further, the effect of a change of x on y at (x0,y0) can beexpressed as

∂g(x)

∂x′= −

[∂f (x,y)

∂y

]−1∂f (x,y)

∂x,

evaluated at (x0,y0).


Implicit Function Theorem IV

Intuitively, the IFT provides a sufficient condition for theexistence of unique solutions (equilibria) and gives us thefirst-order comparative static effects of such a solution.

Example

For the unit circle, x2 + y2 = 1⇔ f (x , y) = x2 + y2 − 1 = 0 canwe solve uniquely for x in terms of y at the point(x0, y0) = (1,0) s.t. f (x ,g(x)) = 0?

Class ExerciseCan we do this at a point (x0, y0) where y0 > 0 or wherey0 < 0?

Class ExerciseTo illustrate that the IFT only gives a sufficient condition for theexistence of unique solutions solve f (x , y) = y3 − x = 0 for y interms of x at the point (x0, y0) = (0,0).


Inverse Function Theorem

Inverse Function TheoremWhen the number of parameters are the same as the numberof endogenous variables, m = n we have that

fi(x,y) = gi(x)− yi , i = 1, . . . ,n.

which is known as the Inverse Function Theorem.

This is a special case of the IFT.


End of Theme 4

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Calculus - DIW fileDifferentiation I Deﬁnition Let f : R !R. Its derivative with respect to a...

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