SECTION 9.6: Constrained Optimization

Graphical Approach: Maximizing Production Subject to a Budget Constraint

If f(x,y) has a global maximum or minimum on the constraint g(x,y) = c, it occurs at a point where the graph of the constraint is tangent to a contour of f, or at an endpoint of the constraint.

Analytical Approach: The Method of Legrange Multipliers

If P0 is a point satisfying the constraint g(x,y) = c,
the function f has a local maximum at P0 subject to the constraint if f(P0) >= f(P) for all points P near P0 satisfying the constraint.
the function f has a global maximum at P0 subject to the constraint if f(P0) >= f(P) for all points P satisfying the constraint.

In this particular reading, I was thoroughly confused by the Lagrangian multiplier and the Lagrangian function. I couldn not blog properly on the reading as I couldnt not understand the significance of most of it.

It is however relevant to economics because budget constraints play a crucial role in the development of a firm at microlevel or an entire country's economy at the macrolevel.

SECTION 4.3: Global Maxima and Minima

The technique/ procedure for finding the global maximum or minimum in a graph is known as optimization. Local maxima or minima occur when a function takes larger or smaler values than nearby points.

A function f is said to have a global maximum at a if f(x) is <= all values of f. It is said to have a global maximum at a if f(x)>= all values of f.

To find the global maximum or minimum of a continuous function on an interval from point a to b, one must compare the values of the function at all critical points in the interval as well as the endpoints, i.e., a and b.

To find the global maximum and minimum of a continuous function on an interval excluding endpoints or on the entire real line, one must find the values of the function at al critical points and then sketch a graph through them.

SECTION 9.5: Critical Points and Optimization

Optimization means finding the largest or smallest values of a function. When interpreting graphs, the method of optimization is used to determine local or global maxima and minima.

A function f is said to have a local maximum at P0 if f(P0) >= f(P) for all points P near P0.
It is said to have a local minimum at Po if f(Po) <= f(P) for all points P near P0.

As explained above, the function has a global maximum at P0 if f(P0) >= f(P) for all points P in R and it has a global minimum at P0 if f(P0) <= f(P) at all points P in R.

If a function f(x,y) has a local maximum or minimum at a point (x0, y0) not in the limits of the domain of f, then either fx (x0,y0) = 0 and fy (x0,y0) = 0 or at least one partial derivative is undefined at the points (x0, y0). Points where each of the partial derivatives is either zero or undefined are called critical points. This is how a critical point of a function can be found analytically.

The part of these two reading sections I found difficult to understand was exactly how one can determine whether a critical point is a local maximum or minimum. The example of the function in the textbook was to confusing.

It is evident that understanding how to determine maximum and minimum poitns whether local or global is useful in economics and the business world because companies need to know how to maximize profits and minimize costs. These can in effect be determined by studying trends on a graph.

SECTION 9.3: Partial Derivatives

For a function. the influence of x and y is studied seperately on the value f (x,y) by keeping one variable fixed and varying the other. For all points (a,b):

The partial derivative of f with respect to x at (a,b) is the derivative of f with y constant.

fx (a,b) = rate of change of f with fixed y fixed at b, at the point (a,b) = lim h-0 ((f(a+(h,b)) - (f(a,b)))/ h

The partial derivative of f with respect to y at (a,b) is the derivative of f with x constant:

fy (a,b) = rate of change of f with x fixed at a, at the point (a,b) = lim h-0 ((f(a,b) + h) - (f(a,b)))/h

There is an alternative notation for Partial Derivatives:

If z = f(x,y) the we can say that,

fx (x,y) = dz/dx and fy (x,y) = dz/dy, where the d is more curvy than a regular derivative.

Partial derivatives can be used to estimate values of functions during research projects etc.

SECTION 9.4: Computing Partial Derivatives Algebraically

The second-order partial derivatives of z = f(x,y)

d^2(z)/ dx^2 = fxx = (fx)x

d^2(z)/ dxdy = fyx = (fy)x

d^2(z)/ dydx = fxy = (fx)y

d^2(z)/ dy^2 = fyy = (fy)y

It is to be noted that the mixed partial derivatives are equal, i.e., if fxy and fyx are continuous at (a,b) then fxy (a,b) = fyx (a,b)

SECTION 3.5: Derivatives of Periodic Functions

It is important to note that since sine and cosine functions are periodic, there derivatives must also be periodic. 

Graphically, it has been proven that if x is the radius,

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

If differentiating a function within a function, assuming the inside function is z and the outside function is sin or cos,

the d/dx (sin z) = cos z * (dz/dx)  and  d/dx (cos z) = - sin z * (dz/dx)

If k is a constant then,

d/dx (sin kx) = k *cos (kx)  and  d/dx (cos kx) = - k*sin (kt)

Again it is difficult to understand exactly why the following functions are applicable without seeing the theoretical proof for the derivations. But it will definitely be useful to know when working with differentiation of functions.

SECTION 3.4: The Product and Quotients Rule

The Product Rule:

If u = f(x) and v= g(x), and they are both differential functions then

(fg)' = f ' g + g ' f

It can also be written as :

d (uv)/ dx = (du/dx)*v + (dv/dx)*u

It is to be noted that the derivative of a product is not simply the product of the derivatives of the two functions involved. 

The Quotient Rule:

If u = f(x) and v = g(x)

then (f/g) ' = (f ' g - g ' f)/ g^2

It can also be written as d/dx(u/v) = ((du/dx)*v - (dv/dx)*u)/(v^2)

The difficulty in understanding this section lies in the usage of the quotient rule and how it was derived as essentially the function could simply be converted into a product rather than a quotient.

Derivatives continue to be important in analyzing smaller and smaller everyday changes. For this reason, both the product rule and the quotient rule serve to simplify the understanding of derivatives in everyday applicability.