MathCamp98 class 4

Class 4.1 The chain rule.

Class 4.2 Partial derivatives.

Last time

Introduction and motivation for derivatives.

Maxima and minima.

Recap example.

The inventory problem:

Inventory costs are the ordering costs + the carrying costs.

A retailer expects 1200 cases of OJ to be sold in a year.

Each order costs $75.

It costs $8 to carry one case in inventory for 1 year.

Find the order size that minimizes the total inventory costs.

Call x the order size, and r the number of orders made in the year. Then

Minimize this cost ... differentiate and set equal to zero.

If sales goes up by a factor of 4, what happens to order size?

Today's class

Composition of functions

Take a function f(x), and another, g(t), then the composition of the functions is written as

Work out g(t) first, then whatever the answer is, put that into f(x).

Example:

First apply g, then apply f to g itself.

Say the number of labor hours to produce x units is

and costs depend on the driver D through the equation

then costs depends on units produced through the equation

We write

We want to find how the cost function changes with respect to units produced, that is to find

so we need to differentiate this composition of functions.

The chain rule.

Example: differentiate .

For the cost function the chain rule says

which equals

Alternative ways of presenting changes.

Straight lines and exponential functions.

The characteristic of a straight line is that a one unit change in x always gives rise to the same change in y (constant slope). That is the absolute change in y is a constant.

Consider the relative rate of change in a function, the slope divided by the value of a function.

For the exponential function , we have

which does not depend on x and is therefore a constant.

The exponential function is characterized by having a constant relative rate of change.

Another way of expressing changes is as percent change.

If we add a small quantity to x then the percent change is

Example: if we add 1 to 50, then we get 51 and the percent change is

If we change x by a little bit then we also change y, call the change in .

Look at the ratio of the percent changes:

As the change in x becomes small we call the ratio of this quantity the elasticity of y with respect to x.

Interpretation: if the elasticity of y with respect to x is a, then a 1% change in x results in an a% change in y.

We can re-express this elasticity as

which in the limit becomes:

Try the elasticity definition out on our simple cost function (find the elasticity of the cost with respect to the driver).

The elasticity is by definition

This special cost model is characterized by having a constant elasticity. In this model a 1% change in the driver always leads to a % change in the cost.

Example: Set equal to 0.5. Then the elasticity says that a 1% change in the driver results in a half percent change in the cost.

Set equal to 1 for simplicity.

Consider the costs when the driver equals 100. The cost equals Now increase the driver by 1%, that is consider the costs when the driver equal 101.

The costs equal , so that costs have risen by which is very close to the half a percent.

Partial derivatives

So far all functions have been of a single variable.

For many models the relationship may be involve more than a single variable:

Your propensity to shop at a store may be a function of both the product assortment at the store and the distance you live from it.

The effectiveness of a medical treatment may depend on the strength of the dosage and the age of the patient.

The fuel economy of a car may depend on the the size of the engine and the weight of the car.

The output of a production process may depend on the amount of labor and the amount of capital inputs.

These were all functions of two variables but you can have more.

We write a function of two variables as

Since the function depends on two variables it makes sense to ask how it changes as x changes or as y changes, so we have derivatives of the function with respect to x and with respect to y.

The simplest multi-variable relationship is the ``plane'':

The manipulation rules for derivatives of a multivariable function are the same as those for single variable functions.

To differentiate f(x,y) with respect to x, you treat y as a constant and proceed as before. The same goes for differentiating with respect to y (hold x constant).

These derivatives are called partial derivatives and are written

For the ``plane'' equation we have:

and

The interpretation of partial derivatives are that they give the slopes of the tangent plane in the x and y directions.

To find maxima/minima you calculate the partial derivatives, set both equal to zero and solve the resulting simultaneous equations.

Examples from the old exams.

A cost function for two products.

Say we produce 2 products, A and B. Each unit of A takes 3 labor hours to produce and each unit of B takes 2 labor hours to produce. We produce units of A and units of b. Then the total labor hours is

Say the cost driver is total labor hours, that is

Now use the usual cost function

What is the marginal cost with respect to product B?