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This week’s assignmentDid not materialize due to flu
It would have been busy work anyway just to get you used to going to the webpage to find material and begin working on your own
Next labI’ll give you instructions on Monday
Lab will be posted by 12 on Tuesday
Questions on discussion must be answered before the 1:20 timeframe to get credit
Some will be volunteer, some not…◦ The only wrong answers are 1) no response and
2) I don’t know. Take your best guess at the simplest short explanation and we’ll work from there…
Functions
A function f(.) takes numerical input and evaluates to a single value◦This is just a different notation◦Y = aX + bZ … is no different than◦f(X,Z) = aX + bZ
For some higher mathematics, the distinction may be more important
An implicit function like G(X,Y,Z)=0
Basic Calculusy=f(x)= x2 -2x + 4
◦This can be evaluated for any value of x
f(1) = 3f(2) = 4
We might be concerned with how y changes when x is changed◦When ∆X = 1, ∆Y = 1, starting from
the point (1,3)
An ExampleUnits Sold
Total Revenue
Total Cost
Change in
Revenue
Change in Cost
1 5 5.0 -- --
2 10 6.5 5.0 1.5
3 15 9.0 5.0 2.5
4 20 13.0 5.0 4.0
5 25 18.5 5.0 5.5
6 30 26.0 5.0 7.5
7 35 40 5.0 14.0
Differentiation (Derivative)
Instead of the average change from x=1 to x=2
Exact change from a tiny move away from the point x = 1◦We call this an instantaneous rate of
change◦Infinitesimal change in x leads to
what change in y?
Power rule for derivativesBasic rule
◦Lower the exponent by 1◦Multiply the term by the original
exponentIf f(x) = axb
Then f’(x) = bax(b-1)
E.g.◦f(x) = 6x3
◦f’(x) = 18x2
Applied Calculus: Optimization
If we have an objective of maximizing profits
Knowing the instantaneous rate of change means we know for any choice◦If profits are increasing◦If profits are decreasing◦If profits are neither increasing nor
decreasing
A Decision Maker’s InformationObjective is to maximize profits
by sales of product represented by Q and sold at a price P that the producer sets
1. Demand is linear2. P and Q are inversely related3. Consumers buy 10 units when
P=04. Consumers buy 5 units when
P=5
More information**Demand must be Q = 10 – P
The producer has fixed costs of 5The constant marginal cost of
producing Q is 3
More informationCost of producing Q (labeled C)**C = 5 + 3Q
So◦1) maximizing: profits◦2) choice: price level◦3) demand: Q = 10-P◦4) costs: C= 5+3Q
What next?
We need some economics and algebraDefinition of ‘Profit’?
How do we simplify this into something like the graph below?
p
Profits
Graphically the producer’s profit function looks like this
-40
-35
-30
-25
-20
-15
-10
-5
0
5
10
0 1 2 3 4 5 6 7 8 9 10
Pro
fits
Price Charged
Applied calculusSo, calculus will let us identify the
exact price to charge to make profits as large as possible
Take a derivative of the profit function
Solve it for zero (i.e. a flat tangent)That’s the price to charge given
the function
Relating this back to what you have learnedWe wrote a polynomial function
for profits and took its derivativeOur rule: Profits are maximized
when marginal profits are equal to zero
Profits = Revenue – Costs0 = Marginal Profits = MR – MC
◦Rewrite this and you have MR = MC