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CALCULUS REVIEW

In this course we will encounter problems requiring calculus, especially dif-
ferentiation. I have prepared the following as a review of the material you have
learnt in a basic course on calculus.

1. Simple Power Functions

For example if a = 1, then . If a = 10, then :

2. Sums, Differences and Constants :
Example 1:

Example 2:

Note: The derivative of a constant is zero .

3. Product Rule

Note: The hi-ho rule can be used as an aid. The derivative of hi-ho equals,
hi dee ho plus ho dee hi. Here, hi is the function , ho is the function
and dee refers to taking the derivative.

Example:

Similarly, using the simple power function example where y = x9, we have:

We will be seeing a lot of functions where there maybe two different variables,
and , for example : Here, we will not be able to use the simple
power function rule as adding and is like adding apples and oranges.

So now we take the derivatives with respect to and separately, and for
each case we treat the other variable as a constant

, where is a constant

, where is a constant

4. Quotient Rule

Note: In alot of instances you will encounter y'(x): This is used instead of


Example 1:

Example 2:

5. Chain Rule
For example if we have : Here, we will need to introduce a new
variable, z:
So,

,where

Using the simple power function rule, we obtain

Using the rule for sums :

We have the derivative of y w.r.t z, and the derivative of z w.r.t x:

Substituting for z , we get

6. Logarithmic Functions
We have a simple logarithmic function y = ln x. The derivative, , is equal
to :(This can be considered as the derivative of x divided by x)
Example 1:
y = ln 2x,then : (The derivative of 2x divided by 2x)

Example 2:
y = ln x2, then :(The derivative of x2 divided by x2)

7. Solving Linear Equations
Economics requires solving systems of linear equations with unknown vari-
ables.
For example, if we have the following system of linear equations:

Equating x in terms of y : (You can also try the reverse)

To check our answers, input the two values into the system of linear equa-
tions:

Remember, calculus makes economics easier, not harder. Once you master
the concepts, you will realize how much fun economics can actually be.

8. Some Practice Questions:


Differentiate Q(K,L) with respect to K and L.

Answer:


Differentiate F(x, y) with respect to x and y:

Answer:

We will be covering pro…t maximization and cost minimization problems in
the course. Although, there are many ways to solve them, one of the most useful
tool is the Lagrange multiplier . Use the note below as a guide to help you solve
the problems.

A note on using Lagrange Multiplier

Maximize the utility function, U(x, y) = xy subject to a budget constraint

Using Lagrange multiplier, we can write the utility maximization problem
as follows:

x and y are quantities of goods, is the price of good x, and is the price
of good y, and m is the income level, and λ is the marginal utility of income.

In order to find our first order conditions, we would differentiate with respect
to x, y and λ and then solve the equations to obtain the utility maximizing
quantities of x and y:

Now substituting for λ, we can use from the first equation and plug
into the second equation to get
Substituting x of equation 3 with , we can get
and :

Thus our utility maximization bundle is :
Practice Problem:
Maximize utility subject to the following budget constraint
constraint: 400 - 4x - 4y

First order conditions:

Substituting for λ  we get:
Input the equation for λ into the 2nd equation:

We get

Now inputting into our 3rd equation, (the budget constraint)

We get 52x = 400


To check our answer:
So we maximize our utility with the following bundle of good x and y:


Good luck and welcome to Econ 100A

Prepared by: Aadil Nakhoda
 

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