# 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
= x^{9}, 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 x^{2}, then :(The derivative of
x^{2}
divided by x^{2})

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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