Math Stuff Flashcards
what is a square root as an exponent?
X^(1/2)
What is 1/x as an exponent?
x^-1
How do we simplify x^a/x^b
=x^(a-b)
What does concave mean?
Peaks at the top, open at the bottom like n
What does convex mean?
Shaped like a U
Expand (a+b)^2
= a^2+2ab+b^2
expand (a-b)^2
a^2-2ab+b^2
Expand (a+b)(a-b)
= a^2-b^2
Expand (1-b^2)
= (1+b)(1-b)
(n-1)h
What is the power rule?
f(x)=x^n becomes f’(x)=nx^(n-1)
What is the constant rule?
The derivative of a constant is zero.
What is the derivative of the sum or difference of two functions?
The sum or difference of their derivatives.
Use the product rule to find the derivative of
u(x)v(x)
u(x)v’(x)+u’(x)v(x)
Use the quotient rule to find the derivative of
u(x)/v(x)
[u’(x)v(x)-u(x)v’(x)]/[v(x)]^2
Use the chain rule to find the derivative of
y(u(x))
(dy/du)*(du/dx)
If the derivative of a function is negative, will the curve be concave or convex?
Concave for a negative.
If the derivative of a function is positive, will the associated curve be concave or convex?
Convex for positive.
What does ln mean?
Natural Logarithm
What is a logarithm?
Definition: The logarithm of a positive real number x to the base b, denoted as log_b(x), is the exponent to which the base b must be raised to obtain the number x. In mathematical notation:
log_b(x) = y if and only if b^y = x
When the base of a log/ ln is not specified, what do we assume it to be?
for ln: e or 2.71828
for log: 10
ln(1) = ?
ln(1) = 0
ln(e)=?
ln(e)=1
What is the product rule with regards to ln?
ln(xy) = ln(x) + ln(y)
What is the quotient (division) rule for ln?
ln(x/y) = ln(x) - ln(y)
What is the change of base formula with regards to both ln and log?
log_b(x) = (log_c(x)) / (log_c(b))
same for ln
Use the chain rule to find the derivative of
y(u(x))
(dy/du)*(du/dx)
e = ?
e = 2.71828
What does this symbol mean?
Nash Equilibrium
If we have the function Y=H^(1/2) then what can we say about the relationship of Y to H?
The rate at which Y increases will drop as H gets larger - i.e. decreasing returns to scale.
What does FOC mean?
First Order Condition, which is where we set the first derivative to zero, and there may be other constraints on the equation. It specifically has to do with optimization.
Is e/2 a term or a constant for the purposes of a derivative of f?
a constant, and therefore gets eliminated.
Is a^2ez a constant or a term for the purposes of a derivative of f?
This would be a term and should remain.
