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ECON 3620: Midterm Review
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Terms in this set (69)
(x^a)(x^b) = ?
x^a+b
1 / (x^a) = ?
x^-a
(x^a) / (x^b) = ?
x^a-b
(x^a)^b = ?
x^ab
(xy)^a = ?
(x^a)(y^a)
(x/y)^a = ?
(x^a) / (y^a)
sqr(x) = ?
x^(1/2)
sqr(b) of x = ?
x^(1/b)
sqr(b) of x^a = ?
x^(a/b) = (x^1/b)^a
x^(-ab)= ?
1 / (x^(a/b))
Simplify: (7x^3) +(5x^2) + (11x^3) - (9x^2) = ?
(18x^3) - (4x^2)
expand: (a + b)(c + d)
ab + ad + bc +bd
expand: (a + b)(c + d + e)
ac + ad + ae + bc + bd + be
What is the quadratic formula?
x = (-b ± √(b^2 - 4ac))/2a
(will get 2 answers)
Equilibrium in supply and demand analysis occurs when ...
Quantity supplied equals quantity demanded (Qs = Qd)
What does each variable in Y = C + I + G + (X - Z) represent?
Respectively; Income, consumption, investment, government expenditure, exports and imports.
What does this function represent? Y = C + I + G + (X - Z)
Income determination
Are matrices defined as rows by columns or columns by rows?
Rows by columns
What element would you look to for Matrix B, element B12
Row 1 column 2
Can you perform addition/subtraction on matrices of different sizes?
No. They must have the same dimensions.
How do you perform addition/subtraction between matrices?
You perform the operation between each corresponding element.
Define "scalar" multiplication
Multiplying each element of a matrix by the same, given, number.
Can you multiply matrices?
Yes. But only if they have inverse number of rows and columns. Example: Matrix A (2X3) and matrix B (3X2).
How do you perform vector multiplication?
You multiply the first item in a row, by the first item in a column, followed by the second in each, and so on. Summing each product.
Define commutative property
Meaning that the order of the operation doesn't matter
Is matrix addition commutative?
Yes it is
Is matrix multiplication commutative?
No. The order matters. AB does not equal BA
Is matrix multiplication distributive?
Yes. A(B+C) = AB +AC
What is an identity matrix?
A matrix with each element along the principal diagonal, from left to right, being 1. With the remaining elements being 0.
What does I(subscript)n mean?
Identity matrix square with dimensions n.
What happens if you multiply a matrix by a corresponding identity matrix?
Nothing. Identity matrix multiplications are the equivalent of linear algebra's 1 multiplier.
Define null matrix
A matrix, of any dimension, in which every element is 0.
How would the following system be represented via matrix?
7x + 3y = 45
4x + 5y = 29
A(2X2) = [7 3 / 4 5] -- X(2X1) = [x / y] -- B(2X1) = [45 / 29] where AX = B
What does |A| mean?
It means you're taking the determinant of Matrix A.
What is a second order determinant? Third order?
It is the determinant of a 2X2 matrix and 3X3 respectively.
If |A| = 0, what does this mean?
It means the system if singular, has linear dependence, and has no unique solutions possible.
How do you find |A| for a 2X2 matrix?
(a11
a22) - (a12
a21)
How do you find a third order determinant?
[a11(a22*a33 - a23*a32)] - [a12(a21
a33 - a23
a33 - a23
3 - a23*a32)] - [a12(a21*a33 - a23*a31)] + [a13(a21
a32 - a22
a31)]
You multiply the top three elements by the determinants of their corresponding lower "squares" that result from drawing a line down from the top element.
What is a minor?
The sub-matrix that results from the deletion steps taken place to determine a third order determinant.
What would |Msub(11)| mean?
The sub-matrix that results from the deletion process using element 1,1.
Define: Cofactor
A minor with its prescribed sign. (As denoted by the odd/even sum of the rows/columns.
Define: Laplace expansion
The sum of any outside row/column with their respective cofactor's equals the entire matrix determinant.
Define: Cofactor matrix
A matrix in which each element is replaced with its respective cofactor.
Define: Adjoint Matrix
A transpose of the given matrix. (where the row and column identifiers switch. ex: element 1,1 stays 1,1. Element 1,2 becomes element 2,1 ect.
How do you find the inverse of a matrix?
1. Find the determinant of your given matrix.
2. Find the cofactor matrix from your given matrix.
3. Take the transpose of the cofactor matrix.
4. Multiply the transpose by 1/the determinant from step 1.
How do you solve a system of equations using inverse?
1. Set the equations to standard matrix format.
2. Find the inverse
2. Multiply the inverse matrix by the solutions (B)
How do you solve a system of equations using Cramer's rule?
1. Find the determinant of your given matrix.
2. Create a new matrix for each variable by replacing the corresponding column with the B matrix.
3. Solve for determinant for each new matrix
4. Divide each new determinant by the original determinant.
What is the definition of a function being continuous?
If the f(x) lim as x > a exists and f(x) lim as x > a = f(x)
Define: Tangent line
The line that touches a point on a graph exactly once and represents the slope of that exact point. (Think, lie tangent to your curves"
Define: Secant line
The line that represents the average slop of a curve and crosses through two points.
What is the name of the equation that finds the slope of the secant line?
the difference quotient
What is the formula for the difference quotient?
(f(x1 + deltax) - f(x)) / delta x
What is the h form of the difference quotient?
function prime of x = lim h > 0 of ((f(x-h) - f(x)) / h
What are the requirements for a point on a line to be considered differentiable?
1. Derivative exists at that point
2. Continuous
3. Unique tangent line at the point
What is the derivative of a constant?
0
What is the derivative of y=mx+b?
m
What is the power rule?
from the equation fprime(x) = kx^n the derivative is equal to k
n
x^(n-1)
What is the derivative product rule?
fprime(x) of gh = ghprime +hgprime
What is the derivative quotient rule?
fprime(x) of g/h = (hgprime - ghprime) / h^2
What is the generalized power function rule?
fprime of (x) = n[g(x)]^(n-1) * gprime(x)
What is the formula for the chain rule?
dy/dx = dy/du * du/dx
What are the steps used in using the chain rule?
1. Break apart the function into being a nested function.
2. Take the derivative of each piece.
3. Multiply the two pieces.
4. Substitute the variable back in for uni-variable function.
How do you know if a function is increasing at a given point?
If the first derivative is greater than zero at a, then the function is increasing at a.
How do you know if a function is concave at a given point?
if the second derivative is less than 0 at a, then the function is concave at 0.
How do you find a relative maximum?
Find where the first derivative is zero and the second derivative is less than zero (convex)
How do you find an inflection point?
find where the second derivative is undefined, concavity changes at a, and the crosses its tangent line at x = a.
How would you find the relative min or max of a function?
1. Take the first derivative of the function.
2. Set equal to zero and solve for the critical points.
3. Take the second derivative of the function.
4. Evaluate the function at the critical points.
5. if < 0, relative maximum. if > 0 then relative minimum.
6. If the second derivative is zero, you continue taking the derivative until the function equals a non-zero. If the n of derivatives is odd, then the point is an inflection point and is not min/max. If even, then a negative number reflects a maximum and a positive number a minimum.
The marginal concept of any economic function can be expressed as the what?
The derivative of the total function.
Total revenue is equal to?
price times quantity
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