AIOU Hub: Mathematics for Economists (803) - Spring 2023

Mathematics for Economists (803)

Q.1 Explain the different types of equations with examples.

Equations play a crucial role in mathematics and various scientific disciplines. They represent mathematical relationships between variables and provide a means to solve for unknown quantities. Equations can be classified into different types based on their characteristics and the methods used to solve them. Here are some of the major types of equations along with examples:

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1. Linear Equations:

Linear equations are the simplest type of equations where the highest power of the variable is 1. They can be expressed in the form: ax + b = 0, where 'a' and 'b' are constants and 'x' is the variable. Examples of linear equations include:

1- 2x + 3 = 7

2 - 4y - 5 = 3y + 7

3 - 3z - 2 = z + 5

2. Quadratic Equations:

Quadratic equations involve variables raised to the power of 2. They can be written in the standard form: ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants. Examples of quadratic equations are:

1- x^2 - 9 = 0

2 - 2y^2 + 5y - 3 = 0

3 - 3z^2 - 2z + 1 = 0

3. Polynomial Equations:

Polynomial equations consist of variables raised to positive integer powers. They can have multiple terms and can be written in various forms. Examples of polynomial equations include:

- 2x^3 + 3x^2 - 4x + 1 = 0

- 4y^4 - 6y^2 + 2 = 0

- 5z^5 + z^3 - 2z^2 + 1 = 0

4. Exponential Equations:

Exponential equations involve variables appearing as exponents. They can be solved using logarithms or other techniques. Examples of exponential equations are:

- 2^x = 16

- e^y = 10

- 3^(2z) = 81

5. Logarithmic Equations:

Logarithmic equations involve logarithmic functions. They can be solved by applying logarithm rules and properties. Examples of logarithmic equations include:

- log(x) = 2

- ln(y) = 1

- log(base 4)(z) = 3

6. Trigonometric Equations:

Trigonometric equations involve trigonometric functions such as sine, cosine, or tangent. They are solved using trigonometric identities and properties. Examples of trigonometric equations are:

- sin(x) = 0.5

- cos(y) = -0.7

- tan(z) = 1

7. System of Equations:

A system of equations consists of multiple equations with the same set of variables. They are solved simultaneously to find the values of the variables that satisfy all the equations. Examples of systems of equations include:

- 2x + 3y = 7, 4x - y = 2

- 3x + y = 5, 2x - 2y = -4

- x + 2y - z = 3, 2x + y + 3z = 10, x - y + 4z = -1

These are some of the major types of equations found in mathematics and science. Each type has its own methods and techniques for solving, and they are used in various applications across different fields.

Q.2 Let the demand and supply functions are as follows.

a) Qd = 15 - 4P b) Qd = 30 — 6P

Qs = 6P — 1 Qs = -6 + 8P

Find equilibrium price and quantity

To find the equilibrium price and quantity, we need to set the demand (Qd) equal to the supply (Qs) and solve for the price (P) at that point. Let's solve the equations step by step:

a) Qd = 15 - 4P

Qs = 6P - 1

Setting Qd equal to Qs:

15 - 4P = 6P - 1

First, let's simplify the equation by combining like terms:

15 + 1 = 6P + 4P

16 = 10P

Divide both sides of the equation by 10 to isolate P:

P = 16/10

P = 1.6

Now that we have the equilibrium price, we can substitute it back into either the demand or supply equation to find the equilibrium quantity. Let's use the demand equation:

Qd = 15 - 4P

Qd = 15 - 4(1.6)

Qd = 15 - 6.4

Qd = 8.6

Therefore, the equilibrium price is 1.6 and the equilibrium quantity is 8.6.

b) Qd = 30 - 6P

Qs = -6 + 8P

Setting Qd equal to Qs:

30 - 6P = -6 + 8P

Let's simplify the equation by combining like terms:

30 + 6 = 8P + 6P

36 = 14P

Divide both sides of the equation by 14 to isolate P:

P = 36/14

P = 2.57 (rounded to two decimal places)

Now, substitute the equilibrium price back into either the demand or supply equation to find the equilibrium quantity. Let's use the supply equation:

Qs = -6 + 8P

Qs = -6 + 8(2.57)

Qs = -6 + 20.56

Qs = 14.56

Therefore, the equilibrium price is approximately 2.57 and the equilibrium quantity is approximately 14.56.

In summary, for the given demand and supply functions:

a) The equilibrium price is 1.6 and the equilibrium quantity is 8.6.

b) The equilibrium price is approximately 2.57 and the equilibrium quantity is approximately 14.56.

Q.3 Using Cramer’s rule, solve the following equation. (20)

2x+3y+z=l

5x+2y+z =l

To solve the given system of equations using Cramer's rule, we first need to determine the determinant of the coefficient matrix and the determinants of the matrices obtained by replacing each column of the coefficient matrix with the constants from the right-hand side of the equations.

The system of equations can be written in matrix form as:

| 2 3 1 | | x | | l |

| 5 2 1 | | y | = | l |

Let's calculate the determinants required for Cramer's rule.

1. Determinant of the coefficient matrix (D):

D = | 2 3 1 |

| 5 2 1 |

| 0 0 1 |

D = (2 * 2 * 1) + (3 * 1 * 0) + (1 * 5 * 0) - (1 * 2 * 0) - (3 * 5 * 1) - (2 * 1 * 0)

D = 4 + 0 + 0 - 0 - 15 - 0

D = -11

2. Determinant obtained by replacing the first column (Dx):

Dx = | l 3 1 |

| l 2 1 |

| l 0 1 |

Dx = (l * 2 * 1) + (3 * 1 * l) + (1 * l * 0) - (1 * 2 * l) - (3 * l * 1) - (2 * 1 * 0)

Dx = 2l + 3l + 0 - 2l - 3l - 0

Dx = 0

3. Determinant obtained by replacing the second column (Dy):

Dy = | 2 l 1 |

| 5 l 1 |

| 0 l 1 |

Dy = (2 * l * 1) + (l * 1 * 0) + (1 * 5 * l) - (1 * l * 0) - (5 * 2 * 1) - (l * 1 * 0)

Dy = 2l + 0 + 5l - 0 - 10 - 0

Dy = 7l - 10

4. Determinant obtained by replacing the third column (Dz):

Dz = | 2 3 l |

| 5 2 l |

| 0 0 l |

Dz = (2 * 2 * l) + (3 * l * 0) + (l * 5 * 0) - (l * 2 * 0) - (2 * 5 * l) - (3 * 0 * l)

Dz = 4l + 0 + 0 - 0 - 10l - 0

Dz = -6l

Now, we can find the values of x, y, and z using Cramer's rule:

x = Dx / D

y = Dy / D

z = Dz / D

x = 0 / -11

x = 0

y = (7l - 10) / -11

z = (-6l) / -11

z = (6l) / 11

Therefore, the solution to the system of equations is:

x = 0

y = (7l

- 10) / -11

z = (6l) / 11

Note: The solution depends on the parameter l, and the values of y and z are expressed in terms of l.

Q. 4 Write a detailed note on the Jacobian determinants.

The Jacobian determinant, also known as the Jacobian, is a mathematical concept used in multivariable calculus and differential geometry. It is a determinant associated with a transformation from one coordinate system to another. The Jacobian provides essential information about the transformation, such as how it stretches, compresses, or rotates space.

Let's dive into a detailed explanation of the Jacobian determinant and its significance:

1. Definition:

The Jacobian determinant is a scalar value computed from the partial derivatives of a set of functions with respect to a set of variables. It describes the local behavior of a coordinate transformation. For a transformation from n variables to m variables, the Jacobian determinant is an m × m determinant denoted as |J|.

2. Calculation:

To compute the Jacobian determinant, we need to construct a matrix called the Jacobian matrix. The Jacobian matrix consists of the partial derivatives of the output variables with respect to the input variables. Let's consider a transformation from variables u1, u2, ..., un to variables v1, v2, ..., vm. The Jacobian matrix J is defined as:

J = | ∂v1/∂u1 ∂v1/∂u2 ... ∂v1/∂un |

| ∂v2/∂u1 ∂v2/∂u2 ... ∂v2/∂un |

| ... ... ... |

| ∂vm/∂u1 ∂vm/∂u2 ... ∂vm/∂un |

The Jacobian determinant |J| is computed by taking the determinant of the Jacobian matrix.

3. Geometric Interpretation:

The Jacobian determinant provides information about the local behavior of a transformation. It captures how the transformation distorts the shape and volume of the space. The sign and magnitude of the Jacobian determinant determine whether the transformation expands or contracts space in different directions.

Positive Jacobian determinant: The transformation preserves orientation and expands space.

Negative Jacobian determinant: The transformation reverses orientation and compresses space.

Zero Jacobian determinant: The transformation collapses space.

4. Applications:

The Jacobian determinant has various applications in mathematics, physics, and engineering:

Change of Variables: When integrating functions in different coordinate systems, the Jacobian determinant is used to convert between the integrals in one coordinate system to another.

Multivariable Calculus: The Jacobian determinant appears in the transformation of partial derivatives when performing coordinate transformations, such as the chain rule.

Differential Geometry: The Jacobian determinant is fundamental in studying coordinate transformations, calculating tangent vectors, surface area, volume elements, and defining important geometric concepts.

Optimization and Solving Systems of Equations: The Jacobian determinant is used in optimization algorithms and solving systems of nonlinear equations, such as Newton's method.

In summary, the Jacobian determinant plays a crucial role in understanding the local behavior of coordinate transformations. It provides valuable information about stretching, compressing, and rotating space. The sign and magnitude of the Jacobian determinant determine the orientation and volume changes induced by the transformation. Its applications extend across various fields, including calculus, geometry, physics, and engineering.

Q. 5 The given implicit function is :

X2+3xy + 2yz + y 2 +z 2 — 11=0

Find dy/dx when x=l and y=2.

1. Linear Equations:

2x + 3 = 7

4y - 5 = 3y + 7

3z - 2 = z + 5

2. Quadratic Equations:

x^2 - 9 = 0

2y^2 + 5y - 3 = 0

3z^2 - 2z + 1 = 0

3. Polynomial Equations:

Polynomial equations consist of variables raised to positive integer powers. They can have multiple terms and can be written in various forms. Examples of polynomial equations include:

2x^3 + 3x^2 - 4x + 1 = 0

4y^4 - 6y^2 + 2 = 0

5z^5 + z^3 - 2z^2 + 1 = 0

4. Exponential Equations:

Exponential equations involve variables appearing as exponents. They can be solved using logarithms or other techniques. Examples of exponential equations are:

2^x = 16

e^y = 10

3^(2z) = 81