FYBCOM Sem 1 Basic Tools in Economics Chapter 7 Notes 2024 | FYBAF | FYBMS – Mumbai University

FYBCOM Sem 1 Basic Tools in Economics Chapter 7 Notes 2024 – Lines and Curves

FYBCOM Sem 1 Basic Tools in Economics Chapter 7 Notes is Lines and Curves.
In this chapter A line depicts the underlying functional relationship between two variables. The line only suggests but does not prove the underlying relationship between the variables. The line is called a curve in economics. A curve can be linear (straight line) or non-linear, depending on the data used.

The notes we provide for FYBCOM students are not tied to a certain course. Students pursuing FYBAF and FYBMS study similar subjects in their first semester, such as Basic Economics Tools, so these notes are equally beneficial to them. Our detailed notes are meant to make it easier for all students in these courses to understand the important concepts. Whether you are in FYBCOM, FYBAF, or FYBMS, these notes will help you succeed academically.

Q.1 A) Select the most appropriate alternative and rewrite the statement:

1) A ________ equation is a function of one variable in which the highest exponent of a variable is 1.
a) quadratic
b) cubic
c) linear
d) exponential

2) Identify the general form of a linear equation form the options given:
a) y = b + mx
b) y = a
c) y = a + bx + cx²
d) y = a + bx + cx² + dx³

3) Which of the following general form of equations represents the total fixed cost function?
a) x = b
b) x = 0
c) y = a
d) y = 0

4) The equation x = b will derive a slope with
a) positive slope
b) zero slope
c) infinite slope
d) negative slope

5) In case of a perfectly in elastic supply curve the slope is
a) m = 1
b) m = 0
c) m < 0
d) m = ∞

6) In economics a/ an ______ is used to express the equation y = x
a) identity line
b) vertical line
c) horizontal line
d) parabola

7) Choose the correct statement:
a) The slope of a linear curve demand curve is < 0
b) The slope of a non-linear curve demand curve is > 0.
c) The slope of an identity line is > 0.
d) The slope of a non-linear curve is a constant.

8) In the quadratic equation, y = a + bx + ex², if c is negative, the resulting curve will be
a) a continuous downward sloping curve
b) a continuous upward sloping curve
c) U-shaped curve
d) inverted U-shaped curve

Q.1 B) State whether the following statements are true or false.

1. A line or a curve proves the underlying relationship between the variables. – False
2. A linear equation is a function of one variable in which the highest exponent of a variable is 1. – True
3. In a linear equation the variables are multiplied to each other. – False
4. In the equation y = b + mx, m is the slope of the line. – True
5. Direct functional relationships are represented by upward sloping curves. – True
6. Quadratic functions give rise S shaped curves. – True

Answer the following question

1) Explain with the help of diagrams, how linear functions give rise to straight line curves.

Answer: A linear equation is a function of one variable in which the highest exponent of a variable is 1 and the variables are not multiplied by each other. Linear equations consist of only sums of variables and variables which are multiplied by a constant, with no products of variables. All of the variables in the equation must be raised to the power of one.

Definition of Linear Functions

A linear function is a mathematical function that can be expressed in the form: y = b + mx

where:

• y is the dependent variable
• x is the independent variable
• m is the slope of the line (indicating the rate of change),
• b is the y-intercept (the value of y when x = 0).

Characteristics:

The highest exponent of the variable x is 1
The graph of a linear function is a straight line

Graphical Representation:

Straight Line Curves
When we graph a linear function, it produces a straight line. the slope m determines the angle at which this line inclines or decline:

1. Lines with Positive Slope: f m > 0, the line rises from left to right, indicating a direct relationship between x and y. For example, the equation y = 10 + 2x:
   Slope: +2
   Y-intercept: 10

1. Lines with Negative Slope: If m < 0, the line falls from left to right, indicating an inverse relationship. For example, in the equation y = 10 – 7x:
   Slope: -7
   Y-intercept: 10

Key Points on Straight Lines

• The slope (m) is constant across all points on the line.
• The y-intercept (b) where the line crosses the y-axis.

2) Discuss the lines representing constant functions.

Answer: The simplest type of real-valued function is a constant function. A constant function is a function which takes the same value for f (x) value of x is. A general constant function is: expressed as f (x) = c, where c is some unspecified constant.

Definition and Characteristics

General Form: The general expression for a constant function can be represented as f (x) = c, where c is a constant number. This means that for any input x, the output will always be c.

Graphical Representation: When graphed, constant functions produce horizontal function lines on a coordinate plane. The equation of such line is typically written as y = a, where a is the constant value. For example, if y = 3, this line will be horizontal at the height of 3 on the y-axis, extending infinitely in both directions along the x-axis

Slope: The slope of a horizontal line is zero, indicating that there is no change in the y-values changes. This reflects the nature of constant functions, where the output does not depend on the input.

Examples of Constant Functios

Demand Curve in Perfect Competition: In economics, a firms demand curve under perfect competition can be represented regardless of price fluctuations.

Total Fixed Cost (TFC): The total fixed cost curve for a firm is another example of a constant function, where costs do not vary with output levels.

Vertical Constant Functions

While horizontal lines represent functions where the output remains constant, vertical lines can also represent that constant equations in terms of inputs:

General Form: A vertical line can be expressed as x = b, where b is constant. This indicates that no matter what value y takes x will always equal b.

Slope: The slope of vertical line is considered undefined or infinite since there is no horizontal change as you move vertically.

3) Explain the concept of identity lines

Answer: Algebraic identities are equations where the value of the left-hand side of the equation is identically equal to the value of the right- hand side of the equation. It is expressed in the form x = y.

An identity line is a special type of line in a graph that represents a situation where two variables are equal to each other. It is mathematically expressed as y = x. This means that for every value of x, the value of y is the same.

Key Characteristics

Slope: The slope of an identity line is 1, indicating that any increase in x results in an equal increase in y (i.e., they change by the same proportion). This relationship can be represented as △y = △x.

Graphical representation: On a graph, an identity line forms a 45-degree angle with both axes, indicating that both variables increase at the same rate.

Importance in Economics

Identity lines are particularly useful in economic models, such as Keynesian Cross Diagram, Where they illustrate equilibrium points. In this context, the identity line shows where aggregate demand equals aggregate supply. When plotted, it helps economists visualize and analyze situations where the outputs or incomes are balanced (i.e., AD =Y).

For Example, if we have an identity line represented by the equation y = x:

If x = 10, then y must also equal 10.
if x increases to 20, then y will also increases to 20.

This direct relationship is crucial for understanding how changes in one variable directly affect another in economic models.

4) What is meant by non-linear functions? Give examples.

Answer: Non linear functions are mathematical equations where the relationship between the variables cannot be represented by a straight line. Instead, their graphs are curves that can take various shapes depending on the specific nature of the function. Unlike linear functions, which have a constant slope (the rate of change is consistent), non linear functions have slopes that very at different points along the curve.

Characteristics of Non linear Functions

• Curved Graphs: The graph of a non linear function is not straight line; it can be parabolic, exponential, logarithmic, or take other forms.
• Variable slope: The slope of a non linear function changes depending on the point on the courve, meaning the rate of change is not constant.
• Higher Degrees: Non linear function often involve variables raised to powers greater than one. (e.g., quadratic functions wit x², cubic equations x³).

Examples of a Non Linear Function

1. Quadratic Function: These are expressed in the form y = ax3 + bx + c and produce a parabolic graph. An example is the average cost function in econommics which can exhibit U -shaped curves depending on costs and output levels.
2. Exponential Functions: Represented as y = a . e^bx , these functions model growth or decay processes such as population growth or compound interest. Their graphs show rapid increases or decreases.
3. Cubic Functions: These take the form y = ax³ + bx² + cx + d and can produce S-shaped curves. An example in economics is the total cost function, which may reflect increasing and then decreasing marginal costs.
4. Logarithmic Functions: Expressed as y = log(x), these functions are useful for modeling phenomena that grow rapidly initially and then level off, such as diminishing returns in production.
5. Reciprocal Functions: Given by y = 1/x, these represent inverse relationships where one variable increases while another decreases.

5) Explain how quadratic functions are represented by curves.

Answer: Quadratic functions are mathematical expressions that can be represented graphically as curves, specially parabolas. Here’s a simple explanation of how they are represented.

Definition of Quadratic Functions

A quadratic functions is typically expressed in the standard form:

y = ax² + bx + c

where:

• a, b, and c are constants,
• x is the independent variable
• y is the dependent variable

The key characteristics of quadratic functions is that the highest exponent of the variable x is 2, which contributes to their unique curved shape.

Shape of the curve

The graph of a quadratic function is a parabola. The orientation and postion of the parabola depend on the coefficient a:

• if a > 0, the parabola opens upward, resembling a “U” shape.
• if a < 0, it opens downward, resembling an inverted “U”.

Vertex and Axis of Symmetry

The vertex of the parabola is its highest or lowest point, depending on its orientation. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. The vertex can be found using the formula: x = – b/2a

Once x is determined, substituting it back into quadratic equation gives the corresponding y coordinate of the vertex.

Example

Consider the quadratic function y = 2x² – 4x + 1

In this case:

The coefficient a = 2 (positive), so the parabola opens upward.
The vertex can be calculated as follows:

First, find x = -4/2×2 = 1

Then substitute back into the equation: y = 2(1)² – 4(1) + 1 = -1.

Thus, the vertex is at (1, -1).

6) Explain the curves representing exponential and cubic functions.

Answer: Definition and Characteristics

Exponential functions are type of non-linear function that can be model growth or decay at a constant percentage change. They typically take the form y = a. bx, Where:

• a is constant
• b is the base of the exponential, and
• x is the exponent.

Graphical representation

The graph of an exponential function is characterized by its rapid increase (or decrease) as x moves away from zero. For example:

• if b > 1, the function exhibits exponential growth, rising steeply as x increases.
• if 0 < b < 1, it shows exponential decay, falling sharply as x increases.

Key Features:

Y-intercept: The value of y when x = 0 is equal to a.
Asymptote: The graph approaches th x-axis but never touches it, indicating that it never reaches zero

Definition and Characteristics

Cubic functions are polynomial functions of degree three, represented by the general form y = ax³ + bx² + cx + d.

• The leading coefficient a determines the direction of the curve (upward or downward).
• The presence of both positive and negative powers allows for more complex shapes compared to linear or quadratic functions.

Graphical Representation

The graph of a cubic function can exhibit various shapes, including:

S-shaped curves, which can have one or two turning points depending on the coefficients. this means they can rise and fall in one continuous motion, making them suitable for modeling scenarios with increasing and decreasing trends.

Key Features:

• Inflecting Point: The point where the curve changes concavity (from concave up to concave down or vice versa)
• Roots: A cubic function can have one real root (crossing the x-axis once) or three real roots (crossing three times), depending on its coefficients.

Comparison of Exponential and Cubic Functions

Feature	Exponential Function	Cubic Function
General form	y = a . bx	y = ax3 + bx2 + cx + d
Growth/Decay	constant percentage rate	Varies; can increase/decrease
Shapes	Rapidly increasing/decreasing	S-shaped, can have multiple turning points
Asymptotic Behavior	Approaches zero (horizontal asymptote)	No horizontal asymptotic; crosses x-axis
Y-intercept	Equal to a	Depends on coefficients

Chapters-1–2–3–4–5–6–7–8–9

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