Functions of one Real Variable

WHAT'S INSIDE THE TOPIC

Bhavya Shree

CSE (Bharti Vidyapeeth, Pune)

Select Langauge

Topic – Functions of One Real Variable (Notes)

Subject – Economics

(Quantitative and Mathematical Methods in Economics)

Quadratic function

1.Quadratic function :

A quadratic function is a polynomial function of degree 2, represented as:

f(x)=ax²+bx+c

where a, b, and c are constants. The graph of a quadratic function is a parabola.

2.Polynomial function :

A polynomial function is a function composed of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents. Examples include:

f(x)=x³-2x-4(quadratic )

f(x)=x³-2x²+x-1(cubic)

3. Power Function

A power function is a function where a single variable (base) is raised to a power (exponent). It has the form:

f(x) = x^n

where n is a constant. Power functions can be even (e.g., x²) or odd (e.g., x³).

4. Exponential Function

An exponential function has the form:

f(x) = a^x

where a is a positive constant (base) and x is the variable. Exponential functions exhibit rapid growth or decay.

5. Logarithmic Function

A logarithmic function is the inverse of an exponential function, represented as:

f(x) = logₐ(x)

where a is the base. Logarithmic functions are used to solve equations involving exponents.

Graphs:

– Quadratic: Parabola (U-shaped)

Polynomial: Varies depending on degree and coefficients
– Power: Depends on exponent (e.g., x² is a parabola, x³ is a cubic curve)
– Exponential: Rapidly increasing or decreasing curve
– Logarithmic: Slowly increasing curve

These functions are fundamental in mathematics and have numerous applications .

Continuous Function

Definition

A function is said to be continuous at a point if:

lim f (x)= f(a) =lim f(x)

(f \cdot g)(x) = f(x) \cdot g(x) \quad \text{is continuous at } x=a

4. Division

\left(\frac{f}{g}\right)(x) \quad \text{is continuous at } x=a \quad \text{if } g(a) \neq 0

5. Composition

If is continuous at , and is continuous at , then

(g \circ f)(x) = g(f(x)) \quad \text{is continuous at } x=a

Applications of Continuous Functions

1. Intermediate Value Theorem (IVT)

If is continuous on and ( f(a) <
This means: no breaks, no jumps, no holes at .

Continuous on an interval: A function is continuous at every point in that interval.

Examples: are continuous for all real .

Properties of Continuous Functions under Various Operations

If and are continuous at , then:

1. Addition

(f+g)(x) = f(x) + g(x) \quad \text{is continuous at } x=a

2. Subtraction

(f-g)(x) = f(x) – g(x) \quad \text{is continuous at } x=a

(f \cdot g)(x) = f(x) \cdot g(x) \quad \text{is continuous at } x=

4. Division

\left(\frac{f}{g}\right)(x) \quad \text{is continuous at } x=a \quad \text{if } g(a) \neq 0

5. Composition

(f \cdot g)(x) = f(x) \cdot g(x) \quad \text{is continuous at } x=a

4. Division

\left(\frac{f}{g}\right)(x) \quad \text{is continuous at } x=a \quad \text{if } g(a) \neq 0

5. Composition

If is continuous at , and is continuous at , then

(g \circ f)(x) = g(f(x)) \quad \text{is continuous at }

Applications of Continuous Functions

1. Intermediate Value Theorem (IVT)

The Intermediate Value Theorem (IVT): This theorem states that if a function $f$ is continuous on a closed interval $[a, b]$ , it takes on every value between $f (a)$ and $f (b)$ .
- Economic Application: Proving the existence of a market-clearing equilibrium price. If excess demand $E D (P) = D (P) - S (P)$ is continuous, and is positive at a low price and negative at a high price, the IVT guarantees there is some price $P^{*}$ in between where $E D (P^{*}) = 0$ . This is a fundamental existence proof in general equilibrium theory.
The Extreme Value Theorem: This theorem states that a continuous function on a closed and bounded interval must attain both a maximum and a minimum value.
- Economic Application: Guaranteeing that a firm’s profit-maximization problem or a consumer’s utility-maximization problem actually has a solution. If the feasible set (e.g., the budget set) is closed and bounded (compact), and the objective function (utility, profit) is continuous, a solution (an optimum) is guaranteed to exist.

3. Important Caveat: When Continuity is Not Assumed

It’s crucial for economics students to understand that continuity is an assumption, not a universal law. There are important cases of discontinuities:

Fixed Costs: A firm’s total cost function has a discontinuity at zero output if there are fixed costs $(T C (0) = F C, b u t T C (Q) = F C + V C (Q) f o r Q > 0)$ .
Leontief Production Function: This function $Q = \min (a L, b K)$ has right-angled isoquants and is not differentiable at the corner.
Sparsity and Indivisibilities: Buying a car or a house is a discrete, not continuous, decision.
Financial Economics: “Jumps” in stock prices due to major news events are modeled using discontinuous processes (like Poisson jumps).

Summary for the Economics Student

As an economics student, you should think of continuous functions as the primary modeling tool that allows you to:

Analyze Marginal Changes: Understand the effect of a small change in one variable (price, income, capital) on another.
Solve Optimization Problems: Use derivatives to find where firms maximize profits or consumers maximize utility.
Prove Existence: Use theorems like the Intermediate Value Theorem to prove that an equilibrium price exists.
Build Dynamic Models: Describe how economies evolve smoothly over time using differential equations.

Power Function:

A power function is a mathematical expression of the form:

y = kxⁿ

Where:

y is the dependent variable (e.g., Total Cost, Output).
x is the independent variable (e.g., Units Produced, Labour).
k is a constant (a scaling factor). It represents the value of y when x=1.
n is the exponent (or power). This is the most important part as it determines the behaviour and shape of the function.

2. The Importance of the Exponent (`n`)

The value of the exponent n defines the type of relationship between the variables x and y.

Case	Value of `n`	Relationship	Graphical Shape (Curve)	Economic Example
1	`n = 1`	Linear & Proportional	A straight line through the origin	Simple Revenue: `TR = p*x` (where `p` is price per unit, `x` is quantity). Revenue increases at a constant rate.
2	`n > 1`	Increasing Returns	A curve that slopes upward and becomes steeper (Convex)	Production Function: `Output = (Labour)²`. Doubling labour more than doubles output. This indicates economies of scale.
3	`0 < n < 1`	Decreasing Returns	A curve that slopes upward but becomes flatter (Concave)	Production/Cost Function: `Output = √(Input)` or `Cost = (Output)^(0.8)`. Increasing input leads to output increasing, but at a slower rate. This is very common (e.g., diminishing marginal returns).
4	`n = 0`	Constant Function	A horizontal straight line	Fixed Cost: `TFC = k`. No matter the level of output (`x`), the fixed cost remains constant.
5	`n < 0`	Inverse Relationship	A downward-sloping curve	Demand Curve: `Qd = k / P` or `Qd = k * P⁻¹`. As price (`P`) increases, quantity demanded (`Qd`) decreases.

3. Key Concept: Elasticity and Power Functions

This is a crucial application for economics students. In a power function, the exponent n is directly the elasticity of y with respect to x.

Elasticity measures the percentage change in one variable in response to a percentage change in another.

Example: If a demand function is written as Q = 100 * P⁻⁰·⁵
- The price elasticity of demand is -0.5.
- This is found simply by looking at the exponent of P! This is why power functions are preferred in econometric models.

4. Why are Power Functions Important in Economics?

Realistic Modelling: They effectively model phenomena like diminishing returns (0 < n < 1) and economies of scale (n > 1), which are fundamental concepts in economics.
Constant Elasticity: They assume elasticity is constant at all points on the curve, which is a useful simplification for analysis.
Linearizable: A major advantage is that they can be easily transformed into a linear form using logarithms, allowing for estimation using linear regression techniques.
- y = kxⁿ becomes log(y) = log(k) + n*log(x)

5. Summary Table for Quick Revision

Function Form	Exponent (`n`)	Economic Interpretation
`y = 5x`	`n = 1`	Constant returns to scale.
`y = 2x²`	`n = 2 > 1`	Increasing returns to scale.
`y = 10√x = 10x⁰·⁵`	`n = 0.5 < 1`	Decreasing returns to scale (Diminishing marginal returns).
`y = 500`	`n = 0`	Constant function (e.g., Fixed Cost).
`y = 50/x = 50x⁻¹`	`n = -1 < 0`	Inverse relationship (e.g., Unitary elastic demand).

Key Takeaway: When you see a function like y = kxⁿ, immediately look at the power n.

Exponential Functions :

1. What is an Exponential Function?

An exponential function is a mathematical function where the independent variable (usually x or t) appears in the exponent.

The general form is:
f(x) = a * b^x

Where:

a: The initial value or starting amount (when x=0). It’s the y-intercept.
b: The base of the exponential. This is the most important part. It must be a positive number not equal to 1 (b > 0 and b ≠ 1).
x: The exponent, which often represents time in economic contexts (e.g., years, months).

2. The Key Feature: Constant Proportional Growth

The defining characteristic of an exponential function is that it models a quantity that grows or shrinks by a constant percentage rate over equal time periods.

If b > 1, the function represents exponential growth. The quantity increases by a fixed percentage each period.
- Example: f(x) = 100 * (1.05)^x grows by 5% each period.
If 0 < b < 1, the function represents exponential decay. The quantity decreases by a fixed percentage each period.
- Example: f(x) = 100 * (0.85)^x decays by 15% each period.

3. The Special Base ‘e’

In finance and economics, the most common base is the irrational number e (approximately 2.71828). Functions with base e are written as:

A(t) = P * e^(rt)

This form is called continuous growth/decay and is used for calculations where growth is theoretically happening constantly (every instant), rather than at discrete intervals (like annually).

4. Economic & Financial Applications (VERY IMPORTANT)

A. Compound Interest

This is the classic example. The formula for compound interest is exponential.

Discrete Compounding: A = P (1 + r/n)^(n*t)
- A = Future Value, P = Principal, r = annual rate, n = compounding periods per year, t = time in years.
Continuous Compounding: A = P * e^(r*t)
- This is used for theoretical models and many financial derivatives.

B. Economic Growth

A country’s GDP often grows at an approximately exponential rate.
Population growth (according to the Malthusian model) is exponential.

C. Depreciation

The value of assets like machinery, vehicles, or equipment decreases (depreciates) exponentially over time.

Example: A car valued at ₹5,00,000 depreciating at 10% per year: V(t) = 500000 * (0.90)^t

D. The Rule of 72

A quick, handy rule derived from exponential growth to estimate the time for an investment to double.

Time to double ≈ 72 / (Interest Rate %)
*Example: At an interest rate of 8%, it takes about 72/8 = 9 years to double your money.*

5. Graphing an Exponential Function

The graph is a smooth curve, not a straight line.
It always crosses the y-axis at (0, a) because when x=0, f(0) = a * b^0 = a * 1 = a.
Growth (b>1): The curve slopes upward steeply to the right. It increases slowly at first and then very rapidly.
Decay (0<b<1): The curve slopes downward to the right. It decreases rapidly at first and then levels off, getting closer to zero but never touching the x-axis (it has a horizontal asymptote at y=0).

6. Key Takeaways for B.Com Students

Think in Percentages: Exponential functions model constant percentage change, not constant absolute change.
Time is the Exponent: The variable x or t usually represents time. Small changes in the rate (r) or time (t) can lead to huge changes in the final outcome due to the power of compounding.
e is Everywhere: The base e is fundamental in finance for continuous growth models.
Power of Compounding: Exponential growth explains why starting to save early for retirement is so powerful—your money earns interest on the interest, leading to dramatic growth over long periods.

Quick Revision Summary

Feature	Exponential Growth	Exponential Decay
Base (b)	`b > 1`	`0 < b < 1`
Economic Meaning	Appreciation, Growth	Depreciation, Decay
Formula (General)	`f(x) = a * b^x`	`f(x) = a * b^x`
Formula (Continuous)	`A = P * e^(rt)`	`A = P * e^(-rt)`
Examples	Compound Interest, GDP	Asset Depreciation, Radioactive Decay
Graph	Upward-sloping curve	Downward-sloping curve

Logarithmic Functions :

1. What is a Logarithm?

A logarithm answers the question: “To what power must we raise a base number to get another number?”

Formula: If $b^{y} = x$ , then $\log_{b} (x) = y$ .
Components:
- $b$ is the base (must be positive and not equal to 1).
- $x$ is the argument (must be positive).
- $y$ is the exponent or the logarithm.

Example: $10^{2} = 100$ , so $\log_{10} (100) = 2$ . We raise the base (10) to the power of 2 to get 100.

2. Common Types of Logarithms

In economics, you will mostly encounter two types:

Natural Logarithm (ln): Logarithm with base $e$ , where $e \approx 2.71828$ .
- Written as $\ln (x)$ .
- Extremely important in economics for modeling continuous growth.
Common Logarithm (log): Logarithm with base 10.
- Written as $\log (x)$ . Often used in scales (like Richter scale) and some financial formulas.

3. Why are Logarithms Crucial in Economics?

Logarithms are not just a mathematical tool; they help us understand economic relationships better.

Non-Linear Relationships: Many economic relationships are not straight lines (e.g., diminishing marginal utility, production functions). Logarithms can transform these curved relationships into straight lines, making them easier to analyze.
Growth Rates: This is the most important application.
- The change in the natural log of a variable approximates its percentage growth rate.
- Formula: $\ln (X_{t}) - \ln (X_{t - 1}) \approx \frac{X_{t} - X_{t - 1}}{X_{t - 1}} = Growth Rate of X$
- Example: If a country’s GDP grows from 100 to 110 units, the exact growth rate is 10%. Using logs: $\ln (110) - \ln (100) \approx 4.7005 - 4.6052 = 0.0953$ , which is approximately 9.53%. This approximation is very good for small changes.
Elasticity: In econometrics (economic statistics), when both the dependent and independent variables in a regression model are in logarithmic form, the estimated coefficient is interpreted directly as an elasticity.
- Model: $\ln (Y) = α + β \ln (X)$
- Interpretation: A 1% increase in X is associated with a β% change in Y. This is incredibly useful for estimating price elasticity of demand, income elasticity, etc.
Handling Large Numbers: Economic data like GDP, national debt, or company revenues can be very large. Taking the log compresses the scale, making graphs easier to read and data easier to handle statistically.

4. Key Rules of Logarithms (Must Know)

These rules are essential for simplifying and solving economic models.

Let $M$ and $N$ be positive numbers.

Product Rule: $\log_{b} (M N) = \log_{b} (M) + \log_{b} (N)$
Quotient Rule: $\log_{b} (M / N) = \log_{b} (M) - \log_{b} (N)$
Power Rule: $\log_{b} (M^{p}) = p \cdot \log_{b} (M)$ (This is why logs are used for growth rates!)
Base Change: $\log_{b} (M) = \frac{\ln (M)}{\ln (b)}$ (Useful for calculations)

5. Economic Applications & Examples

Compound Interest/Growth: The formula for continuous compounding, $A = P e^{r t}$ , can be solved for the growth rate (r) using natural logs: $\ln (A / P) = r t$ .
Cobb-Douglas Production Function: A standard production function is $Y = A L^{α} K^{β}$ .
- Taking the natural log of both sides gives: $\ln (Y) = \ln (A) + α \ln (L) + β \ln (K)$ .
- This transforms a multiplicative function into a linear one, where $α$ and $β$ are the output elasticities of labor (L) and capital (K), which can be estimated using linear regression.
Data Visualization: A time series graph of a variable (like GDP) that is growing at a constant percentage rate will curve upwards. However, a graph of $\ln (G D P)$ over time will be a straight line. The slope of this line is the approximate constant growth rate.

Summary for Quick Revision

Aspect	Key Takeaway for Economics
What is it?	The power to which a base is raised to get a number.
Most Used Type	Natural Logarithm (ln) with base $e$ .
Primary Use	To calculate growth rates and model elasticities.
Key Rule	Power Rule: $\ln (X^{p}) = p \cdot \ln (X)$
Economic Meaning	The difference in logs ( $\ln (X_{t}) - \ln (X_{t - 1})$ ) ≈ Percentage Change in X.
Econometrics	In a log-log model, coefficients represent elasticities.

Sequences and Series :

1. Basic Definitions

Sequence: An ordered list of numbers following a specific rule.
- Example: 2, 4, 6, 8, … (Rule: Multiples of 2)
- Denoted as: $a_{1}, a_{2}, a_{3}, . . ., a_{n}$ where $a_{n}$ is the $n^{t h}$ term.
Series: The sum of the terms of a sequence.
- Example: 2 + 4 + 6 + 8 + … (Sum of the sequence above)
- Denoted as: $S_{n} = a_{1} + a_{2} + . . . + a_{n}$ .
Key Difference: A sequence is a list; a series is a sum.

2. Arithmetic Progression (AP) – Linear Growth/Decay

An AP is a sequence where the difference between consecutive terms is constant. This models situations with constant absolute change.

Common Difference (d): $d = a_{2} - a_{1} = a_{3} - a_{2}$
$n^{t h}$ Term: $a_{n} = a_{1} + (n - 1) d$
Sum of first $n$ terms: $S_{n} = \frac{n}{2} [2 a_{1} + (n - 1) d]$ or $S_{n} = \frac{n}{2} (a_{1} + a_{n})$

Economic Applications of AP:

Straight-Line Depreciation: The value of an asset decreases by a fixed amount each year.
Simple Interest: Interest earned is constant every period because it’s calculated only on the principal.
Constant Increment in Costs/Wages: A fixed annual salary increase or a fixed cost increment.

Example (Simple Interest): If you invest ₹10,000 at a simple interest of 5% per year, the interest each year is constant (₹500). The total interest earned after $n$ years forms an AP: 500, 1000, 1500, …

3. Geometric Progression (GP) – Exponential Growth/Decay

A GP is a sequence where the ratio between consecutive terms is constant. This is crucial for modelling compound growth, which is very common in economics.

Common Ratio (r): $r = \frac{a_{2}}{a_{1}} = \frac{a_{3}}{a_{2}}$
$n^{t h}$ Term: $a_{n} = a_{1} \cdot r^{n - 1}$
Sum of first $n$ terms:
- If $r > 1$ : $S_{n} = a_{1} \frac{(r^{n} - 1)}{r - 1}$
- If $r < 1$ : $S_{n} = a_{1} \frac{(1 - r^{n})}{1 - r}$

Economic Applications of GP:

Compound Interest: Interest is calculated on both the principal and accumulated interest, leading to exponential growth.
Population Growth: Populations often grow at a rate proportional to their current size.
Economic Growth (GDP): Often measured as a percentage growth rate year-on-year.
Inflation: The rise in the price level is usually a compounding process.

Example (Compound Interest): ₹10,000 invested at 10% p.a. compounded annually.
Value after 1 year: $10000 \times (1.1)^{1}$
Value after 2 years: $10000 \times (1.1)^{2}$
The sequence of values is a GP: 10000, 11000, 12100, 13310, … with $r = 1.1$ .

4. Infinite Geometric Series

This is a special case of a GP where the number of terms ( $n$ ) approaches infinity ( $n \to \infty$ ).

Sum to Infinity ( $S_{\infty}$ ): This sum converges (reaches a finite value) only if the common ratio is between -1 and 1 ( $∣ r ∣ < 1$ or $- 1 < r < 1$ ).
Formula: $S_{\infty} = \frac{a_{1}}{1 - r}$

Economic Applications of Infinite Series:

The Multiplier Effect in Economics: An initial injection of spending (e.g., government investment) leads to more income, which leads to more spending, and so on. The total increase in national income is the sum of an infinite geometric series with $r$ = Marginal Propensity to Consume (MPC).
Perpetuities (Finance): A financial instrument that pays a fixed amount forever. Its present value is calculated using the infinite series formula.
Depreciation with a Constant Ratio: Some models of depreciation assume an asset loses a fixed percentage of its value each year.

Example (Multiplier): If MPC = 0.8, the multiplier is $\frac{1}{1 - M P C} = \frac{1}{1 - 0.8} = \frac{1}{0.2} = 5$ . A ₹100 crore investment increases total income by $100 \times 5 = ₹ 500$ crore.

5. Important Relationships and Comparison

Feature	Arithmetic Progression (AP)	Geometric Progression (GP)
Nature of Change	Constant Absolute Change (`+d` or `-d`)	Constant Relative/Percentage Change (`×r`)
General Term	$a_{n} = a_{1} + (n - 1) d$	$a_{n} = a_{1} \cdot r^{n - 1}$
Sum of $n$ terms	$S_{n} = \frac{n}{2} [2 a_{1} + (n - 1) d]$	$S_{n} = a_{1} \frac{(r^{n} - 1)}{r - 1}$ (for r≠1)
Key Concept	Linear Growth	Exponential Growth
Economic Example	Simple Interest, Straight-Line Depreciation	Compound Interest, GDP Growth, Inflation

Key Takeaways for Exams

Identify the Pattern: Is the change a fixed amount (AP) or a fixed percentage/ratio (GP)?
AP Formula: Use $a_{n} = a_{1} + (n - 1) d$ for specific terms and $S_{n} = \frac{n}{2} (a_{1} + a_{n})$ for sum.
GP Formula: Use $a_{n} = a_{1} \cdot r^{n - 1}$ for specific terms. For sum, choose the formula based on whether $r > 1$ or $r < 1$ .
Infinite Series: Remember the crucial condition $∣ r ∣ < 1$ for the sum to infinity to exist. The formula is $S_{\infty} = \frac{a_{1}}{1 - r}$

Continuous Functions :

1. What is a Continuous Function? (The Intuitive Idea)

Think of drawing a graph without lifting your pen from the paper. A function is continuous if its graph is an unbroken curve. Formally, a function f(x) is continuous at a point x = a if:

f(a) is defined (the point exists on the graph).
The limit of f(x) as x approaches a exists.
The limit equals the function value: lim (x→a) f(x) = f(a).

Economic Interpretation: In economics, continuity implies that a small change in the input (e.g., price, income) leads to a small, predictable change in the output (e.g., quantity demanded, consumption). There are no sudden, unexpected jumps.

2. Properties of Continuous Functions (The “Rules”)

Continuous functions behave very nicely under standard mathematical operations. If f(x) and g(x) are both continuous at a point x=a, then the following are also continuous at x=a:

Operation	New Function	Economic Example
Sum/Difference	`f(x) + g(x)` or `f(x) - g(x)`	Total Cost (TC) = Fixed Cost (FC) + Variable Cost (VC). If FC and VC are continuous, so is TC.
Scalar Multiplication	`k * f(x)` (where k is a constant)	Calculating GST on a continuous revenue function. The taxed revenue function remains continuous.
Product	`f(x) * g(x)`	*Total Revenue (TR) = Price (P) Quantity (Q)**. If both P and Q are smooth functions, TR is continuous.
Quotient	`f(x) / g(x)`	Average Cost (AC) = Total Cost (TC) / Quantity (Q). This is continuous as long as Q ≠ 0 (you are producing something).
Composition	`f(g(x))`	Utility from hours of leisure: If Utility U is a function of leisure (L), and leisure L is a function of work hours (H), then U is a function of H. If both relationships are smooth, the overall function is continuous.

3. Key Theorems & Their Economic Applications

These are powerful results that are fundamental to economic reasoning.

a) The Intermediate Value Theorem (IVT)

Statement: If a function f is continuous on a closed interval [a, b], then it takes on every value between f(a) and f(b).
Economic Application: Finding Equilibrium Prices
- Scenario: Consider the market price (P) and excess demand function, ED(P) = Q_d(P) – Q_s(P).
- At a very high price (P_high), excess demand is negative (surplus).
- At a very low price (P_low), excess demand is positive (shortage).
- If the demand and supply functions are continuous (a standard assumption), then ED(P) is continuous.
- The IVT guarantees that there exists a price P* between P_low and P_high where ED(P*) = 0. This is the equilibrium price. The theorem proves that an equilibrium must exist under these reasonable conditions.

b) The Extreme Value Theorem

Statement: If a function f is continuous on a closed (and bounded) interval [a, b], then f must attain both a maximum and a minimum value on that interval.
Economic Application: Profit Maximization
- A firm’s profit function, π(Q), is often assumed to be continuous.
- A firm typically operates between a minimum output (maybe 0) and a maximum feasible output (say, Q_max due to factory capacity). This defines a closed interval [0, Q_max].
- The Extreme Value Theorem guarantees that there is at least one level of output Q within this range that maximizes profit* (and one that minimizes it). This justifies the process of “finding the maximum” using calculus.

4. Why is Continuity Important for Economics Students?

Realism: Most economic relationships are continuous in the short run. A ₹1 change in price doesn’t cause a million-unit drop in demand; the change is gradual.
Mathematical Tractability: Continuity allows us to use the powerful tools of calculus (differentiation and integration). We can find marginal cost, marginal revenue, and elasticities only if the underlying functions are continuous.
Modeling Equilibrium: As shown by the IVT, continuity is crucial for proving the existence of market equilibria.
Optimization: The Extreme Value Theorem is the foundation for solving optimization problems (maximizing profit, utility, or minimizing cost).

5. Important Non-Continuous Functions in Economics

While continuity is common, discontinuities are also important:

Piecewise Functions: A tax function might be continuous until a specific income bracket, then jump to a new rate (creating a discontinuity).
Fixed Costs: The total cost function often has a discontinuity at Q=0 because of fixed costs (TC jumps from 0 to a positive number as soon as you produce the first unit).
Indifference Curves: While the curves themselves are continuous, the utility function for perfect complements is not differentiable at the kink.

Differentiable Functions :

1. What is a Differentiable Function?

In simple terms, a function is differentiable at a point if it has a well-defined, non-vertical tangent at that point. This means the function is “smooth” at that point, with no sharp corners, breaks, or jumps.

Mathematical Definition: A function $f (x)$ is differentiable at a point $x = a$ if the following limit exists:
$f^{'} (a) = \lim_{h \to 0} \frac{f (a + h) - f (a)}{h}$
This limit, if it exists, is called the derivative of $f$ at $x = a$ .
Key Implication: If a function is differentiable at a point, it is automatically continuous at that point. However, the reverse is not always true (a continuous function may not be differentiable).

2. Characterization: How to Recognize Differentiability

A function is differentiable on an interval if:

Its graph is a smooth curve with no discontinuities (gaps or jumps).
It has no sharp corners or cusps.
Its derivative function, $f^{'} (x)$ , exists for every point in that interval.

Economic Example:

Differentiable: A production function showing output against labor input is typically smooth. Adding one more worker leads to a small, predictable change in output.
Not Differentiable: A tax function with different brackets might have a kink at the income threshold where the tax rate changes. The slope (marginal tax rate) changes abruptly, making it non-differentiable at that exact income point.

3. Properties under Various Operations (The “Calculus” of Derivatives)

If $f (x)$ and $g (x)$ are differentiable functions, and $c$ is a constant, then the following are also differentiable (with the noted rules):

Operation	Function	Derivative Rule	Economic Interpretation
Scalar Multiplication	$c \cdot f (x)$	$c \cdot f^{'} (x)$	Scaling output scales marginal effects. E.g., Doubling production doubles marginal cost.
Sum/Difference	$f (x) \pm g (x)$	$f^{'} (x) \pm g^{'} (x)$	The marginal effect of a combined activity is the sum of the marginal effects.
Product Rule	$f (x) \cdot g (x)$	$f^{'} (x) g (x) + f (x) g^{'} (x)$	The marginal revenue from selling two complementary goods.
Quotient Rule	$\frac{f (x)}{g (x)}$	$\frac{f^{'} (x) g (x) - f (x) g^{'} (x)}{[g (x)]^{2}}$	The change in average cost (Total Cost/Quantity) as output changes.
Chain Rule	$f (g (x))$	$f^{'} (g (x)) \cdot g^{'} (x)$	The effect of a change in an underlying variable. E.g., How a change in raw material costs (g) affects overall profit (f) through its impact on production cost (g(x)).

4. Applications in Economics (This is the most important part!)

In economics, the derivative $f^{'} (x)$ represents a marginal concept. It is the instantaneous rate of change, which approximates the change resulting from a one-unit increase in the variable $x$ .

Concept	Function, $f (x)$	Derivative, $f^{'} (x)$	Economic Meaning
Cost	Total Cost (TC): $C (q)$	Marginal Cost (MC): $C^{'} (q)$	The approximate cost of producing one more unit. Crucial for pricing and output decisions.
Revenue	Total Revenue (TR): $R (q)$	Marginal Revenue (MR): $R^{'} (q)$	The additional revenue from selling one more unit. Profit is maximized where $M R = M C$ .
Production	Production Function: $Q (L)$	Marginal Product of Labor (MPL): $Q^{'} (L)$	The additional output gained by employing one more unit of labor.
Utility	Utility Function: $U (x)$	Marginal Utility (MU): $U^{'} (x)$	The additional satisfaction gained from consuming one more unit of a good. Explains the law of diminishing returns.
Consumption	Consumption Function: $C (Y)$	Marginal Propensity to Consume (MPC): $C^{'} (Y)$	The fraction of an additional rupee of income that is spent on consumption. Central to Keynesian economics.

5. Why is Differentiability So Important for Economists?

Optimization: The fundamental tool for finding maximum profit or minimum cost is to take the derivative and set it to zero ( $f^{'} (x) = 0$ ). This only makes sense if the derivative exists.
Marginal Analysis: Economics is built on the idea of thinking “at the margin.” Differentiability provides the precise mathematical tool for this analysis.
Modeling Behavior: Smooth, differentiable functions are often realistic for modeling economic relationships (like production and utility) over most of their range.
Comparative Statics: It allows us to study how a change in an exogenous parameter (like tax rates) affects an endogenous variable (like equilibrium price) by using calculus.

Second-Order Derivatives :

1. What is a Second-Order Derivative?

First Derivative (f'(x) or dy/dx): It represents the rate of change of a function. In economics, this is often the marginal concept (e.g., Marginal Cost, Marginal Revenue).
Second Derivative (f''(x) or d²y/dx²): It is the derivative of the first derivative. It tells us about the rate of change of the rate of change. In simple terms, it describes how the slope of the original function is itself changing.

2. The Key Property: Concavity and Convexity

The most important economic interpretation of the second derivative is whether a function is concave or convex at a point.

Concave Downward (Maximum Point):
- Condition: f''(x) < 0 (Negative Second Derivative)
- Shape: The curve is bending downwards, like a hill (∩).
- Implication: The slope (f'(x)) is decreasing. As you move to the right, the function increases at a slower rate or decreases at a faster rate.
Convex Upward (Minimum Point):
- Condition: f''(x) > 0 (Positive Second Derivative)
- Shape: The curve is bending upwards, like a valley (∪).
- Implication: The slope (f'(x)) is increasing. As you move to the right, the function decreases at a slower rate or increases at a faster rate.

3. Application 1: Profit Maximization for a Firm (The Most Important)

A firm maximizes its profit where Marginal Revenue (MR) = Marginal Cost (MC). However, this condition alone could also indicate a point of minimum profit. The second derivative acts as the test to confirm we have found a maximum.

Profit Function: π(Q) = R(Q) - C(Q) where π is profit, R is revenue, C is cost, and Q is quantity.
First-Order Condition (FOC):
- dπ/dQ = MR - MC = 0 => MR = MC
- This finds critical points (possible max or min).
Second-Order Condition (SOC):
- d²π/dQ² = d(MR)/dQ - d(MC)/dQ < 0
- This simplifies to: Slope of MR < Slope of MC
- Interpretation: For profit to be maximized, the Marginal Revenue curve must be cutting the Marginal Cost curve from above. This ensures the point where MR=MC is a maximum.

Example:
Let π(Q) = -2Q² + 20Q - 10

FOC: dπ/dQ = -4Q + 20 = 0 => Q = 5
SOC: d²π/dQ² = -4 (which is < 0)
Conclusion: Since the second derivative is negative, profit is maximized at Q = 5.

4. Application 2: Diminishing Marginal Utility

The law of diminishing marginal utility states that as a consumer consumes more units of a good, the additional utility (satisfaction) from each extra unit eventually decreases.

Total Utility Function: TU(X), where X is quantity consumed.
Marginal Utility (MU): MU = d(TU)/dX (First derivative).
Diminishing MU: This means that Marginal Utility is falling as X increases.
Second Derivative: The rate at which MU is falling is given by d²(TU)/dX².
- If d²(TU)/dX² < 0, it means Marginal Utility is decreasing. This confirms the law of diminishing marginal utility.

5. Application 3: Determining Returns to Scale in Production

While often analyzed using elasticity, the second derivative can help describe the nature of a production function Q = f(L),

Marginal Product of Labour (MPL): MPL = dQ/dL
Second Derivative: d²Q/dL²
- If d²Q/dL² < 0: It indicates diminishing marginal returns. The addition of one more worker increases output, but at a decreasing rate.
- If d²Q/dL² > 0: It indicates increasing marginal returns. The addition of one more worker increases output at an increasing rate (rare in the long run).

6. Application 4: Optimizing Inventory (EOQ-like concepts)

While calculating the Economic Order Quantity (EOQ), the total cost function (ordering cost + holding cost) is minimized. The second derivative test confirms that the critical point found gives the minimum cost.

If the second derivative of the total cost function with respect to quantity is positive (> 0), the critical point is indeed a minimum.

Summary Table for Quick Revision

Second Derivative (`f''(x)`)	Shape of Function	Economic Meaning	Example
`f''(x) < 0` (Negative)	Concave (∩)	Maximum Point	Profit Maximization (confirmed when SOC is negative). Diminishing Marginal Utility.
`f''(x) > 0` (Positive)	Convex (∪)	Minimum Point	Cost Minimization. Confirming a point is a minimum, like in EOQ models.

Key Takeaway: the second derivative is crucial for confirming whether an optimal value (found by setting the first derivative to zero) is a maximum or a minimum, which is fundamental to decision-making in economics and business.

Economic Applications: MR, MC, and Total, Average, Marginal Relationships:

1. Marginal Revenue (MR)

Concept: Marginal Revenue is the additional revenue earned by selling one more unit of a product.

Formula: MR = ΔTR / ΔQ
- ΔTR = Change in Total Revenue
- ΔQ = Change in Quantity Sold

Application & Interpretation:

For a Competitive Firm (Price Taker): The firm can sell any quantity at the prevailing market price (P). Therefore, the revenue from each additional unit is constant.
- MR = Price (P). The MR curve is a horizontal straight line equal to the market price.
For a Monopoly Firm (Price Maker): To sell more units, the firm must lower the price for all units. Therefore, the revenue from each additional unit is less than the previous one.
- MR < Price (P). The MR curve is downward sloping and lies below the demand (Average Revenue) curve.

Key Insight: MR indicates the revenue contribution of the last unit sold. It is crucial for deciding the level of output.

2. Marginal Cost (MC)

Concept: Marginal Cost is the additional cost incurred for producing one more unit of a product.

Formula: MC = ΔTC / ΔQ
- ΔTC = Change in Total Cost (which includes both fixed and variable costs)
- ΔQ = Change in Quantity Produced

Application & Interpretation:

The MC curve is typically U-shaped due to the law of variable proportions.
- Initially, MC falls because of increasing returns from specialization.
- Eventually, MC rises due to diminishing returns (e.g., overcrowding of inputs).
MC is a critical concept for supply decisions. A firm will only be willing to supply an additional unit if the revenue from it (MR) covers its cost (MC).

3. The Profit-Maximizing Rule: The Core Application

The most important application of MR and MC is determining the profit-maximizing level of output for a firm.

The Rule: A firm maximizes its profit where MR = MC.
Logic:
- If MR > MC: Producing and selling one more unit adds more to revenue than to cost, so profit increases. The firm should increase production.
- If MR < MC: Producing one more unit adds more to cost than to revenue, so profit decreases. The firm should decrease production.
- If MR = MC: The revenue from the last unit exactly covers its cost. At this point, profit is maximized. The firm should maintain this output level.

Why not produce where MC is minimum? Minimum MC is about cost-efficiency, not profit. Profit is about the difference between revenue and cost. The goal is not to minimize cost per unit, but to maximize the total profit.

4. Relationship between Total, Average, and Marginal Concepts:

This relationship is universal and applies to Total Product (TP), Average Product (AP), Marginal Product (MP) as well as Total Cost (TC), Average Cost (AC), Marginal Cost (MC).

General Rules:

The Marginal (M) leads the Average (A):
- When Marginal > Average, the Average is rising.
  - Example: If your test score (marginal) is above your class average, the class average rises.
- When Marginal < Average, the Average is falling.
- When Marginal = Average, the Average is constant (at its maximum or minimum point). The marginal curve always cuts the average curve at its peak (for product) or its lowest point (for cost).
The link between Total (T) and Marginal (M):
- The Marginal value is the slope of the Total curve.
- When the Total curve is increasing at an increasing rate, the Marginal curve is rising.
- When the Total curve is increasing at a decreasing rate, the Marginal curve is falling.
- Peak of Total Curve: When the Total curve reaches its maximum, the Marginal value is zero. (If producing one more unit adds nothing, the marginal gain is zero).

Application to Cost Curves:

If…	Then…	And…
MC < ATC	ATC is falling	The cost of the new unit is pulling the average down.
MC > ATC	ATC is rising	The cost of the new unit is pulling the average up.
MC = ATC	ATC is at its minimum	This is the most efficient scale of output.

The same logic applies to Average Variable Cost (AVC). The MC curve always cuts both the AVC and ATC curves at their minimum points.

Summary :

Marginal Revenue (MR): Extra money from selling one more unit.
Marginal Cost (MC): Extra cost of producing one more unit.
Profit Maximization: The golden rule is to produce up to the point where MR = MC. This is the fundamental decision-making tool for firms.
Total-Average-Marginal Relationship:
- Marginal drives the Average.
- Marginal is the slope of the Total.
- The Average is at its max/min when it is equal to the Marginal.

Exam Tip: Always draw the diagrams (Cost Curves and Revenue Curves) to visualize these relationships. It makes the concepts much easier to understand and remember.

Quantitative/Mathematical Methods Membership Required

You must be a Quantitative/Mathematical Methods member to access this content.

Join Now

Already a member? Log in here

Table of Contents

Quadratic function

1.Quadratic function :

f(x)=ax²+bx+c

2.Polynomial function :

f(x)=x³-2x-4(quadratic )

f(x)=x³-2x²+x-1(cubic)

3. Important Caveat: When Continuity is Not Assumed

Summary for the Economics Student

Build Dynamic Models: Describe how economies evolve smoothly over time using differential equations.

Power Function:

2. The Importance of the Exponent (n)

3. Key Concept: Elasticity and Power Functions

4. Why are Power Functions Important in Economics?

5. Summary Table for Quick Revision

Exponential Functions :

1. What is an Exponential Function?

2. The Key Feature: Constant Proportional Growth

3. The Special Base ‘e’

4. Economic & Financial Applications (VERY IMPORTANT)

A. Compound Interest

B. Economic Growth

C. Depreciation

D. The Rule of 72

5. Graphing an Exponential Function

6. Key Takeaways for B.Com Students

Quick Revision Summary

Logarithmic Functions :

1. What is a Logarithm?

2. Common Types of Logarithms

3. Why are Logarithms Crucial in Economics?

4. Key Rules of Logarithms (Must Know)

5. Economic Applications & Examples

Summary for Quick Revision

Sequences and Series :

1. Basic Definitions

2. Arithmetic Progression (AP) – Linear Growth/Decay

Economic Applications of AP:

3. Geometric Progression (GP) – Exponential Growth/Decay

Economic Applications of GP:

4. Infinite Geometric Series

Economic Applications of Infinite Series:

5. Important Relationships and Comparison

Key Takeaways for Exams

Continuous Functions :

1. What is a Continuous Function? (The Intuitive Idea)

2. Properties of Continuous Functions (The “Rules”)

3. Key Theorems & Their Economic Applications

4. Why is Continuity Important for Economics Students?

5. Important Non-Continuous Functions in Economics

Differentiable Functions :

1. What is a Differentiable Function?

2. Characterization: How to Recognize Differentiability

3. Properties under Various Operations (The “Calculus” of Derivatives)

4. Applications in Economics (This is the most important part!)

5. Why is Differentiability So Important for Economists?

Second-Order Derivatives :

1. What is a Second-Order Derivative?

2. The Key Property: Concavity and Convexity

3. Application 1: Profit Maximization for a Firm (The Most Important)

4. Application 2: Diminishing Marginal Utility

5. Application 3: Determining Returns to Scale in Production

6. Application 4: Optimizing Inventory (EOQ-like concepts)

Summary Table for Quick Revision

Economic Applications: MR, MC, and Total, Average, Marginal Relationships:

1. Marginal Revenue (MR)

2. Marginal Cost (MC)

3. The Profit-Maximizing Rule: The Core Application

4. Relationship between Total, Average, and Marginal Concepts:

Summary :

Quantitative/Mathematical Methods Membership Required

2. The Importance of the Exponent (`n`)