Source: ECON 323 Preliminary, Microeconomic Theory (Texas A&M University)
Tags: functions, one variable, linear functions, slope, y-intercept, derivatives, marginal cost, rate of change, inverse demand, power rule, product rule, sum rule, tangent line, microeconomic math
This chapter is the mathematical toolkit you need before the economics starts. It covers how functions of one variable work (input in, output out), how linear functions are defined by their slope and intercept, and how derivatives let you measure the rate of change for non-linear functions. The derivative of a cost function is called marginal cost, a concept that runs through the entire course.
Function of one variable
A rule that assigns each number x in the domain to exactly one value f(x). Written y = f(x), where x is the independent variable and y is the dependent variable.
Independent variable
The input to a function, the value you choose or control. Typically x or q.
Dependent variable
The output of a function, determined by the independent variable. Typically y or C(q).
Linear function
A function of the form y = mx + b, whose graph is a straight line.
Slope (m)
The rate at which the function's value changes per one-unit increase in x. Calculated as m = Δy / Δx.
Y-intercept (b)
The value of y where the line crosses the vertical axis, i.e. the output when x = 0.
Inverse demand function
The demand relationship rewritten with price (P) as a function of quantity (Q) rather than the other way round. Used because economists plot P on the vertical axis by convention.
Derivative (f'(x) or df/dx)
The slope of the tangent line to f(x) at a given point. Measures the instantaneous rate of change of the function.
Marginal cost
The derivative of the total cost function. Roughly measures how much additional cost the firm pays when it increases output by one unit.
Power rule
If f(x) = x^k, then f'(x) = kx^(k−1).
A function of one variable takes an input x and produces a single output f(x).
The notation y = f(x) tells you x is the independent variable (you pick it) and y is the dependent variable (the function determines it).
Example: taco restaurant cost function
A restaurant producing q tacos has a total cost of C(q) = 1000 + q².
Here q is independent and C(q) is dependent.
When q = 0, the cost is C(0) = 1000. This is the fixed cost the restaurant pays regardless of output.
General form: y = mx + b.
b (the y-intercept) is where the graph hits the vertical axis, at x = 0.
m (the slope) tells you how much y changes for each one-unit increase in x.
Slope formula: m = Δy / Δx (rise over run).
Graphing examples
f₁(x) = 3 is a horizontal line at y = 3. Slope is 0, intercept is 3.
f₂(x) = 1 − 0.5x has intercept 1 and slope −0.5 (the line falls as x increases).
Example: demand and inverse demand
Demand function: Q(P) = 8 − 2P, valid for 0 ≤ P ≤ 4.
Economists plot P on the vertical axis and Q on the horizontal axis, so you need to rearrange.
Solving for P gives the inverse demand function: P = 4 − 0.5Q.
The P-intercept is 4 (price at which quantity demanded drops to zero).
The Q-intercept is 8 (quantity demanded when price is zero).
For a linear function, the rate of change is the same everywhere (it is just the slope m).
For a non-linear function, the rate of change varies from point to point. The derivative captures the instantaneous rate at a specific point.
Definition
For a smooth function f at the point x₀, the derivative f'(x₀) is the slope of the tangent line to the graph at (x₀, f(x₀)).
Equivalent notations: f'(x₀) and df/dx(x₀).
Think of the "d" in df/dx as "delta" (a small change): f'(x₀) ≈ Δf / Δx.
Using the derivative to approximate change
If you know f'(x₀) and x changes by a small amount Δx, then the change in f is approximately Δf ≈ f'(x₀) × Δx.
Example: f'(5) = 3 and x increases from 5 to 7, so Δx = 2. Then Δf ≈ 3 × 2 = 6.
Constant rule: if f(x) = c, then f'(x) = 0.
Power rule: if f(x) = x^k (k any real number), then f'(x) = kx^(k−1).
Special case: f(x) = 1/x = x^(−1), so f'(x) = −x^(−2) = −1/x².
Special case: f(x) = √x = x^(1/2), so f'(x) = (1/2)x^(−1/2) = 1/(2√x).
Constant multiple rule: (kf)'(x) = k · f'(x).
Sum rule: (f + g)'(x) = f'(x) + g'(x).
Product rule: (fg)'(x) = f'(x)g(x) + f(x)g'(x).
Given a total cost function C(q) = 0.5q² + 10:
C'(q) = [0.5q²]' + [10]' = 0.5 · 2q + 0 = q.
At q = 4, the marginal cost is C'(4) = 4.
The derivative of the total cost function is called the marginal cost. It tells you, roughly, how much extra it costs to produce one more unit of output.
Name | Formula |
|---|---|
Linear function | y = mx + b |
Slope | m = Δy / Δx |
Inverse demand (from the example) | P = 4 − 0.5Q |
Power rule | f(x) = x^k → f'(x) = kx^(k−1) |
Constant multiple rule | (kf)'(x) = k · f'(x) |
Sum rule | (f + g)'(x) = f'(x) + g'(x) |
Product rule | (fg)'(x) = f'(x)g(x) + f(x)g'(x) |
Derivative approximation | Δf ≈ f'(x₀) × Δx |
⚠️ The derivative of total cost = marginal cost. This concept reappears throughout the entire course. Know it cold.
⚠️ Economists flip the axes: P goes on the vertical axis, Q on the horizontal. You will need to rearrange demand functions into inverse form regularly.
⚠️ The derivative approximation (Δf ≈ f'(x₀) × Δx) is a common exam setup. They give you f'(x₀) and Δx, and you multiply.
⚠️ When differentiating 1/x or √x, rewrite them as x^(−1) or x^(1/2) first, then apply the power rule. Forgetting to rewrite is a common slip.
⚠️ The product rule has two terms. Do not confuse it with the sum rule, which just adds the derivatives.
Q: A function f has f'(10) = 5. If x increases from 10 to 12, what is the approximate change in f(x)?
A: Δx = 2, so Δf ≈ 5 × 2 = 10.
Q: What is the inverse demand function if Q(P) = 12 − 3P?
A: Solve for P: 3P = 12 − Q, so P = 4 − (1/3)Q.
Q: Find the derivative of f(x) = 4x³ + 7.
A: f'(x) = 4 · 3x² + 0 = 12x².
Q: If C(q) = 2q² + 50, what is the marginal cost at q = 6?
A: C'(q) = 4q, so C'(6) = 24. The marginal cost at 6 units of output is 24.
Q: What does it mean for f'(x₀) to equal zero?
A: The function is flat at that point. The tangent line is horizontal, meaning the function is neither increasing nor decreasing at x₀.
Q: Differentiate f(x) = √x and evaluate at x = 9.
A: Rewrite as x^(1/2). f'(x) = (1/2)x^(−1/2) = 1/(2√x). At x = 9, f'(9) = 1/(2 · 3) = 1/6.
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