Source: Lecture 1B, Math Review
Tags: math review, lines, curves, slope, y-intercept, derivative, three-step rule, power rule, tangent line, finding the maximum, optimisation, profit maximisation, elasticity, percentages, demand curve, slope-intercept form, calculus review, intermediate micro
Economics uses words, graphs, and mathematics together. This math review covers the essentials you need before the course moves forward: how lines and curves behave, how to find and interpret slopes (using derivatives or the three-step rule), and how to find the maximum of a function by setting the derivative equal to zero. These tools come up repeatedly in demand, supply, cost, and profit analysis.
Slope (m)
The rate of change of Y with respect to X. For a line, slope is constant. For a curve, slope varies depending on where you are. Slope = rise / run. In economics, slope captures the relationship between two variables (e.g. how price changes as quantity changes).
Y-intercept (b)
The value of Y when X = 0. In the equation Y = mX + b, the y-intercept is b. It is the point where the line crosses the vertical axis.
Slope-intercept form
The standard form Y = mX + b. Any linear equation can be rearranged into this form. If an equation cannot be written this way, it is a curve, not a line.
Derivative (∂Y/∂X)
The mathematical tool for finding the slope of a curve at a specific point. For lines, the derivative equals the constant slope. For curves, the derivative is a function of X, meaning the slope changes as X changes.
Three-step rule (power rule for derivatives)
A simplified method for taking derivatives of polynomials:
Step 1: Eliminate (drop) all constants (terms with no variable).
Step 2: Bring the power down and put it in front of the variable as a coefficient.
Step 3: Reduce the power by one.
This works for basic polynomials. It does not work for special functions like log(x).
Tangent line
A straight line that touches a curve at exactly one point and has the same slope as the curve at that point. Drawing a tangent line is the graphical way to find a curve's slope at a given location.
Concave function
A curve that opens downward (like an upside-down bowl). Concave functions have a maximum point where the slope equals zero.
Maximum of a function
The highest point on a concave curve. Found by taking the derivative, setting it equal to zero, and solving for the variable. At the maximum, the slope of the function is zero.
Every line can be written in Y = mX + b form, where m is the slope and b is the y-intercept.
Example: Y = 5X + 1. The slope is 5 (for every one-unit increase in X, Y increases by 5). The y-intercept is 1.
The slope of a line is constant everywhere. This is the defining feature of a line.
In economics, the same structure appears with different variable names. A demand curve written as P = -(5/2)Qd + 5 is still a line: slope is -5/2 and y-intercept is 5. For every one-unit increase in quantity demanded, price falls by 5/2.
If an equation cannot be written in Y = mX + b form, it is a curve.
Example: Y = X² + 5. The squared term means the slope is not constant.
At higher values of X, the slope gets smaller (for a curve like Y = √X) or larger (for Y = X²), depending on the function's shape.
A curve can increase at a decreasing rate (slope is positive but shrinking) or increase at an increasing rate (slope is positive and growing).
The slope at any point on a curve is found using the derivative.
Graphical method: draw a tangent line at the point of interest. The slope of that tangent line is the slope of the curve at that point.
Algebraic method (three-step rule):
Drop all constant terms.
Bring the exponent down as a coefficient.
Reduce the exponent by one.
Example: Y = X² + 5
Drop the constant 5.
Bring the power (2) down: 2X².
Reduce the power by one: 2X¹ = 2X.
Result: ∂Y/∂X = 2X.
The slope depends on X. At X = 1, slope = 2. At X = 3, slope = 6. The slope changes, which is what makes it a curve rather than a line.
Slope-intercept form
Y = mX + b
Derivative (three-step rule) example
Y = X² + 5 → ∂Y/∂X = 2X
Finding the maximum of a profit function
∏ = 2Q - 0.1Q² - 3.6
Step 1: Take the derivative. ∂∏/∂Q = 2 - 0.2Q
Step 2: Set the derivative equal to zero. 2 - 0.2Q = 0
Step 3: Solve. 0.2Q = 2 Q* = 10
Profit is maximised at Q = 10.
Note: at a minimum, the slope is also zero. The second derivative test distinguishes maxima from minima, but you can ignore that for this course.
⚠️ You must be able to convert any linear equation into Y = mX + b form and immediately identify the slope and y-intercept.
⚠️ Know the three-step rule cold. If you cannot take a basic derivative, every optimisation problem in the course becomes a wall.
⚠️ "Find the maximum" is the single most repeated mathematical task in intermediate micro: profit maximisation, utility maximisation, cost minimisation all use the same method (derivative = 0, solve).
⚠️ Be able to interpret what a slope means in economic context, not just compute it. "For every one-unit increase in Q, price falls by 5/2" is the kind of interpretation the exam expects.
⚠️ Know the difference between a line (constant slope, Y = mX + b form) and a curve (changing slope, cannot be written in that form). If you are given a graph, you should be able to identify which is steeper at a given point.
⚠️ Students must score at least 8 out of 10 on the math quizzes (best of three attempts) to continue in the course. Take the practice quizzes seriously.
Q: What is the slope and y-intercept of the equation P = -(5/2)Qd + 5?
A: The slope is -5/2 and the y-intercept is 5. For every one-unit increase in quantity demanded, price falls by 2.5.
Q: Use the three-step rule to find the derivative of Y = 3X² + 7X - 4.
A: Drop the constant (-4). Bring powers down and reduce by one: 3·2·X^(2-1) + 7·1·X^(1-1) = 6X + 7. So ∂Y/∂X = 6X + 7.
Q: Find the quantity that maximises the profit function ∏ = 2Q - 0.1Q² - 3.6.
A: Take the derivative: ∂∏/∂Q = 2 - 0.2Q. Set equal to zero: 2 - 0.2Q = 0, so Q* = 10. Profit is maximised when Q = 10.
Q: Why is the slope of a curve different from the slope of a line?
A: A line has a constant slope everywhere. A curve's slope changes depending on the value of X, because the derivative is itself a function of X rather than a fixed number.
Q: How do you graphically determine the slope of a curve at a particular point?
A: Draw a tangent line at that point. The slope of the tangent line equals the slope of the curve at that location.
Q: If Y = X² + 5, what is the slope at X = 4?
A: The derivative is 2X. At X = 4, the slope is 2(4) = 8.
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