Calculus & Statistics
Welcome to Calculus & Statistics — the most powerful and modern tools in mathematics. Calculus lets you understand how things change (derivatives) and how things accumulate (integrals). Statistics and probability let you make sense of data and uncertainty.
You will begin with limits and continuity, master derivative rules including the chain rule, apply derivatives to real-world optimisation problems, learn integration basics, and then switch gears to probability and statistics — counting techniques, distributions, and data analysis. The syllabus lists the basics of derivatives and calculus under Functions, and classical probability, mean and variance, and the normal distribution under Probability and Statistics. A few topics here go beyond that, such as L'Hôpital's rule, integration by substitution, volumes of revolution and the Poisson distribution, and the sections mark them.
This module bridges pure math and the real world. Let's begin!
What the Exam Tests
This module joins two syllabus areas. Derivatives and integration belong to Functions, whose syllabus line is the basics of derivatives and calculus: the definition, the geometric meaning and simple applications such as tangents, monotonicity and extrema. Probability and Statistics is an area of its own: classical probability, the numerical characteristics of data such as mean and variance, and the basic concepts of the normal distribution.
Questions can ask for the slope of a tangent, the interval on which a function increases, a maximum or minimum, a simple definite integral, a probability found by counting equally likely outcomes, a mean or a variance, or a probability read off a normal curve by symmetry.
In our mock papers Probability and Statistics gets 8 of the 48 questions, and the derivative lessons count toward the 16 of Functions. The CSCA publishes the topics but not how many questions each gets, so this split is our own reading of the syllabus. A few practice questions go beyond the syllabus (L'Hôpital's rule, moment generating functions, hypothesis tests): they are marked and left out of mocks. Some lessons also include enrichment beyond the syllabus (L'Hôpital's rule, inverse-trigonometric derivatives, related rates, points of inflection, integration by substitution, volumes of revolution, the Poisson distribution, standard errors and confidence intervals), each marked where it appears.
A limit describes the value a function approaches as x gets closer to a certain number. We write lim(x→a) f(x) = L, meaning f(x) gets as close to L as we want when x is near a.
Evaluating Limits
- Direct substitution: Try plugging in x = a. If you get a number, that is the limit.
- 0/0 indeterminate form: Factor, simplify, then substitute again.
- Rationalise: If there is a square root, multiply by the conjugate.
- L'Hôpital's Rule: If you get 0/0 or ∞/∞, take the derivative of top and bottom separately, then try again. This rule is beyond the CSCA syllabus and Chinese senior-high textbooks, so treat it as enrichment or a check.
The syllabus does not list limits as a topic of their own; it needs only the intuitive idea of a limit behind the definition of the derivative (导数的定义), so treat this section as background for the derivative.
One-Sided Limits
lim(x→a⁺)= limit from the right.lim(x→a⁻)= limit from the left.- The two-sided limit exists only if both one-sided limits exist and are equal.
Limits at Infinity
For rational functions, compare the degrees of numerator and denominator:
- Degree of top < bottom → limit = 0.
- Degrees equal → limit = ratio of leading coefficients.
- Degree of top > bottom → limit = ±∞.
Continuity
f is continuous at x = a if: (1) f(a) exists, (2) lim(x→a) f(x) exists, (3) they are equal. Informally, you can draw the graph without lifting your pen.
Key takeaway: A limit is what f(x) approaches as x→a. Try direct substitution first; if 0/0, factor (or rationalise) and substitute again. L'Hôpital's rule is beyond the syllabus, only a check. Continuous = no holes, jumps, or breaks.
Key Formulas
0/0 → factor or rationalise (L'Hôpital: beyond the syllabus)
Degrees: top < bottom → 0 | equal → leading ratio | top > bottom → ±∞
lim(x→0) sin(x)/x = 1
Worked Example 1
Example 1: Find lim(x→2) (x² - 4)/(x - 2).
Direct sub gives 0/0. Factor: (x-2)(x+2)/(x-2) = x+2. Limit = 2+2 = 4.
Worked Example 2
Example 2: Find lim(x→∞) (3x² + 1)/(5x² - 2).
Degrees equal (both 2). Ratio of leading coefficients: 3/5 = 0.6.
Worked Example 3
Example 3: Find lim(x→0) sin(x)/x.
This is a famous limit = 1 (x in radians). It is quoted, not derived, at this level; you can check it numerically: sin(0.01)/0.01 ≈ 0.99998.
Exam tip: Always try direct substitution first — many exam limits are designed to work immediately.
Exam tip: For 0/0 forms, factoring is usually faster than L'Hopital's Rule.
Common mistake: Dividing by zero and writing ∞: 1/0 is undefined, not ∞. Only limits can equal ∞.
Common mistake: Cancelling without checking: (x-2)/(x-2) = 1 only when x ≠ 2. For limits this is fine, but f(2) is still undefined.
Try This Yourself
Find lim(x→3) (x² - 9)/(x - 3).
Module Summary
In this module you conquered calculus — from limits and continuity through derivatives (power, chain, product, quotient rules) and their applications (tangents, optimisation), to integration and area calculations. You then explored probability (counting, conditional probability, binomial distribution) and statistics (mean, median, variance, standard deviation, quartiles, the normal distribution). These are powerful tools, and the syllabus lists them under Functions (derivatives) and Probability and Statistics. Practise these skills daily!
Open and read all sections to complete this module