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. These topics represent some of the highest-value questions on the CSCA exam.
This module bridges pure math and the real world. Let's begin!
Calculus and statistics typically account for 8-10 questions worth 30-40 marks on the CSCA exam — the largest single module. Expect: limit evaluation, differentiation with chain/product/quotient rules, finding and classifying turning points, optimisation word problems, definite integral calculations, area between curves, probability computations (including binomial), and statistics interpretation. These are high-value questions that reward methodical working.
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
One-Sided Limits
lim(x→a⁺) = limit from the right.lim(x→a⁻) = limit from the left.Limits at Infinity
For rational functions, compare the degrees of numerator and denominator:
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.
💡A limit is what f(x) approaches as x→a. Try direct substitution first; if 0/0, factor or use L'Hopital's. Continuous = no holes, jumps, or breaks.
📋 Key Formulas
0/0 → factor or L'Hôpital
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. (Can verify with L'Hôpital: cos(x)/1 at x=0 = 1.)
🧠Always try direct substitution first — many exam limits are designed to work immediately.
🧠For 0/0 forms, factoring is usually faster than L'Hopital's Rule.
⚠️Dividing by zero and writing ∞: 1/0 is undefined, not ∞. Only limits can equal ∞.
⚠️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).
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). These are the most powerful tools in the CSCA toolkit and carry the highest exam marks. Practise these skills daily!
Open and read all sections to complete this module