The ideas behind engineering, data science and economics: functions, limits, derivatives, integrals, matrices and vectors. Kanakku keeps a picture in front of each idea, such as a curve with its tangent or sets drawn as overlapping circles, so the formulas mean something before you start drilling them.
Union, intersection, Venn diagrams and counting subsets.
A set is a collection of things. The Venn diagram shows how two sets overlap.
Values, domain and range, and counting relations.
A function gives exactly one output for each input. The domain is the inputs allowed; the range is the outputs you can get.
Radians, all four quadrants, and values beyond 90°.
On the unit circle, cos θ is the x-coordinate and sin θ is the y-coordinate. That works for any angle, not just up to 90°.
i² = −1, adding, multiplying and the modulus.
A complex number a + bi has a real part a and an imaginary part b. Work with i like a letter, then replace i² by −1.
Solve inequalities, and remember to flip the sign when dividing by a negative.
Solve an inequality like an equation, with one extra rule: multiplying or dividing by a negative number flips the sign.
Counting arrangements and selections.
Permutations count arrangements, where order matters. Combinations count selections, where order does not.
Expanding (a + b)ⁿ with Pascal’s triangle and ⁿCᵣ.
(a + b)ⁿ expands into n + 1 terms. The coefficients are ⁿC₀, ⁿC₁, …, ⁿCₙ: a row of Pascal’s triangle.
Geometric progressions, their sums, and AM and GM.
In a geometric progression, each term is the one before multiplied by the same ratio r.
Slope, intercepts, and parallel and perpendicular lines.
The slope m tells how steep a line is: rise ÷ run. Lines in the form y = mx + c have slope m and cross the y-axis at c.
The equation of a circle: centre and radius.
(x − h)² + (y − k)² = r² is a circle with centre (h, k) and radius r. It comes straight from the distance formula.
Limits, the derivative as slope, and the power rule.
A limit is the value a function approaches. The derivative is the slope of the curve at a point: the limit of rise ÷ run as the run shrinks to 0.
Mean, variance and standard deviation.
Variance measures how spread out data is: the average of the squared distances from the mean. Standard deviation is its square root.
Sample spaces, the addition rule and “at least one”.
P(A or B) = P(A) + P(B) − P(A and B). For “at least one”, it is often easiest to use 1 − P(none).
