Japan 高校3年 Mathematics III

Chapters: 3

1. Limits

Limits of sequences · Infinite series · Rational and irrational functions · Composite and inverse functions · Limits and continuity of functions

Limits (Class 11): What a Function Gets Close To – A limit tells us which number f(x) gets close to when x gets close to a point. We walk towards the point from the left and from the right. If both sides reach the same number, that number is the limit. You will learn limits of polynomial, rational, trigonometric, exponential and log functions, with the standard results you must know.
Infinite Series – An infinite series adds the terms of a sequence forever: a₁ + a₂ + a₃ + … We study it through its partial sums Sₙ. If Sₙ settles at a number S, the series converges to S; otherwise it diverges. A geometric series a + ar + ar² + … converges to a/(1 − r) when |r| < 1. Terms going to 0 is needed but not enough: the harmonic series 1 + 1/2 + 1/3 + … diverges. Tests (nth-term, p-series, comparison, integral, ratio, alternating) tell us which series converge.
Relations and Functions (Class 12) – A relation R on a set A is any set of pairs (a, b) taken from A × A. R is reflexive if every element is related to itself, symmetric if (a, b) in R always brings (b, a), and transitive if (a, b) and (b, c) always bring (a, c). A relation with all three is an equivalence relation; it cuts A into separate equivalence classes. A function f: A → B sends every element of A to exactly one element of B. It is one-one (injective) if different inputs give different outputs, onto (surjective) if every element of B is hit, and bijective if it is both.

2. Differentiation

Differentiability; product and quotient rules · Chain rule · Derivatives of trig, exponential and log functions · Applications: graphs, rates, approximations

Continuity and Differentiability – A function is continuous at a point when its graph has no break there: the left limit, the right limit and the value are all equal. The derivative is the slope of the tangent line. With the chain rule, implicit differentiation, the rules for eˣ, ln x and inverse trig functions, logarithmic differentiation and parametric forms, you can differentiate almost any Class 12 function. The second derivative tells how the slope itself changes, that is, how the curve bends.
Application of Derivatives – The derivative measures how fast one quantity changes compared with another. Its sign tells whether a function goes up (f′ > 0, increasing) or down (f′ < 0, decreasing). Where f′ = 0 the tangent is flat: these critical points may be a local maximum or minimum, checked by the first derivative test (sign change) or the second derivative test (sign of f″). This lets us solve real problems like the biggest box or the cheapest tank.

3. Integration

Basic integrals · Substitution and by parts · Areas, volumes, curve lengths

Integrals – Integration undoes differentiation: if F′(x) = f(x), then ∫f(x) dx = F(x) + C. We integrate with standard formulas, substitution (replace an inside part by u), partial fractions (split a fraction) and by parts (∫u dv = uv − ∫v du). A definite integral ∫ₐᵇ f(x) dx is the signed area under the curve from a to b, and the fundamental theorem says it equals F(b) − F(a). Its properties make many hard integrals easy.
Application of Integrals: Area Under Curves – The area between a curve y = f(x), the x-axis and the lines x = a and x = b is ∫ₐᵇ |f(x)| dx: we add thin vertical strips of height y and width dx. For curves given as x = g(y) we use horizontal strips. Symmetry saves work: find one part of a circle, parabola or ellipse and multiply. A circle of radius r gives πr² and an ellipse with semi-axes a, b gives πab.