South 고등학교 3학년 Mathematics II
Chapters: 3
1. Limits and continuity
Limits of functions · Limit laws · Continuity · Continuous 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.
- 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.
2. Differentiation
Derivative at a point · Geometric meaning · Differentiability and continuity · Derivative of xⁿ · Differentiation rules · Tangent lines · Mean value theorem · Increase, decrease, extrema · Graph sketching · Equations and inequalities · Velocity and acceleration
- 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.
- Mean Value Theorem and Rolle's Theorem – If a function is continuous on [a, b] and differentiable on (a, b), there is at least one point c between a and b where the tangent slope equals the average slope: f'(c) = (f(b) − f(a)) / (b − a). Rolle's theorem is the special case f(a) = f(b), where f'(c) = 0.
- Curve Sketching Using Derivatives – To sketch y = f(x): find the domain and intercepts, solve f'(x) = 0 for stationary points, use the sign of f'(x) to see where the curve rises or falls, use f''(x) for concavity and points of inflection, check asymptotes and end behaviour, then join everything smoothly. Example: y = x³ − 3x has a maximum at (−1, 2), a minimum at (1, −2) and an inflection at (0, 0).
- Kinematics with Calculus: Velocity, Acceleration, Distance – If position is s(t), then velocity is v = ds/dt and acceleration is a = dv/dt. Going back, displacement is the integral of v dt. Distance travelled uses |v|. In the plane, speed = √(x′² + y′²), and the length of a curve is the integral of that speed.
3. Integration
Antiderivatives · Integration rules · Definite integrals · Definite integrals of polynomials · Areas · Velocity and distance
- 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.
- Kinematics with Calculus: Velocity, Acceleration, Distance – If position is s(t), then velocity is v = ds/dt and acceleration is a = dv/dt. Going back, displacement is the integral of v dt. Distance travelled uses |v|. In the plane, speed = √(x′² + y′²), and the length of a curve is the integral of that speed.