South 고등학교 2학년 Calculus II
Chapters: 3
1. Limits of sequences
Convergence of sequences · Limit laws for sequences · Geometric sequences' limits · Series · Geometric series
- Limits of Sequences: Where Do the Terms Go? – A sequence converges to a limit L if its terms get as close to L as we like and stay close from some term onward. If it does not settle on one number, it diverges. Limit laws let us add, multiply and divide limits. The geometric sequence rⁿ converges to 0 when |r| < 1, to 1 when r = 1, and diverges otherwise.
- 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.
2. Differentiation techniques
Limits and derivatives of exponentials and logs · Trig addition formulas · Limits and derivatives of sine and cosine · Quotient rule · Chain rule · Parametric differentiation · Implicit and inverse differentiation · Tangents to curves · Graph shapes and concavity · Equations and inequalities · Velocity and acceleration in the plane
- 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.
- Trigonometric Functions: Radians, Unit Circle, Graphs and Identities – An angle of one radian cuts an arc equal to the radius, so π radians = 180°. On a unit circle, the point at angle x is P = (cos x, sin x), which gives sin²x + cos²x = 1 and extends sine and cosine to every real number. The signs follow 'All, Sin, Tan, Cos' in quadrants I to IV; sin x and cos x repeat every 2π and stay between −1 and 1. Compound-angle formulas such as cos(A + B) = cos A cos B − sin A sin B lead to tan(A + B), cot(A + B), sum-to-product, double-angle and triple-angle identities.
- 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.
- 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 techniques
Integrals of xⁿ, exponentials, trig · Integration by substitution · Integration by parts · Definite integrals as sums · Area problems · Volumes of solids · Distance travelled
- 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.
- Solids of Revolution – A solid of revolution is the 3D shape you get when a flat shape turns a full 360° around a straight line (the axis). A rectangle makes a cylinder, a right triangle makes a cone, a half-circle makes a sphere and a trapezium makes a frustum. Their volumes are V = πr²h, V = ⅓πr²h, V = ⁴⁄₃πr³ and V = ⅓πh(R² + Rr + r²). In calculus, any curve y = f(x) turned about the x-axis gives V = π∫y² dx.
- 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.