716 marks

Limit, continuity and differentiability

This chapter covers the fundamental concepts of calculus: limits, continuity, and differentiability, forming the basis for differentiation and its applications.

21 Topics

45h prep

5% subject weight

21 Topics

Real valued functions

2m2/10

📌 Key FormulaDomain and range of functions like polynomial, rational, trigonometric, etc.

Algebra of functions

2m2/10

📌 Key Formula(f ± g)(x) = f(x) ± g(x), (fg)(x) = f(x)g(x), (f/g)(x) = f(x)/g(x), domain considerations.

Polynomials functions

2m2/10

📌 Key Formulaf(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ, degree n, behavior, zeroes.

Rational functions

2m2/10

📌 Key Formulaf(x) = p(x)/q(x), p and q polynomials, domain excludes points where q(x)=0.

Trigonometric functions

2m3/10

📌 Key Formulaf(x) = sin x, cos x, tan x etc., their domains, ranges, periods.

Logarithmic functions

2m3/10

📌 Key Formulaf(x) = logₐ x, domain (0, ∞), range (-∞, ∞), a > 0, a ≠ 1.

Exponential functions

2m2/10

📌 Key Formulaf(x) = aˣ, a > 0, a ≠ 1, domain (-∞, ∞), range (0, ∞).

Inverse function

2m4/10

📌 Key Formulaf⁻¹ defined if f is bijective, f(f⁻¹(x)) = f⁻¹(f(x)) = x.

Graphs of simple functions

2m3/10

📌 Key FormulaGraph shapes of linear, quadratic, cubic, sin, cos, tan, exponential, logarithmic functions.

Basics of Limits, continuity and differentiability

2m3/10

📌 Key FormulaLimit definition, LHL = RHL for existence. Continuity: f(a) = lim_{x→a} f(x). Differentiability: derivative exists.

Differentiation of the sum, difference, product and quotient of two functions

2m3/10

📌 Key Formulad/dx (u ± v) = u' ± v', d/dx (uv) = u'v + uv', d/dx (u/v) = (u'v - uv')/v²

Differentiation of trigonometric, inverse trigonometric functions

2m3/10

📌 Key Formulad/dx sin x = cos x, d/dx cos x = - sin x, d/dx tan x = sec²x, d/dx sin⁻¹x = 1/√(1-x²), etc.

Differentiation of logarithmic and exponential functions

2m3/10

📌 Key Formulad/dx eˣ = eˣ, d/dx aˣ = aˣ ln a, d/dx ln x = 1/x, d/dx logₐ x = 1/(x ln a)

Differentiation of composite functions

2m4/10

📌 Key FormulaChain rule: d/dx f(g(x)) = f'(g(x)) g'(x)

Derivatives of order up to two

2m3/10

📌 Key FormulaSecond derivative: d²y/dx² = f''(x)

Lagrange's Mean value Theorems

2m5/10

📌 Key FormulaIf f is continuous on [a,b] and differentiable on (a,b), then ∃ c ∈ (a,b): f'(c) = (f(b)-f(a))/(b-a)

Applications of derivatives

2m3/10

📌 Key FormulaRate of change, increasing/decreasing functions, tangents and normals.

Rate of change of quantities

2m3/10

📌 Key Formulady/dx as rate of change of y with respect to x.

Monotonic Increasing and decreasing functions

2m3/10

📌 Key Formulaf'(x) > 0 ⇒ increasing, f'(x) < 0 ⇒ decreasing

Maxima and minima of functions of one variable

2m5/10

📌 Key FormulaFirst derivative test, second derivative test: f'(c)=0, f''(c)>0 ⇒ local min, f''(c)<0 ⇒ local max.

Tangents and normal

2m3/10

📌 Key FormulaTangent: y - y₀ = f'(x₀)(x - x₀), Normal: y - y₀ = -1/f'(x₀)(x - x₀)

716 marks

Limit, continuity and differentiability

This chapter covers the fundamental concepts of calculus: limits, continuity, and differentiability, forming the basis for differentiation and its applications.

21 Topics

45h prep

5% subject weight

21 Topics

Real valued functions

2m2/10

📌 Key FormulaDomain and range of functions like polynomial, rational, trigonometric, etc.

Algebra of functions

2m2/10

📌 Key Formula(f ± g)(x) = f(x) ± g(x), (fg)(x) = f(x)g(x), (f/g)(x) = f(x)/g(x), domain considerations.

Polynomials functions

2m2/10

📌 Key Formulaf(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ, degree n, behavior, zeroes.

Rational functions

2m2/10

📌 Key Formulaf(x) = p(x)/q(x), p and q polynomials, domain excludes points where q(x)=0.

Trigonometric functions

2m3/10

📌 Key Formulaf(x) = sin x, cos x, tan x etc., their domains, ranges, periods.

Logarithmic functions

2m3/10

📌 Key Formulaf(x) = logₐ x, domain (0, ∞), range (-∞, ∞), a > 0, a ≠ 1.

Exponential functions

2m2/10

📌 Key Formulaf(x) = aˣ, a > 0, a ≠ 1, domain (-∞, ∞), range (0, ∞).

Inverse function

2m4/10

📌 Key Formulaf⁻¹ defined if f is bijective, f(f⁻¹(x)) = f⁻¹(f(x)) = x.

Graphs of simple functions

2m3/10

📌 Key FormulaGraph shapes of linear, quadratic, cubic, sin, cos, tan, exponential, logarithmic functions.

Basics of Limits, continuity and differentiability

2m3/10

📌 Key FormulaLimit definition, LHL = RHL for existence. Continuity: f(a) = lim_{x→a} f(x). Differentiability: derivative exists.

Differentiation of the sum, difference, product and quotient of two functions

2m3/10

📌 Key Formulad/dx (u ± v) = u' ± v', d/dx (uv) = u'v + uv', d/dx (u/v) = (u'v - uv')/v²

Differentiation of trigonometric, inverse trigonometric functions

2m3/10

📌 Key Formulad/dx sin x = cos x, d/dx cos x = - sin x, d/dx tan x = sec²x, d/dx sin⁻¹x = 1/√(1-x²), etc.

Differentiation of logarithmic and exponential functions

2m3/10

📌 Key Formulad/dx eˣ = eˣ, d/dx aˣ = aˣ ln a, d/dx ln x = 1/x, d/dx logₐ x = 1/(x ln a)

Differentiation of composite functions

2m4/10

📌 Key FormulaChain rule: d/dx f(g(x)) = f'(g(x)) g'(x)

Derivatives of order up to two

2m3/10

📌 Key FormulaSecond derivative: d²y/dx² = f''(x)

Lagrange's Mean value Theorems

2m5/10

📌 Key FormulaIf f is continuous on [a,b] and differentiable on (a,b), then ∃ c ∈ (a,b): f'(c) = (f(b)-f(a))/(b-a)

Applications of derivatives

2m3/10

📌 Key FormulaRate of change, increasing/decreasing functions, tangents and normals.

Rate of change of quantities

2m3/10

📌 Key Formulady/dx as rate of change of y with respect to x.

Monotonic Increasing and decreasing functions

2m3/10

📌 Key Formulaf'(x) > 0 ⇒ increasing, f'(x) < 0 ⇒ decreasing

Maxima and minima of functions of one variable

2m5/10

📌 Key FormulaFirst derivative test, second derivative test: f'(c)=0, f''(c)>0 ⇒ local min, f''(c)<0 ⇒ local max.

Tangents and normal

2m3/10

📌 Key FormulaTangent: y - y₀ = f'(x₀)(x - x₀), Normal: y - y₀ = -1/f'(x₀)(x - x₀)