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Calculus

Limits, continuity, differentiation, integration, and differential equations

Key Concepts
  • Limits and continuity form the foundation of calculus
  • Differentiation gives the instantaneous rate of change
  • Applications: maxima, minima, tangents and rates of change
  • Integration is the reverse of differentiation (area under a curve)
  • Definite integrals evaluate accumulated change over an interval
Important Formulae
Power rule d/dx(xⁿ) = n·xⁿ⁻¹
Derivatives of sin/cos d/dx(sin x) = cos x; d/dx(cos x) = −sin x
Product rule (uv)' = u'v + uv'
Chain rule dy/dx = (dy/du)(du/dx)
Power rule (integration) ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
Integral of 1/x ∫(1/x) dx = ln|x| + C
Quick Tips
  • At a maximum or minimum, the first derivative is zero; use the second derivative to classify it.
  • Always add the constant of integration C to indefinite integrals.
  • Check whether a limit is an indeterminate form before applying L'Hôpital's rule.
Sample Practice Questions

  1. The derivative of a constant is:

    • 0
    • 1
    • The constant itself
    • Undefined
    Show answer

    Answer: 0

  2. The derivative of x³ with respect to x is:

    • 3x²
    • x²
    • 3x
    • x⁴/4
    Show answer

    Answer: 3x²

  3. The derivative of sin x with respect to x is:

    • cos x
    • −cos x
    • −sin x
    • tan x
    Show answer

    Answer: cos x

  4. The indefinite integral ∫ x dx is:

    • x² + C
    • x²/2 + C
    • 1 + C
    • 2x + C
    Show answer

    Answer: x²/2 + C

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Practice Questions

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