Class 12 Maths Calculus Master Guide – Continuity, Differentiation & Integration

Class 12 Maths mein Calculus ka weightage 35 marks hai — total 80-marks paper ka 44%! Agar aap Calculus strong kar leta hain, toh 90+ almost guaranteed hai.

Marks Distribution

• Continuity and Differentiability – 8 marks
• Application of Derivatives – 6 marks
• Integrals – 9 marks (highest)
• Application of Integrals – 5 marks
• Differential Equations – 7 marks

Total = 35 marks (44% of 80).

Continuity & Differentiability

f(x) continuous at x=a if f(a) exists, lim_x→a f(x) exists, aur dono equal. Differentiable if LHD = RHD = finite. Every differentiable function is continuous, but not vice versa (|x| at 0).

Rules of Differentiation

• d/dx(xⁿ) = nxⁿ⁻¹
• d/dx(sin x) = cos x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x
• Chain: d/dx f(g(x)) = f'(g(x))·g'(x)
• Product: (uv)' = u'v + uv'; Quotient: (u/v)' = (u'v − uv')/v²

Special: implicit, parametric (dy/dx = (dy/dt)/(dx/dt)), logarithmic differentiation.

Application of Derivatives

Rate of change, increasing/decreasing (f'>0 / f'<0), tangent & normal, maxima & minima (first & second derivative test). Board mein 5-mark maxima-minima word problem — systematic 5-step approach.

Integrals (9 Marks — Highest)

• ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
• ∫1/x dx = ln|x| + C; ∫eˣ dx = eˣ + C
• ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C

Techniques: substitution, integration by parts (ILATE), partial fractions, trig substitutions. Definite integral = F(b) − F(a), plus 8 NCERT properties.

Applications & Differential Equations

Area under curve ∫f(x)dx; area between curves ∫(upper − lower)dx. DE: order & degree, variable separable, linear (IF = e^∫P dx), homogeneous (y = vx).

Common Mistakes

• '+C' bhoolna in indefinite integrals.
• Chain rule skip — d/dx(sin 2x) = 2cos 2x.
• Definite integral limits — substitution ke saath naye limits.
• Diagrams nahi banana in area problems.

Frequently Asked Questions