Calculus Practice Questions

### 1. Definition Study of continuous change — limits, differentiation, integration, and their applications. ### 2. Limits & Continuity **Standard Limits (Very Important)** - lim(x→0) sin x / x = 1 - lim(x→0) (1 + x)^(1/x) = e - lim(x→0) (eˣ − 1) / x = 1 - lim(x→0) ln(1 + x) / x = 1 - lim(x→∞) (1 + 1/x)ˣ = e **Continuity**: f is continuous at x=a if lim(x→a) f(x) = f(a). Must check: limit exists, f(a) defined, both equal. **L'Hôpital's Rule**: If limit gives 0/0 or ∞/∞ form → differentiate numerator and denominator separately. ### 3. Differentiation (Very Important) **Key Rules** - Product: (uv)' = u'v + uv' - Quotient: (u/v)' = (u'v − uv') / v² - Chain: d/dx[f(g(x))] = f'(g(x))·g'(x) **Standard Derivatives** - d/dx(xⁿ) = nxⁿ⁻¹ - d/dx(eˣ) = eˣ - d/dx(ln x) = 1/x - d/dx(sin x) = cos x, d/dx(cos x) = −sin x - d/dx(tan x) = sec²x - d/dx(sin⁻¹x) = 1/√(1−x²) - d/dx(tan⁻¹x) = 1/(1+x²) ### 4. Partial Derivatives For f(x,y): ∂f/∂x treats y as constant. ∂²f/∂x∂y = ∂²f/∂y∂x (Clairaut's theorem, if continuous). **Total Derivative**: df = (∂f/∂x)dx + (∂f/∂y)dy **Euler's Theorem** (for homogeneous function of degree n): x·(∂f/∂x) + y·(∂f/∂y) = n·f ### 5. Integration (Very Important) **Standard Integrals** - ∫xⁿ dx = xⁿ⁺¹/(n+1) + C - ∫eˣ dx = eˣ + C - ∫1/x dx = ln|x| + C - ∫sin x dx = −cos x + C - ∫cos x dx = sin x + C - ∫sec²x dx = tan x + C - ∫1/(1+x²) dx = tan⁻¹x + C - ∫1/√(1−x²) dx = sin⁻¹x + C **Integration by Parts**: ∫u dv = uv − ∫v du Choose u by ILATE: Inverse trig → Log → Algebraic → Trig → Exponential ### 6. Definite Integrals & Properties - ∫ₐᵇ f(x)dx = −∫ᵦᵃ f(x)dx - ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx (King's property — Very Important) - ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f is even; = 0 if f is odd ### 7. Maxima & Minima - Critical point: f'(x) = 0 - f''(x) > 0 → local minimum - f''(x) < 0 → local maximum - f''(x) = 0 → use higher derivatives (inflection point check) **For f(x,y)**: Let r=fxx, s=fxy, t=fyy. D = rt − s² - D > 0, r > 0 → minimum | D > 0, r < 0 → maximum | D < 0 → saddle point | D = 0 → inconclusive ### 8. Multivariable — Gradient, Divergence, Curl - Gradient: ∇f = (∂f/∂x)î + (∂f/∂y)ĵ + (∂f/∂z)k̂ — points in direction of max increase - Divergence: ∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z — scalar field - Curl: ∇×F — measures rotation. Conservative field → curl = 0 ### 9. GATE Trick King's property solves most definite integral problems in 2 lines. For maxima/minima of 2-variable: always compute D = rt − s² first. L'Hôpital's can be applied repeatedly until indeterminate form is resolved.