Calculus

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Calculus — Key Facts for BUET Derivative: dy/dx = lim(h→0) [f(x+h)−f(x)]/h; represents rate of change and slope of tangent d/dx(x^n) = nx^{n−1}; d/dx(sin x) = cos x; d/dx(e^x) = e^x; d/dx(ln x) = 1/x Integration: reverse of differentiation; ∫ x^n dx = x^{n+1}/(n+1) + C Fundamental theorem: ∫_a^b f(x) dx = F(b) − F(a) where F’ = f ⚡ Exam tip: BUET calculus is the highest-weight section — practice integration by parts and area problems extensively!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Calculus — BUET Study Guide

Derivative formulas:

• d/dx(x^n) = nx^{n−1}
• d/dx(e^x) = e^x
• d/dx(a^x) = a^x ln a
• d/dx(ln x) = 1/x
• d/dx(sin x) = cos x
• d/dx(cos x) = −sin x
• d/dx(tan x) = sec²x

Chain rule: d/dx[f(g(x))] = f’(g(x)) · g’(x) Example: d/dx(sin(x²)) = cos(x²) · 2x

Product rule: d/dx(f·g) = f’g + fg’

Quotient rule: d/dx(f/g) = (f’g − fg’)/g²

Derivative of inverse function: If y = f(x), then dx/dy = 1/f’(x)

Implicit differentiation: Differentiate both sides with respect to x, treating y as function of x For d/dx(y²): use 2y · dy/dx

Parametric differentiation: If x = f(t), y = g(t): dy/dx = (dy/dt)/(dx/dt)

Second derivative: d²y/dx² = d/dx(dy/dx)

Integration formulas:

• ∫ x^n dx = x^{n+1}/(n+1) + C, n ≠ −1
• ∫ 1/x dx = ln|x| + C
• ∫ e^x dx = e^x + C
• ∫ sin x dx = −cos x + C
• ∫ cos x dx = sin x + C

Integration by substitution: Let u = g(x), then du = g’(x)dx; transform integral to simpler form

Integration by parts: ∫ u dv = uv − ∫ v du Choose u using LIATE: Log, Inverse trig, Algebraic, Trigonometric, Exponential

Partial fractions: Decompose rational function into simpler fractions, then integrate term by term

Definite integral: ∫_a^b f(x) dx = F(b) − F(a) Properties: ∫_a^b f(x) dx = −∫_b^a f(x) dx; ∫_a^b f(x) dx + ∫_b^c f(x) dx = ∫_a^c f(x) dx

Area under curve: A = ∫_a^b f(x) dx (above x-axis) For below x-axis: take absolute value

Area between two curves: A = ∫_a^b |f(x) − g(x)| dx

• Key formula: d/dx(x^n) = nx^{n−1}; ∫ x^n dx = x^{n+1}/(n+1) + C; ∫_a^b f(x) dx = F(b) − F(a)
• Common trap: Chain rule: d/dx(sin(x²)) = cos(x²) · 2x, not just cos(x²); always multiply by derivative of inner function
• Exam weight: 5–8 questions per exam (20–32 marks); the highest weight section in BUET

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Calculus — Comprehensive BUET Notes

nth derivative:

• d^n/dx^n (x^m) = m(m−1)…(m−n+1) x^{m−n}
• d^n/dx^n (e^{ax}) = a^n e^{ax}
• d^n/dx^n (sin(ax+b)) = a^n sin(ax+b + nπ/2)

Leibniz theorem for nth derivative of product: d^n/dx^n (uv) = Σ C(n,k) · u^{(n−k)} · v^{(k)}

L’Hôpital’s rule: For 0/0 or ∞/∞ forms: lim f(x)/g(x) = lim f’(x)/g’(x) if limit exists

Mean Value Theorem: If f is continuous on [a,b] and differentiable on (a,b), then ∃ c ∈ (a,b) where f’(c) = [f(b)−f(a)]/(b−a)

Taylor series: f(x) = f(a) + f’(a)(x−a)/1! + f”(a)(x−a)²/2! + …

Maclaurin series: f(x) = f(0) + f’(0)x + f”(0)x²/2! + …

• e^x = 1 + x + x²/2! + x³/3! + …
• sin x = x − x³/3! + x⁵/5! − …
• cos x = 1 − x²/2! + x⁴/4! − …

Standard limits:

• lim(x→0) sin x/x = 1
• lim(x→0) (e^x − 1)/x = 1
• lim(x→0) (1+x)^(1/x) = e

Reduction formulas for integration: For ∫ sin^n x dx: use I_n = −(sin^{n−1}x cos x)/n + (n−1)/n · I_{n−2}

Wallis formula: ∫_0^{π/2} sin^m x cos^n x dx = (Γ((m+1)/2)·Γ((n+1)/2)) / (2·Γ((m+n+2)/2))

Beta and Gamma: B(m,n) = 2∫_0^{π/2} sin^{2m−1}x cos^{2n−1}x dx = Γ(m)Γ(n)/Γ(m+n) Γ(n+1) = n! for positive integer n

Area in polar coordinates: A = ½∫ r² dθ

Volume of revolution: About x-axis: V = π∫ y² dx Using shell method: V = 2π∫ x·f(x) dx (about y-axis with vertical strips)

Arc length: s = ∫ √(1 + (dy/dx)²) dx

Differential equations: First order linear: dy/dx + P(x)y = Q(x); IF = e^{∫P dx}; solution: y·IF = ∫ Q·IF dx + C Variable separable: dy/dx = f(x)g(y) → dy/g(y) = f(x)dx

Exact differential equation: M dx + N dy = 0 is exact if ∂M/∂y = ∂N/∂x

Second order linear with constant coefficients: ay” + by’ + cy = 0 → characteristic equation ar² + br + c = 0

Application problems:

• Rate: dy/dt = (dy/dx) · (dx/dt)
• Maxima/minima: f’(x) = 0; check f”(x)
• Optimization: express quantity in one variable, find max/min

Using substitution in definite integrals: When changing variable, also change limits!

Leibniz rule for differentiation under integral: d/dx [∫_{a(x)}^{b(x)} f(t) dt] = f(b(x))b’(x) − f(a(x))a’(x)

• Remember: Chain rule for derivatives; for integration by parts use LIATE to choose u; definite integral = F(b) − F(a); L’Hôpital for 0/0 and ∞/∞
• Previous years: “Find derivative of x²e^x” [2023 BUET]; “Evaluate ∫ x sin x dx” [2024 BUET]; “Find area under y = x² from x = 0 to x = 2” [2024 BUET]

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📊 BUET Admission Exam Essentials

Detail	Value
Questions	Varies by year (~40-50 MCQ)
Time	Usually 2–3 hours
Marks	Varies by section
Subjects	Mathematics (highest weight), Physics, Chemistry
Negative	Usually no negative marking in BUET
Mode	Written + MCQ depending on year

🎯 High-Yield Topics for BUET Mathematics

• Calculus (Differentiation + Integration) — highest weight (20–30 marks)
• Algebra (Quadratics, AP/GP/HP) — very high weight
• Coordinate Geometry (Circle, Conics) — high weight
• Trigonometry — medium-high weight
• Complex Numbers — medium weight

📝 Previous Year Question Patterns

• Calculus: 5–8 questions per exam, 20–32 marks
• Common patterns: differentiation, integration by parts, area under curve, differential equations
• Weight: very high — must master for BUET admission

💡 Pro Tips

• In differentiation, always apply chain rule — missing the inner derivative is the most common error
• For integration by parts: choose u using LIATE rule (Log, Inverse, Algebraic, Trigonometric, Exponential)
• When integrating rational functions, use partial fractions first
• For area problems, always sketch the curves to identify which is above/below
• Always check domain and range restrictions in calculus problems
• For maximum/minimum, find critical points (f’(x) = 0 or undefined) and evaluate endpoints

🔗 Official Resources

• BUET Official
• BUET Admission Portal

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