Calculus

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Calculus — Key Facts for Kenyatta University Derivative: dy/dx = lim(h→0) [f(x+h)−f(x)]/h; represents rate of change d/dx(x^n) = nx^{n−1}; d/dx(e^x) = e^x; d/dx(ln x) = 1/x; d/dx(sin x) = cos x Integration: reverse of differentiation; ∫ x^n dx = x^{n+1}/(n+1) + C for n ≠ −1 Fundamental theorem: ∫_a^b f(x) dx = F(b) − F(a) ⚡ Exam tip: Kenyatta calculus focuses on differentiation and basic integration — master chain rule and integration by parts!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Calculus — Kenyatta University 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

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

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

Definite integral: ∫_a^b f(x) dx = F(b) − F(a) Area under curve from a to b

Second derivative: d²y/dx² = d/dx(dy/dx) Used for identifying maxima/minima and concavity

• Key formula: d/dx(x^n) = nx^{n−1}; ∫ x^n dx = x^{n+1}/(n+1) + C
• Common trap: Chain rule — always multiply by derivative of inner function
• Exam weight: 3–5 questions per exam; highest weight in most mathematics papers

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Calculus — Comprehensive Kenyatta Notes

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

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

Applications of derivatives:

• Equation of tangent: y − y₁ = f’(x₁)(x − x₁)
• Maximum/minimum: f’(x) = 0; f”(x) > 0 for minimum, < 0 for maximum
• Rate problems: dy/dx = (dy/dt)/(dx/dt)

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

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

Integration of rational functions: Use partial fractions, then integrate term by term

Reduction formulas: ∫ sin^n x dx = −(sin^{n−1}x cos x)/n + (n−1)/n ∫ sin^{n−2}x dx

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

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

Volume of revolution: V = π∫ y² dx (about x-axis)

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

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! + …
• sin x = x − x³/3! + x⁵/5! − …

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

Partial derivatives: For z = f(x,y): ∂f/∂x (treat y constant), ∂f/∂y (treat x constant)

Double integrals: ∫∫ f(x,y) dx dy over region; integrate with respect to one variable at a time with limits

• Remember: Chain rule for derivatives; LIATE for integration by parts; definite integral = F(b) − F(a); L’Hôpital for 0/0 and ∞/∞
• Previous years: “Find derivative of x²e^x” [2023 KU]; “Evaluate ∫ x sin x dx” [2024 KU]; “Find equation of tangent to y = x³ at x=1” [2024 KU]

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📊 Kenyatta University Exam Essentials

Detail	Value
Questions	50 (depending on course)
Time	2–3 hours
Marks	100
Format	Mix of short answer and problem solving

💡 Pro Tips

• Always apply chain rule — missing inner derivative is the most common error
• For integration by parts, use LIATE to choose u
• For area problems, always sketch the curves first
• For maximum/minimum, find critical points and evaluate second derivative or check sign change

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