Common Computation Tricks (No Calculator!)

GRE Math Subject · Axiom Academy

LESSON Common Computation Tricks (No Calculator!) Hand computation techniques that save time and reduce errors Step 1: Fraction Arithmetic Shortcuts Partial Fractions Decomposition: 1/(x² - 1) = 1/((x-1)(x+1)) = A/(x-1) + B/(x+1) Quick cover-up: A = 1/(2) = 1/2, B = 1/(-2) = -1/2 Result: 1/2 · 1/(x-1) - 1/2 · 1/(x+1) Master fractions with common denominators mentally: 1/2 + 1/3 = 3/6 + 2/6 = 5/6 (find LCD = 6 in your head) 1/3 + 1/4 = 4/12 + 3/12 = 7/12 1/6 + 1/8 = 4/24 + 3/24 = 7/24 Example: Compute ∫ 1/(x² - 1) dx = 1/2 ln|x-1| - 1/2 ln|x+1| + C Step 2: Common Series and Limits sin(π/6) = 1/2, cos(π/6) = √3/2 sin(π/4) = cos(π/4) = 1/√2 = √2/2 sin(π/3) = √3/2, cos(π/3) = 1/2 lim(x→0) sin(x)/x = 1 [Don't compute derivatives, just know it] Σ nxⁿ⁻¹ = 1/(1-x)² for |x| < 1 Step 3: Matrix Computation Shortcuts |a b| = ad - bc (diagonal products, subtract) For matrix M = |a b|, trace = a+d, det = ad-bc Characteristic polynomial: λ² - (trace)λ + (det) = 0 λ = [(a+d) ± √((a+d)² - 4(ad-bc))] / 2 Example: For M = |2 1|, trace = 3, det = 3 λ² - 3λ + 3 = 0 → λ = (3 ± √(9-12))/2 = (3 ± √(-3)i)/2 d/dx[xⁿ] = nxⁿ⁻¹ (automatic, no thinking) d/dx[sin(x)] = cos(x), d/dx[cos(x)] = -sin(x) d/dx[f(g(x))] = f'(g(x)) · g'(x) (derivative of outside × derivative of inside) Integration by Parts Recognition: ∫ x·sin(x) dx → u = x, dv = sin(x) dx ∫ x·ln(x) dx → u = ln(x), dv = x dx Recognize the pattern: If you see product of polynomial and transcendental, use integration by parts with polynomial as u.

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