By C. B. Gupta

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1 1. Prove that ∆2 ≡ E2 – 2E + 1. 2. Prove that if f(x) and g(x) are the function of x then (i) ∆[f(x) + g(x)] = ∆f(x) + ∆g(x) (ii) ∆[af(x)] = a ∆f(x) (iii) ∆[f(x) g(x)] = f(x) ∆ g(x) + g(x + 1) f(x) = f(x + 1) ∆ g(x) + g(x) ∆f(x) (iv) ∆ LM f ( x) OP = g( x) ∆ f ( x) − f ( x) ∆ g( x) . g( x ) g( x + h) N g( x) Q 3. Evaluate (i) ∆[sinh (a + bx)] (iii) ∆ [cot 2x ] (ii) ∆[tan ax] (iv) ∆ (x + cos x) 19 CALCULUS OF FINITE DIFFERENCES (v) ∆ (x2 + ex + 2) (vi) ∆ [log x] LM x OP N cos 2 x Q 2 (vii) ∆ [eax log bx] (viii) ∆ 4.

We get new system of equation as b22 44 ADVANCECED MATHEMATICS U| || V| | = b ′′ |W a11 x1 + a12 x 2 + a13 x 3 + ...... + a1n x n = b1 b22 x 2 + b23 x 3 + ...... + b2 n x n = b2′ c33 x3 + ...... + c3 n x n = b3′′ . . 3) cm 3 x 3 + ...... + cmn x n m Proceeding in the same way we eliminate x3 in third step, we eliminate x4 in fourth step and so on. We get new system of equation as a11 x1 + a12 x 2 + a13 x 3 + ...... + a1n x n = b1 b22 x 2 + b23 x 3 + ...... + b2 n x n = b2′ c33 x3 + ...... + c3n x n = b3′′ .

1)n ux. S. ux +n – nC1 ux + n – 1 + nC2 ux + n–2 – ..... + (– 1)n ux Solution. = (En – nC1 En –1 + nC2 En–2 – ..... S. 1 1. Prove that ∆2 ≡ E2 – 2E + 1. 2. Prove that if f(x) and g(x) are the function of x then (i) ∆[f(x) + g(x)] = ∆f(x) + ∆g(x) (ii) ∆[af(x)] = a ∆f(x) (iii) ∆[f(x) g(x)] = f(x) ∆ g(x) + g(x + 1) f(x) = f(x + 1) ∆ g(x) + g(x) ∆f(x) (iv) ∆ LM f ( x) OP = g( x) ∆ f ( x) − f ( x) ∆ g( x) . g( x ) g( x + h) N g( x) Q 3. Evaluate (i) ∆[sinh (a + bx)] (iii) ∆ [cot 2x ] (ii) ∆[tan ax] (iv) ∆ (x + cos x) 19 CALCULUS OF FINITE DIFFERENCES (v) ∆ (x2 + ex + 2) (vi) ∆ [log x] LM x OP N cos 2 x Q 2 (vii) ∆ [eax log bx] (viii) ∆ 4.