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Define the "forward difference operator" Δ acting on a function f(x) by
Δf(x)=f(x+1)−f(x).
1. Let f(x)=x+3. Compute Δf(x).
Solution: By definition,
Δf(x)=f(x+1)−f(x)=[(x+1)+3]−[x+3]=(x+4)−(x+3)=1.
2. Let g(x)=x2+2x−4. Compute Δg(x).
Solution: By definition,
Δg(x)=g(x+1)−g(x)=[(x+1)2+2(x+1)−4]−[x2+2x−4]=[x2+2x+1+2x+2−4]−[x2+2x−4]=2x+3
3. Consider the function h(x)=a(x)b(x), where a(x) and b(x) are some functions. The well-known product rule for differentiation says h′(x)=a′(x)b(x)+a(x)b′(x). Find a product rule for the difference operator by computing Δh(x) and expressing the result in terms of a(x), Δa(x), b(x), and Δb(x).
Solution: Compute
Δh(x)=a(x+1)b(x+1)−a(x)b(x)=a(x+1)b(x+1)−a(x)b(x)+0=a(x+1)b(x+1)−a(x)b(x)+a(x+1)b(x)−a(x+1)b(x)=a(x+1)[b(x+1)−b(x)]+b(x)[a(x+1)−a(x)]=a(x+1)Δb(x)+b(x)Δa(x).
Let a be a number and let k=0,1,2,…. Define the "rising factorial" (a)k by
(a)k=a(a+1)(a+2)(a+3)…(a+k−1).
4. Compute (3)2.
Solution: (3)2=3(4)=12
5. Compute (2)3.
Solution: (2)3=2(3)(4)=24