Define the "forward difference operator" $\Delta$ acting on a function $f(x)$ by $$\Delta f(x) = f(x+1)-f(x).$$ 1. Let $f(x)=x+3$. Compute $\Delta f(x)$.
2. Let $g(x)=x^2+2x-4$. Compute $\Delta g(x)$.
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 $\Delta h(x)$ and expressing the result in terms of $a(x)$, $\Delta a(x)$, $b(x)$, and $\Delta b(x)$.
Let $a$ be a number and let $k=0,1,2,\ldots$. Define the "rising factorial" $(a)_k$ by $$(a)_k = a(a+1)(a+2)(a+3)\ldots (a+k-1).$$
4. Compute $(3)_2$.
5. Compute $(2)_3$.