Compose the two constructions in part (d) to obtain
$$
U_g=U_{f_2}U_{f_1},
\qquad g(x)=f(x+1)-f(x),
$$
using two queries to $U_f$. For
$$
f(x)=ax^2+bx+c
$$
over $\mathbb Z_3$, the <finite difference> is
$$
g(x)=a(2x+1)+b=2ax+(a+b).
$$
This is affine with slope $s_g=2a=-a$. Apply the one-query affine procedure from part (c) to the implemented oracle $U_g$. It returns $s_g$ with certainty, after which the fixed classical computation
$$
\boxed{a=2s_g\pmod3}
$$
recovers the quadratic coefficient. Since one use of $U_g$ consists of one use each of $U_{f_1}$ and $U_{f_2}$, the algorithm makes exactly two calls to the original oracle $U_f$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 15D / e / Solution

{wiki}

A truncation error is the error introduced by replacing an exact mathematical operation with a finite approximation, such as replacing a <derivative> by a <finite difference>.

Truncation error

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A finite difference compares the values of a <function> at finitely separated arguments. For example, the forward finite difference with step $h$ is
$$
\Delta_hf(x)=f(x+h)-f(x).
$$
It is the discrete analogue of a <derivative> and lowers the degree of a nonconstant <polynomial> by one.

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 Incoming links: Finite difference