  We give explicit examples of infinite families of elliptic curves E over Q
with (nonconstant) quadratic twists over Q(t) of rank at least 2 and 3. We
recover some results announced by Mestre, as well as some additional families.
Suppose D is a squarefree integer and let r_E(D) denote the rank of the
quadratic twist of E by D. We apply results of Stewart and Top to our examples
to obtain results of the form #{D : |D| < x, r_E(D) >= 2} >> x^{1/3}, #{D : |D|
< x, r_E(D) >= 3} >> x^{1/6} for all sufficiently large x.
