Final answer:
A sketch of the curve φ(t) for 0≤t≤1 in the complex plane results in a 'snail shell' spiral pattern that scales and rotates counter-clockwise as t increases from 0 to 1.
Step-by-step explanation:
To sketch the curve φ(t) defined as (1+8t2)ei(3π/4)t, where i is the imaginary unit and 0≤t≤1, one needs to understand that this is a parametric equation in the complex plane. The function ei(3π/4)t represents a spiral in the complex plane, and multiplying by (1+8t2) scales the spiral based on the value of t. Since t ranges from 0 to 1, we get a spiraling curve that starts at point (1, 0) at t=0 and scales up with an 8t2 factor as t increases while simultaneously rotating due to the ei(3π/4)t term.
To draw this curve, one can plot discrete points for values of t between 0 and 1 and connect these points smoothly to illustrate the scaling and rotating spiral. The shape will resemble an expanding and rotating 'snail shell' pattern in the complex plane, orientated in a counter-clockwise direction, as ei(3π/4)t indicates a rotation by 3π/4 radians counter-clockwise for each unit of t.