一条曲线的方程为 $y = x^2 + x + 2k - 4$，一条直线的方程为 $y - 2x = k$，其中 $k$ 为常数。求使直线和曲线不相交的 $k$ 的取值集合，并在数轴上表示此集合。 已知方程为 $y = ax^2 + 8x + c$ 的曲线（其中 $a$ 和 $c$ 是常数）完全位于 $x$ 轴下方。
写出适用于 $a$ 和 $c$ 的条件。 如果 $x$ 以每秒 $0.2$ 个单位的速度增加，求在 $y = -4$ 的瞬间 $y$ 的变化率。

1. Express $y$ in terms of $x$ from the linear equation and substitute into the curve equation

From the line equation, $y = 2x + k$.
Substitute into the curve equation:
$2x + k = x^2 + x + 2k - 4$

2. Rearrange into a standard quadratic equation form $ax^2 + bx + c = 0$

$x^2 + x - 2x + 2k - k - 4 = 0$
$x^2 - x + k - 4 = 0$

3. Apply the discriminant condition for no intersection

For the line and curve to not intersect, the quadratic equation must have no real roots, so $b^2 - 4ac < 0$.
$(-1)^2 - 4(1)(k - 4) < 0$

4. Solve the inequality for $k$

$1 - 4k + 16 < 0$
$17 - 4k < 0$
$4k > 17$
$k > 4.25$

5. Represent the solution on a number line

Draw a number line, place an open circle at $4.25$ (or $\frac{17}{4}$), and draw an arrow pointing to the right to indicate values greater than $4.25$.

1. State the condition on the leading coefficient for the curve to lie completely below the $x$-axis

For the curve to lie completely below the $x$-axis, it must open downwards. Therefore, the coefficient of $x^2$ must be negative:
$a < 0$

2. Apply the discriminant condition for the curve not to intersect the $x$-axis

The curve must not intersect the $x$-axis, so the equation $ax^2 + 8x + c = 0$ has no real roots. Thus, the discriminant must be less than zero:
$b^2 - 4ac < 0$
$(8)^2 - 4(a)(c) < 0$
$64 - 4ac < 0$
$4ac > 64$
$ac > 16$