葡萄糖是人体血液中的一种糖，身体利用它来提供能量。如果不进行治疗，长期低血糖会引起健康问题。 在上午 10:00，患者通过静脉滴注以每小时 $r$ 单位葡萄糖的恒定速率摄入葡萄糖。已知血液中的葡萄糖被身体吸收的速率与血液中现存的葡萄糖量成正比。 在治疗开始时，患者血液中有 $x_0$ 单位的葡萄糖。 已知 $x$ 表示患者开始治疗 $t$ 小时后血液中现存的葡萄糖量。

1. Formulate the differential equation based on the given rates.

The rate at which glucose is added is $r$.
The rate at which it is absorbed is proportional to $x$, so the rate out is $kx$, where $k$ is a positive constant.
$\frac{dx}{dt} = \text{Rate in} - \text{Rate out}$
$\frac{dx}{dt} = r - kx$

2. Separate variables and integrate both sides.

$\int \frac{1}{r - kx} \, dx = \int 1 \, dt$
$-\frac{1}{k} \ln |r - kx| = t + C$

3. Express the general solution without logarithms.

$\ln |r - kx| = -kt - kC$
$r - kx = \pm e^{-kC} e^{-kt}$
Let $A = \pm e^{-kC}$.
$r - kx = A e^{-kt}$

4. Apply the initial condition to find the particular solution.

At $t = 0, x = x_0$:
$r - kx_0 = A e^{0} \implies A = r - kx_0$
Substituting $A$ back into the equation:
$r - kx = (r - kx_0) e^{-kt}$
$kx = r - (r - kx_0)e^{-kt}$
$x = \frac{r}{k} - \left(\frac{r}{k} - x_0\right)e^{-kt}$
$x = \left(x_0 - \frac{r}{k}\right)e^{-kt} + \frac{r}{k} \quad \text{(shown)}$

1. Determine the behaviour of $x$ as $t \to \infty$.

As $t \to \infty$, since $k$ is a positive constant, $e^{-kt} \to 0$.

Therefore, $x \to \left( x_0 - \frac{r}{k} \right)(0) + \frac{r}{k} = \frac{r}{k}$.

1. Determine the intercepts and asymptotes of the graph.

At $t = 0$, $x = x_0$. Thus, the vertical intercept is $(0, x_0)$.
From part (b), as $t \to \infty$, the theoretical limiting value is $x = \frac{r}{k}$. Thus, there is a horizontal asymptote at $x = \frac{r}{k}$.

2. Deduce the shape of the curve based on the context and sketch the graph.

Since the patient is receiving treatment for low blood sugar, the initial amount $x_0$ is less than the target stable level $\frac{r}{k}$. This means $\left( x_0 - \frac{r}{k} \right)$ is a negative constant.
As $t$ increases, $e^{-kt}$ decreases, so the expression $\left( x_0 - \frac{r}{k} \right)e^{-kt}$ approaches $0$ from below. Therefore, $x$ increases over time, starting from $x_0$ and asymptotically approaching $\frac{r}{k}$.

Key features for the sketch:

1. Axes correctly labelled $t$ and $x$.
2. A horizontal dotted line for the asymptote at $x = \frac{r}{k}$.
3. A concave-down curve starting at $(0, x_0)$ and increasing towards $x = \frac{r}{k}$.