https://file.aiquickdraw.com/mathbot/file/1726099888351jnr51o5w.jpg

Let's break down the questions from the image you provided:

### Question (a)
- **Description:** The amount of the drug in the body increases at first and then eventually levels off as the body metabolizes it. The function \( Q(t) \), which denotes the quantity of drug in the body over time, increases rapidly at first but at a decreasing rate, eventually leveling off.
- **Answer for (a):** The function \( Q(t) \) is initially increasing and then starts decreasing in its rate of growth (but not the actual amount). Therefore, the correct option is **D. both increasing and decreasing**.

### Question (b)
- **Description:** Since the function \( Q(t) \) starts by increasing rapidly but at a decreasing rate and then levels off, this means that the curve is concave down (i.e., its second derivative is negative during the period of decrease in growth rate).
- **Answer for (b):** The function is concave down as it starts increasing and eventually levels off. So the correct option is **B. Concave Down**.

Let me know if you'd like more details or further clarification.

Here are 5 relative questions that expand on this concept:
1. What happens to the graph of \( Q(t) \) after it levels off?
2. How does the first derivative of \( Q(t) \) behave as time progresses?
3. Can the function \( Q(t) \) ever decrease in absolute quantity?
4. How would the function look if the drug were administered in different dosages over time?
5. How do changes in metabolism rate affect the concavity of the function?

**Tip:** A concave-down graph represents a situation where the rate of change is decreasing, even if the function itself is still increasing, which is typical in cases where there is a saturation limit like drug absorption.

When a drug is injected into a person's bloodstream, the amount of the drug present in the body increases rapidly at first. If the person receives daily injections, the body metabolizes the drug so that the amount of the drug present in the body continues to increase but at a decreasing rate. Eventually the quantity levels off at a saturation level. Let Q(t) denote the quantity of the drug in the body t days after receiving the first injection.

Learn how the function Q(t) models drug absorption, increasing at a decreasing rate and eventually leveling off. Explore concepts of concavity and the behavior of derivatives in this real-life application. Suitable for calculus students in grades 10-12.

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