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Let's go through the questions based on the provided function for glucose concentration \( C(t) = r - (r - k)e^{-t} \), where \( r \) is the constant infusion rate, and \( k \) is a constant.

### Part (a)
To find the initial concentration \( C_0 \), which is the concentration at \( t = 0 \), we can substitute \( t = 0 \) into the expression for \( C(t) \).

\[
C(0) = r - (r - k)e^{0}
\]

Since \( e^{0} = 1 \), this simplifies to:

\[
C(0) = r - (r - k) \cdot 1
\]
\[
C(0) = r - (r - k)
\]
\[
C(0) = r - r + k
\]
\[
C(0) = k
\]

Thus, the initial concentration \( C_0 \) in terms of \( r \) and \( k \) is:
\[
C_0 = k
\]

### Part (b)
To find the rate at which the concentration of glucose is changing at any time \( t \), we need to compute the derivative of \( C(t) \) with respect to \( t \), which we'll denote as \( C'(t) \).

Starting with:
\[
C(t) = r - (r - k)e^{-t}
\]

Take the derivative of \( C(t) \) with respect to \( t \):

\[
C'(t) = 0 - (r - k)(-e^{-t})
\]

This simplifies to:

\[
C'(t) = (r - k)e^{-t}
\]

Therefore, the rate at which the concentration of glucose is changing at time \( t \) is:
\[
C'(t) = (r - k)e^{-t}
\]

### Summary of Answers
- **(a)** \( C_0 = k \)
- **(b)** \( C'(t) = (r - k)e^{-t} \)

Would you like further details on any of these steps, or do you have any specific questions?

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Here are some related questions that can help deepen understanding:

1. How does the concentration \( C(t) \) behave as \( t \) approaches infinity?
2. What happens to \( C(t) \) if \( k = r \)?
3. If the initial concentration \( C_0 \) is known, can we solve for \( k \) in terms of \( C_0 \) and \( r \)?
4. How does the rate \( C'(t) \) change over time, and what does this signify about the glucose concentration?
5. What is the interpretation of the exponential term \( e^{-t} \) in the context of glucose concentration decay?

**Tip:** When solving for derivatives of functions involving exponentials, remember to apply the chain rule correctly, especially with negative exponents like \( e^{-t} \).

A glucose solution is administered intravenously into the bloodstream of a patient at a constant rate of r milligrams/hour. The concentration of the glucose solution after t hours is given by C(t) = r - (r - k)e^{-t}. Given that C_0 is the initial concentration of glucose, find C_0 in terms of r and k, and determine the rate at which the concentration of glucose is changing at time t.

Solve for the initial concentration and rate of change of glucose concentration in the bloodstream, given the exponential decay model C(t) = r - (r - k)e^{-t}. This calculus problem explores differentiation and exponential functions, ideal for undergraduate calculus students.

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