The concentration of antibiotic in the bloodstream t hours after being injected is given by the function C(t) is measured in milligrams per litre of blood. Find the rate of change of  C(t) = \frac{ t^2 + 4}{2t} .Determine the rate of change for t=2 hours. 

To find the rate of change of the function \( C(t) = \frac{t^2 + 4}{2t} \), we need to calculate its derivative, \( C'(t) \), using differentiation rules. Here's the process step-by-step:

### Step 1: Rewrite the function
Rewrite \( C(t) = \frac{t^2 + 4}{2t} \) by splitting it into two simpler terms:
\[
C(t) = \frac{t^2}{2t} + \frac{4}{2t} = \frac{t}{2} + \frac{2}{t}
\]

### Step 2: Differentiate the function
Now, differentiate \( C(t) = \frac{t}{2} + \frac{2}{t} \) with respect to \( t \).

- The derivative of \( \frac{t}{2} \) is \( \frac{1}{2} \).
- The derivative of \( \frac{2}{t} \) can be computed using the power rule. Rewrite \( \frac{2}{t} \) as \( 2t^{-1} \), and differentiate:
\[
\frac{d}{dt}(2t^{-1}) = -2t^{-2} = -\frac{2}{t^2}
\]

Thus, the derivative is:
\[
C'(t) = \frac{1}{2} - \frac{2}{t^2}
\]

### Step 3: Evaluate the derivative at \( t = 2 \)
Substitute \( t = 2 \) into the derivative \( C'(t) = \frac{1}{2} - \frac{2}{t^2} \):
\[
C'(2) = \frac{1}{2} - \frac{2}{2^2} = \frac{1}{2} - \frac{2}{4} = \frac{1}{2} - \frac{1}{2} = 0
\]

### Final Answer:
The rate of change of \( C(t) \) at \( t = 2 \) hours is \( 0 \).

Would you like more details on any part of the solution? Here are some related questions:

1. How do you apply the quotient rule to find the derivative of a function?
2. Why is the rate of change zero at \( t = 2 \) in this case?
3. What does it mean for a function to have a rate of change of zero at a specific point?
4. How would the rate of change differ if the function were given in a different form?
5. Can the function \( C(t) \) have negative rates of change at other values of \( t \)?

**Tip:** To simplify the differentiation process, try to break down complex functions into simpler terms before applying derivative rules.

The concentration of antibiotic in the bloodstream t hours after being injected is given by the function C(t) is measured in milligrams per litre of blood. Find the rate of change of C(t) = \frac{t^2 + 4}{2t}. Determine the rate of change for t=2 hours.

Learn how to calculate the rate of change of the antibiotic concentration function C(t) = (t^2 + 4) / (2t). This problem covers key calculus concepts, such as differentiation and simplifying rational functions. Suitable for Grades 10-12 students.

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