Given `h(r)=pir^(2)`, use the delta method to find `h^(')(r)`. Hence, find `h^(')(5/2)" and "h^(')(pi)`.

Answer

To find the derivative of the function \( h(r) = \pi r^2 \) using the delta method, we will follow these steps:

### Step 1: Define the function and the increment
Let \( h(r) = \pi r^2 \). We will consider a small increment \( \Delta r \) in \( r \), so we define:
\[ h(r + \Delta r) = \pi (r + \Delta r)^2 \]

### Step 2: Expand \( h(r + \Delta r) \)
Now, we expand \( h(r + \Delta r) \):
\[ h(r + \Delta r) = \pi (r^2 + 2r\Delta r + (\Delta r)^2) = \pi r^2 + 2\pi r \Delta r + \pi (\Delta r)^2 \]

### Step 3: Calculate \( \Delta h \)
Next, we find the change in \( h \), denoted as \( \Delta h \):
\[ \Delta h = h(r + \Delta r) - h(r) = (\pi r^2 + 2\pi r \Delta r + \pi (\Delta r)^2) - \pi r^2 \]
This simplifies to:
\[ \Delta h = 2\pi r \Delta r + \pi (\Delta r)^2 \]

### Step 4: Find \( \frac{\Delta h}{\Delta r} \)
Now, we divide \( \Delta h \) by \( \Delta r \):
\[ \frac{\Delta h}{\Delta r} = \frac{2\pi r \Delta r + \pi (\Delta r)^2}{\Delta r} = 2\pi r + \pi \Delta r \]

### Step 5: Take the limit as \( \Delta r \to 0 \)
To find the derivative \( h'(r) \), we take the limit as \( \Delta r \) approaches 0:
\[ h'(r) = \lim_{\Delta r \to 0} \left( 2\pi r + \pi \Delta r \right) = 2\pi r \]

### Step 6: Evaluate \( h'(5/2) \) and \( h'(\pi) \)
Now we can find the values of \( h' \) at specific points:
1. For \( r = \frac{5}{2} \):
\[ h'\left(\frac{5}{2}\right) = 2\pi \left(\frac{5}{2}\right) = 5\pi \]

2. For \( r = \pi \):
\[ h'(\pi) = 2\pi^2 \]

### Final Answers
Thus, the derivative \( h'(r) \) is \( 2\pi r \), and the specific values are:
- \( h'\left(\frac{5}{2}\right) = 5\pi \)
- \( h'(\pi) = 2\pi^2 \)