Ok so how does checking differentiability differs from closed to open intervals

Great question! Understanding the difference between checking differentiability on closed versus open intervals is key to applying calculus theorems correctly. Let's break down the concepts:

### Differentiability on Open vs. Closed Intervals

1. **Differentiability on an Open Interval \((a, b)\):**

   A function \( f(x) \) is said to be differentiable on an open interval \((a, b)\) if the derivative \( f'(x) \) exists for every point \( x \) in that interval. 

   - **Open Interval \((a, b)\):** An interval that does not include its endpoints \( a \) and \( b \). In other words, \( a < x < b \).
   - **Checking Differentiability on \((a, b)\):** To check differentiability on an open interval, you only consider points strictly between \( a \) and \( b \). You do not need to consider the behavior of the function at the endpoints \( a \) and \( b \).

   **Example:** If \( f(x) = x^2 \), then \( f'(x) = 2x \) exists for every \( x \) in the interval \((-1, 1)\). Thus, \( f(x) \) is differentiable on \((-1, 1)\).

2. **Differentiability on a Closed Interval \([a, b]\):**

   A function \( f(x) \) is said to be differentiable on a closed interval \([a, b]\) if the derivative \( f'(x) \) exists for every point in the open interval \((a, b)\) and also if the derivative exists at the endpoints \( a \) and \( b \) as one-sided limits.

   - **Closed Interval \([a, b]\):** An interval that includes its endpoints \( a \) and \( b \). In other words, \( a \leq x \leq b \).
   - **Checking Differentiability on \([a, b]\):** To check differentiability on a closed interval, you must check:
     - The function is differentiable on the open interval \((a, b)\).
     - The derivative \( f'(x) \) exists as a right-hand derivative at \( x = a \) and as a left-hand derivative at \( x = b \).

   **Example:** If \( f(x) = x^2 \), then \( f'(x) = 2x \) exists for every \( x \) in the interval \([0, 1]\) and also at the endpoints \( x = 0 \) and \( x = 1 \) as one-sided derivatives:
   - Right-hand derivative at \( x = 0 \): \( \lim_{h \to 0^+} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^+} \frac{h^2}{h} = 0 \)
   - Left-hand derivative at \( x = 1 \): \( \lim_{h \to 0^-} \frac{f(1+h) - f(1)}{h} = \lim_{h \to 0^-} \frac{(1+h)^2 - 1}{h} = \lim_{h \to 0^-} \frac{2h + h^2}{h} = 2 \)

### Key Differences Between Open and Closed Intervals for Differentiability

- **Open Interval \((a, b)\):** Only considers differentiability for points within the interval, not at the endpoints.
- **Closed Interval \([a, b]\):** Requires checking differentiability at all points within the interval and ensuring that the one-sided derivatives exist at the endpoints.

### Why This Matters for Calculus Theorems

- **Rolle's Theorem and LMVT:** These theorems specifically require that the function be differentiable on the open interval \((a, b)\) and continuous on the closed interval \([a, b]\). This is because the derivative must exist strictly between \( a \) and \( b \), but the behavior at the endpoints doesn't directly affect the result as long as the function is continuous there.
  
- **Checking Differentiability:** In practical terms, checking differentiability on \((a, b)\) is often easier because you don’t need to worry about the boundary behavior. When checking differentiability on \([a, b]\), you need to carefully evaluate the one-sided limits at the endpoints, which can sometimes introduce additional complexity.

**Tip:** Always ensure you understand whether a theorem or a problem is asking for differentiability on an open or closed interval, as this will determine how you check the function's behavior at the endpoints.

Explore the differences in checking differentiability on open versus closed intervals in calculus. This detailed explanation covers key concepts like Rolle's Theorem and the Mean Value Theorem, with examples illustrating the application of derivatives on intervals. Ideal for advanced high school students and anyone studying calculus.

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