Prove That A Function F Is Differentiable At X = A With F'(a)=b, BeR, If And Only If F(x)-f(a)-b(x-a)
Understanding the differentiability of a function at a specific point is fundamental in calculus and analysis. This article provides a comprehensive explanation of the necessary and sufficient conditions for a function \(F\) to be differentiable at \(x = a\) with derivative \(F'(a) = b\), focusing on the equivalence involving the expression \(F(x) - F(a) - b(x - a)\). We will explore the theoretical underpinnings, formal proof, implications, and applications of this important concept.
Introduction to Differentiability
What Does It Mean for a Function to Be Differentiable?
A function \(F: \mathbb{R} \to \mathbb{R}\) is said to be differentiable at a point \(a \in \mathbb{R}\) if the derivative \(F'(a)\) exists at that point. The derivative at \(a\) intuitively measures the instantaneous rate of change of \(F\) at \(a\), or equivalently, the slope of the tangent line to the graph of \(F\) at \(a\).
Mathematically, \(F\) is differentiable at \(a\) if the following limit exists:
\[
F'(a) = \lim_{x \to a} \frac{F(x) - F(a)}{x - a}
\]
If this limit exists, we say that \(F\) is differentiable at \(a\), and the value of the limit is \(F'(a)\).
The Theorem: Characterization of Differentiability
Statement of the Theorem
The theorem provides a criterion for differentiability in terms of the behavior of the difference:
\[
F(x) - F(a) - b(x - a)
\]
where \(b\) is a candidate for the derivative at \(a\). The formal statement is:
> Theorem: A function \(F\) is differentiable at \(a\) with \(F'(a) = b\) if and only if
\[
\lim_{x \to a} \frac{F(x) - F(a) - b(x - a)}{x - a} = 0
\]
which is equivalent to stating that
\[
F(x) - F(a) - b(x - a) = o(x - a) \quad \text{as} \quad x \to a
\]
meaning that the difference between \(F(x)\) and its linear approximation at \(a\) vanishes faster than \(x - a\) as \(x\) approaches \(a\).
Intuitive Explanation
The expression \(F(x) - F(a) - b(x - a)\) measures how well the linear function \(L(x) = F(a) + b(x - a)\) approximates \(F(x)\) near \(a\). If this difference becomes negligible relative to \((x - a)\) as \(x \to a\), then the function is well-approximated by its tangent line, indicating differentiability.
Conversely, if \(F\) is differentiable at \(a\) with derivative \(b\), then the function behaves like its tangent line near \(a\), and the difference \(F(x) - F(a) - b(x - a)\) tends to zero faster than \((x - a)\).
Formal Proof of the Theorem
Necessity: If \(F\) is differentiable at \(a\) with \(F'(a) = b\), then the limit holds
Suppose \(F\) is differentiable at \(a\) with derivative \(b\). By the definition of differentiability:
\[
F'(a) = \lim_{x \to a} \frac{F(x) - F(a)}{x - a} = b
\]
This implies:
\[
\frac{F(x) - F(a)}{x - a} = b + \varepsilon(x)
\]
where \(\varepsilon(x) \to 0\) as \(x \to a\).
Multiplying both sides by \((x - a)\):
\[
F(x) - F(a) = b(x - a) + \varepsilon(x)(x - a)
\]
Rearranged, this becomes:
\[
F(x) - F(a) - b(x - a) = \varepsilon(x)(x - a)
\]
Dividing both sides by \((x - a)\):
\[
\frac{F(x) - F(a) - b(x - a)}{x - a} = \varepsilon(x)
\]
Since \(\varepsilon(x) \to 0\) as \(x \to a\), it follows that:
\[
\lim_{x \to a} \frac{F(x) - F(a) - b(x - a)}{x - a} = 0
\]
which completes the necessity part.
Sufficiency: If the limit holds, then \(F\) is differentiable at \(a\) with \(F'(a) = b\)
Conversely, assume:
\[
\lim_{x \to a} \frac{F(x) - F(a) - b(x - a)}{x - a} = 0
\]
Define:
\[
\varepsilon(x) = \frac{F(x) - F(a) - b(x - a)}{x - a}
\]
so that \(\varepsilon(x) \to 0\) as \(x \to a\).
Rearranged, this yields:
\[
F(x) - F(a) = b(x - a) + \varepsilon(x)(x - a)
\]
Dividing both sides by \((x - a)\):
\[
\frac{F(x) - F(a)}{x - a} = b + \varepsilon(x)
\]
Taking the limit as \(x \to a\), since \(\varepsilon(x) \to 0\), we get:
\[
\lim_{x \to a} \frac{F(x) - F(a)}{x - a} = b
\]
Hence, \(F\) is differentiable at \(a\) with derivative \(b\).
Implications and Applications
Linear Approximation and Differentiability
The theorem essentially states that differentiability at a point is equivalent to the existence of a good linear approximation near that point. The function \(F\) can be approximated by its tangent line \(L(x) = F(a) + b(x - a)\), with the error term \(F(x) - F(a) - b(x - a)\) vanishing faster than \((x - a)\).
This understanding is fundamental in numerical analysis, optimization, and approximation theory. It allows us to approximate complex functions locally with linear functions, simplifying analysis and computations.
Connection to the Definition of the Derivative
The classic definition involves the limit of the difference quotient:
\[
F'(a) = \lim_{x \to a} \frac{F(x) - F(a)}{x - a}
\]
The theorem refines this by considering the remainder term \(F(x) - F(a) - b(x - a)\). If this remainder is negligible compared to \((x - a)\), the function is differentiable with the derivative \(b\).
Practical Examples
- Polynomial functions: They are differentiable everywhere, and the theorem confirms that the difference between the polynomial and its tangent line at a point vanishes faster than \((x - a)\).
- Absolute value function: Not differentiable at \(x=0\), as the difference \(F(x) - F(0) - 0 \cdot (x - 0)\) does not behave as \(o(x)\).
- Exponential and trigonometric functions: These are differentiable everywhere, and the theorem applies straightforwardly to confirm their differentiability.
Summary
The core idea of the theorem is that the differentiability of \(F\) at a point \(a\) with derivative \(b\) is equivalent to the fact that:
\[
F(x) - F(a) - b(x - a) = o(x - a) \quad \text{as} \quad x \to a
\]
or, equivalently,
\[
\lim_{x \to a} \frac{F(x) - F(a) - b(x - a)}{x - a} = 0
\]
This characterization provides a powerful and intuitive criterion for differentiability, linking the limit definition of the derivative to the approximation of the function by its tangent line.
Conclusion
Proving that a function \(F\) is differentiable at a point \(a\) with derivative \(b\) involves understanding the behavior of the difference \(F(x) - F(a) - b(x - a)\). The key equivalence states that this difference must be negligible compared to \((x -
