Problem 15 - A Textbook of Differential Calculus

Question

Let \(f(x)= \begin{cases}\frac{1}{e^{x^{2}}}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{cases}\) \(f^{\prime}(0)\) is equal to (a) 0 (b) 1 (c) \(-1\) (d) Doesn't exist

Step-by-Step Solution

Answer

The derivative \(f'(0)\) is 0.

Step 1: Understand the Problem

We need to find the derivative of the given piecewise function \(f(x)\) at \(x = 0\). The function is defined differently depending on whether \(x\) is zero or not. We will use the limit definition of the derivative because we are asked about \(f'(0)\).

Step 2: Apply the Definition of Derivative

The derivative \(f'(0)\) can be found using: \[ f'(0) = \lim_{{h \to 0}} \frac{f(h) - f(0)}{h} \]Given, \(f(0) = 0\). Thus,\[ f'(0) = \lim_{{h \to 0}} \frac{\frac{1}{e^{h^2}} - 0}{h} = \lim_{{h \to 0}} \frac{1}{h \cdot e^{h^2}} \]

Step 3: Evaluate the Limit

To evaluate \(\lim_{{h \to 0}} \frac{1}{h \cdot e^{h^2}}\), note that both the numerator and denominator approach zero, creating a \(\frac{0}{0}\) situation. Applying L'Hospital's Rule requires differentiating the numerator and the denominator with respect to \(h\).

Step 4: Apply L'Hospital's Rule

Differentiate the numerator (which is \(1\)) and the denominator \(h \cdot e^{h^2}\). The derivative of the denominator is \[ \frac{d}{dh}(h \cdot e^{h^2}) = e^{h^2} + 2h^2e^{h^2} \].Now compute the limit:\[ \lim_{{h \to 0}} \frac{0}{e^{h^2}(1 + 2h^2)} = 0 \].

Step 5: Conclusion

The limit evaluates to zero, so \(f'(0) = 0\). Thus, by using the limit definition of derivative and L'Hospital's Rule, we've determined the derivative at \(x=0\) for the function as \(0\).

Key Concepts

Derivative

In calculus, the derivative of a function is a fundamental concept that measures the rate at which the function value changes as its input changes. It is expressed mathematically as the limit of the average rate of change of the function over a small interval as the interval approaches zero.

The derivative can be thought of as the slope of the tangent line to the graph of the function at a given point. If a function is differentiable at a point, it means that the graph is not "sharp" or "disconnected" at that point.

Applications of derivatives are numerous and include optimization problems, motion analysis, and curve sketching. Understanding derivatives allows us to predict how small changes in input affect the output.

In this context, finding the derivative at a specific point (like finding \(f'(0)\) for a piecewise function) often involves assessing the behavior of the function around that point.

Piecewise Function

Piecewise functions are mathematical functions defined by multiple sub-functions, each applying to a specific interval of the main function's domain. This kind of function is useful for modeling situations where a rule might change depending on the input value. For instance, a piecewise function can simulate different tax rates based on income levels.

The function given in the original exercise is piecewise, meaning it has different expressions based on whether \(x\) equals zero or not. Understanding how a piecewise function behaves at the boundaries, like \(x = 0\), is crucial in calculus to determine continuity and differentiability.

To work with piecewise functions, ensure to evaluate each segment separately and consider the function’s behavior at the points where the definition changes. Often, determining derivatives like \(f'(0)\) requires using the definition that applies to near-zero values of \(x\).

Limit Definition of Derivative

The limit definition of a derivative is a powerful calculus tool used to find the derivative of a function at a particular point. It provides a rigorous approach to understanding the instantaneous rate of change.

Mathematically, it's expressed as:

• \( f'(a) = \lim_{{h \to 0}} \frac{f(a + h) - f(a)}{h} \)

This illustrates how the function \(f\) changes as \(x\) shifts from \(a\) to \(a + h\).

In exercises involving piecewise functions, like our problem with \(f(x)\), this approach is essential. It's used at transition points where direct application of basic differentiation rules may not suffice. This method involves computing the limit as \(h\) approaches zero, ensuring accurate calculation of \(f'(0)\), important when the function behaves differently around certain values.

L'Hospital's Rule

L'Hospital's Rule is a technique used in calculus to evaluate limits that present indeterminate forms, such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These forms often appear in the process of finding derivatives or solving limits where direct substitution isn't possible.

The rule states that if the limit \(\lim_{{x \to c}} \frac{f(x)}{g(x)}\) results in an indeterminate form, then:

• \( \lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)} \), provided this limit exists.

In our original problem, the derivative at \(x = 0\) resulted in a \(\frac{0}{0}\) form, making L'Hospital's Rule an ideal method. Differentiating the numerator and the denominator separately allows simplification of the limit, ultimately providing accurate calculation for the derivative.

Understanding and applying this rule correctly enables solving complex limit problems that occur in calculus, particularly when evaluating the behavior of functions at certain controversial points.