Use zooming to investigate f^'(0), the slope of the curve y=f(x) at x=0, for the following two f

step 1 We want to determine if the functions are differentiable at x equals 0. So we're going to zoom i

Use zooming to investigate f^'(0), the slope of the curve...

Use zooming to investigate $f^{\prime}(0)$, the slope of the curve $y=f(x)$ at $x=0$, for the following two functions. a. $$ f(x)= \begin{cases}x \sin \left(\frac{1}{x}\right), & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{cases} $$ b. $$ f(x)= \begin{cases}x^2 \sin \left(\frac{1}{x}\right), & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{cases} $$

Use zooming to investigate $f^{\prime}(0)$, the slope of the curve $y=f(x)$ at $x=0$, for the following two functions.
a.
$$
f(x)= \begin{cases}x \sin \left(\frac{1}{x}\right), & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{cases}
$$
b.
$$
f(x)= \begin{cases}x^2 \sin \left(\frac{1}{x}\right), & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{cases}
$$

Key Concepts

Derivative

The derivative at a point is the slope of the tangent line and measures the instantaneous rate of change of the function. It is defined as the limit of the difference quotient as the input change approaches zero. This concept underpins many ideas in calculus, particularly when analyzing the behavior of functions locally.

Limit Definition of the Derivative

The derivative is defined by the limit of (f(x + h) - f(x))/h as h approaches zero. This formalism is crucial for determining whether a function has a well?defined tangent at a specific point, especially when the function exhibits unusual behavior or is defined piecewise.

Local Linearization (Zooming In)

Local linearization, or zooming in on a function's graph at a particular point, is a technique for examining whether the function behaves increasingly like a straight line as one gets closer to that point. This method helps to visually and conceptually assess the existence and value of the derivative by investigating the function’s behavior at smaller and smaller scales.

Oscillatory Behavior

Oscillatory behavior refers to the rapid fluctuations of a function’s output values, typically seen with trigonometric components such as sine or cosine. These oscillations can interfere with the straightforward application of the limit definition of the derivative, especially if the oscillations do not diminish in amplitude when approaching the point of interest.

Damping Effects via Multiplicative Factors

Some functions involve a multiplicative factor that 'damps' the oscillations as the input approaches a particular value. When the damping term decreases at a different rate than the oscillatory component, it can lead to the cancellation of wild fluctuations, potentially allowing the function to have a well?defined tangent, as seen when a higher order vanishing term multiplies an oscillatory function.

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