Question

Describe the \(x\) -values at which \(f\) is differentiable. $$ f(x)=\left\\{\begin{array}{ll} x^{2}-3, & x \leq 0 \\ 3-x^{2}, & x>0 \end{array}\right. $$

Short answer

The function \(f\) is differentiable for all real values of \(x\).

Step-by-step solution

Differentiate each part of the function

Differentiate \(x^{2}-3\) with respect to \(x\) to get \(2x\) when \(x \leq 0\) and differentiate \(3-x^{2}\) with respect to \(x\) to get \(-2x\) when \(x>0\).

Check differentiability at the joining point

The two parts of the function meet at \(x=0\). The derivative from the left (using the first part of the function) at \(x=0\) is 0 and the derivative from the right (using the second part of the function) at \(x=0\) is also 0. Since the two one-sided derivatives are equal, \(f\) is differentiable at \(x=0\).

Check differentiability elsewhere

For \(x<0\), the derivative is \(2x\), which is defined for all \(x<0\). For \(x>0\), the derivative is \(-2x\), which is defined for all \(x>0\). Therefore, \(f\) is differentiable for all \(x \neq 0\).

Key concepts

Piecewise Functions

Piecewise functions, like the one provided in the exercise \( f(x) \), are defined by different expressions for different intervals of the domain. They can sometimes present challenges in understanding their behavior at the points where the expression changes, commonly known as 'joining points' or 'boundaries'.

In essence, a piecewise function is like a chameleon, changing its formula based on where you are along the number line. For instance, the function given in the exercise has two 'personalities': for \( x \leq 0 \), it behaves like \( x^2-3 \), and for \( x > 0 \), it transforms into \( 3 - x^2 \). Students often wonder if these pieces fit together smoothly or if there is a 'seam' or 'gap' where the pieces meet, which can affect the overall continuity and differentiability of the function.

Derivative of a Function

The derivative of a function represents the rate at which the function's value is changing at any given point. Think of it as a snapshot of a car's speedometer – it tells you how fast the car is going at a precise moment in time.

To find a function's derivative, like in the step-by-step solution for \( f(x) \), you apply the power rule. For the first piece, \( x^{2}-3 \) when \( x \leq 0 \) the derivative is \( 2x \) using the power rule which states that the derivative of \( x^n \) is \( nx^{n-1} \). Similarly, for the second piece, \( 3 - x^{2} \) when \( x > 0 \) the derivative is \( -2x \) following the opposite sign due to the power rule applied to \( -x^{2} \). Determining the derivative for each 'piece' of the function is like decoding its behavior – how it increases or decreases – within its designated zone.

Continuity and Differentiability

Continuity and differentiability are two concepts that dance together in the world of calculus. A function is continuous if there are no breaks, jumps, or holes in its graph. Differentiability takes it a step further, implying that not only is the function smooth, but its rate of change is also well defined everywhere.

To verify continuity at the joining point (in this case, \( x = 0 \)), we typically check if the left-hand limit equates to the right-hand limit. For differentiability, a continuous function needs a derivative that doesn’t make any abrupt changes. As seen in the provided steps, the derivative is consistent from the left and the right at \( x = 0 \) since both approach a derivative of 0. This smooth passing of the baton from the left to the right of \( x = 0 \) signifies not just continuity, but also differentiability at that point. Essentially, differentiability is the 'gold standard' – if a function is differentiable at a point, then it’s guaranteed to be continuous there as well. However, the converse isn't always true - a continuous function isn't always differentiable at every point.