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Piecewise functions are distinctive types of functions that have different expressions or rules for different parts of their domains. The graph of a piecewise function is often composed of separate pieces that connect under specific rules.
In mathematical notation, a piecewise function is typically written with a bracket and formulas for each segment of its domain. For example, the function in the exercise is given as:

• When \(x < 0\), the function is expressed as \(x + b\).
• When \(x \geq 0\), the function switches to a new rule, \(\cos x\).

This particular piecewise function changes its rule at \(x = 0\). The challenge with piecewise functions is ensuring continuity and differentiability at the points where the rules switch, especially at those transition points like \(x = 0\) in this question.

Limits are foundational concepts in calculus and are used to describe the behavior of functions as they approach a certain point. In examining a piecewise function, limits are essential when determining if a function is continuous at the transition between different pieces.
For continuity at a point, the left-hand limit (approaching from the left) and the right-hand limit (approaching from the right) must be equal to the function's value at that point.

• In the exercise, we calculated \(\lim_{x \to 0^-} g(x) = b\) when approaching from the left with the function \(x + b\).
• The right-hand limit \((\lim_{x \to 0^+} g(x) = \cos(0) = 1)\) was calculated using \(\cos x\) for \(x \geq 0\).

To find a value of \(b\) that makes the function continuous, we equate the left and right hand limits: \(b = 1\). This ensures that the function’s behavior remains consistent as \(x\) approaches 0 from both directions.

Derivatives represent the rate of change of a function. They are crucial in determining how smooth a function is at a given point, particularly for piecewise functions. A function is said to be differentiable at a point if its derivative exists at that point and matches from both directions.
For the piecewise function discussed:

• On the left side \(x < 0\), the function \(g(x) = x + b\) has a derivative \(\frac{d}{dx}(x + b) = 1\).
• While on the \(x \geq 0\) side, the function \(g(x) = \cos x\) has a derivative \(\frac{d}{dx}(\cos x) = -\sin x\), which evaluates to 0 at \(x = 0\).

To check for differentiability, these derivatives must be equal when evaluated at \(x = 0\). Since the left-hand derivative (1) and right-hand derivative (0) do not match, the function is not differentiable at \(x = 0\). This means there is a sharp point or corner at this transition.

Calculus encompasses the study of limits, derivatives, integrals, and infinite series, and is a core mathematical discipline employed in a variety of applications from engineering to economics. In our specific exercise, we utilize two essential concepts of calculus: limits and derivatives.
These concepts allow us to analyze the behavior and properties of piecewise functions around troublesome points, such as discontinuities or points of non-differentiability.

• Limits help us determine whether a piecewise function is continuous at a transition point by comparing the behavior of the function on both sides of the point.
• Derivatives provide insight into the smoothness and slope of the function curve, indicating how well-connected the transitions in the function are.

Together, these tools of calculus help us design and analyze mathematical models that reflect real-world phenomena. Understanding how functions behave at every point, especially tricky ones, is an integral component of problem-solving in calculus.