Problem 70 - Calculus for Biology and Medicine

Question

Suppose the function \(f(x)\) is piecewise defined; that is, \(f(x)=f_{1}(x)\) for \(x \leq a\) and \(f(x)=f_{2}(x)\) for \(x>a\). Assume that \(f_{1}(x)\) is continuous and differentiable for \(xa\). Sketch graphs of \(f(x)\) for the following three cases: (a) \(f(x)\) is continuous and differentiable at \(x=a\). (b) \(f(x)\) is continuous, but not differentiable, at \(x=a\). (c) \(f(x)\) is neither continuous nor differentiable at \(x=a\).

Step-by-Step Solution

Answer

(a) Smooth graph; (b) Continuous with a corner; (c) Discontinuous with a gap.

Step 1: Understand Continuity and Differentiability

To sketch the graphs, we need to understand the conditions for continuity and differentiability. A function is continuous at a point if the left hand limit, right hand limit, and the function's value at that point are equal. It is differentiable at a point if there is a defined derivative (or tangent) at that point, which implies both continuity and no sharp corner or cusp.

Step 2: Case (a) - Continuous and Differentiable

For part (a), both the continuity and differentiability conditions must be met. This implies that at the point \(x = a\), the left hand limit \(\lim_{{x \to a^-}} f_1(x)\) must equal \(f_1(a)\) and the right hand limit \(\lim_{{x \to a^+}} f_2(x)\) must equal \(f_2(a)\). Also, the derivatives of \(f_1(x)\) and \(f_2(x)\) at \(x = a\) must be equal, i.e., \(f_1'(a) = f_2'(a)\). Graphically, \(f(x)\) is a smooth, unbroken curve at \(x = a\).

Step 3: Case (b) - Continuous but Not Differentiable

For part (b), the function needs to be continuous but not differentiable. At \(x = a\), the left and right limits are equal, \(f_1(a) = f_2(a)\), but the derivatives are different, \(f_1'(a) eq f_2'(a)\). This situation results in a corner or cusp at \(x = a\). On the graph, \(f(x)\) will be a continuous line that changes direction sharply at \(x = a\).

Step 4: Case (c) - Neither Continuous nor Differentiable

For part (c), the function is neither continuous nor differentiable at \(x = a\). This means either \(\lim_{{x \to a^-}} f_1(x) eq \lim_{{x \to a^+}} f_2(x)\) or there is no limit at all, indicating a discontinuity. As \(f(x)\) is not continuous, it cannot be differentiable. Graphically, there is a gap or jump at \(x = a\).

Key Concepts

Understanding Continuity

Continuity is a crucial property that indicates a function behaves in an uninterrupted manner at a given point. When a piecewise function like \(f(x)\) is continuous at a point \(x = a\), three critical conditions need to be satisfied:

• The left-hand limit \(\lim_{{x \to a^-}} f(x)\) exists.
• The right-hand limit \(\lim_{{x \to a^+}} f(x)\) exists.
• The function's value at the point, \(f(a)\), equals both of these limits.

These conditions together imply that the graph of the function is unbroken at \(x = a\).
This smooth transition without jumps or gaps is what ensures continuity. If any of these conditions fail, the function loses its continuity at that point.
Visualizing continuity can help, so imagine drawing the graph of \(f(x)\) without lifting your pencil off the paper at \(x = a\). This indicates the seamless connection across the point.

Differentiability and Its Implications

Differentiability adds an extra layer to the behavior of a function. For a function to be differentiable at a point, it does not just have to be continuous at that point—there additionally needs to be a well-defined tangent or slope without sharp edges.
Importantly, we have:

• \(f(x)\) is differentiable at \(x = a\) if both \(f_1'(a)\) and \(f_2'(a)\) exist and are equal.
• Differentiability implies continuity, but not vice versa: if \(f(x)\) has a sharp corner or cusp at a point, it won't be differentiable there.

Recognizing differentiability graphically means ensuring not just that the function does not exhibit gaps, but also that it does not suddenly shift direction sharply at that point.
When drawing these functions, think of the differentiable part as a "smooth slope" at the point, continuing without any kinks or abrupt direction changes.

Graph Sketching for Piecewise Functions

When dealing with piecewise functions, the art of graph sketching becomes crucial in visualizing the behavior at certain key points like \(x = a\). Consider the cases:

• Case (a): The graph is continuous and differentiable. Expect a seamless, smooth curve at \(x = a\), indicating both sides slope into each other perfectly.
• Case (b): Continuous but not differentiable; visualize a corner more than a smooth curve. The function does not break, but it sharply changes direction at \(x = a\).
• Case (c): Neither continuous nor differentiable results in a disconnect at \(x = a\). Here, imagine a jump or a gap in the graph.

Crafting these graphs helps in anticipating the underlying function's definition's impact on its continuity and differentiability.
Always pay attention to the visual cues: whether they are an unbroken road (continuity) or a smooth path without corners (differentiability). These sketches are key to mastering piecewise function evaluations.