Problem 75 - Calculus an Applied Approach

Question

Writing Use a graphing utility to graph the two function \(f(x)=x^{2}+1\) and \(g(x)=|x|+1\) in the same viewin window. Use the zoom and trace features to analyze the graphs near the point \((0,1) .\) What do you observe? Whic function is differentiable at this point? Write a short paragraph describing the geometric significance of differentiability at a point.

Step-by-Step Solution

Answer

The function \(f(x)=x^{2}+1\) is differentiable at the point (0,1) while \(g(x)=|x|+1\) is not. The geometric significance of differentiability at a point is that the function has a definite slope at that point, which means the function is smooth and without corners or bends.

Step 1: Graph of the Functions

Start by plotting both \(f(x)=x^{2}+1\) and \(g(x)=|x|+1\) on the same graph. By observing the graph, it is possible to see that \(f(x)\) is a smooth curve and that \(g(x)\) has a corner at the point (0,1).

Step 2: Zoom and Trace

Now use the zoom and trace features of the graphing tool to get a better view of the functions around the point (0,1). It will be observed that \(f(x)\) is smooth at this point, indicating that it might be differentiable there. On the other hand, \(g(x)\) has a sharp 'turn' at this point, indicating non-differentiability.

Step 3: The Concept of Differentiability

Differentiability at a point essentially means that the function has a defined slope at that particular point. In this case, \(g(x)\) has two different slopes for the immediate left and right of \(x=0\), hence it is not differentiable at \(x=0, y=1\). But for \(f(x)\), there is only one possible tangent line at \(x=0, y=1\), thus it is differentiable.

Key Concepts

Graphing Functions

When exploring the differentiability of functions, it's crucial to start with graphing. Using a graphing utility, we plot two functions:

• The quadratic function: \(f(x) = x^2 + 1\)
• The absolute value function: \(g(x) = |x| + 1\)

Both functions are displayed within the same window to allow for direct comparison. This side-by-side comparison is helpful to visually analyze the behavior of each function around any given point, such as \((0,1)\).
The graph of \(f(x) = x^2 + 1\) forms a smooth parabolic curve, indicating a continuous and smooth transition in its slope throughout its domain. In contrast, the graph of \(g(x) = |x| + 1\) reveals a V-shape, with a noticeable corner where the two linear pieces meet. This graphical observation is vital in understanding where differentiability might break down.

Smooth Curves

The concept of smoothness in a curve relates directly to differentiability. A smooth curve is one where there are no abrupt changes in direction. This is evident in the graph of \(f(x) = x^2 + 1\), where the curve flows continuously without any sharp turns or corners. In mathematical terms, this means that the derivative, or the slope of the tangent line at any point on the curve, changes at a steady rate.
At the point \((0,1)\), both functions intersect. To determine the smoothness around this point, we can zoom in using our graphing tool. Zooming in on \(f(x)\) reveals that it remains a smooth, continuous curve. Any tangent line we draw will meet the curve at only one point with a single slope, indicating smoothness and confirming differentiability.
Alternatively, \(g(x) = |x| + 1\) has a sharp vertex at \((0,1)\). This vertex breaks the smoothness since the direction of the function changes abruptly, leading to multiple potential tangent lines with different slopes, demonstrating non-differentiability.

Geometric Significance

Understanding the geometric significance of differentiability involves recognizing how a function's graph behaves at a particular point. When a function is differentiable at a point, like \(f(x) = x^2 + 1\) at \((0,1)\), it has a clear and unique tangent line that touches the curve at that point and nowhere else. This tangent line represents the instantaneous rate of change at that exact location.
On the other hand, at the same point for \(g(x) = |x| + 1\), the concept of a single tangent breaks down because of the sharp corner. Here, the function does not have a single slope but rather two distinct slopes as you approach \(x = 0\) from either side. This lack of a unique tangent line is a key geometric indicator of non-differentiability. Therefore, observing the graph helps to determine where a function fails to be smooth and why it is not differentiable at that point.