Problem 35 - Calculus for Biology and Medicine

Question

In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=\left|2 x^{2}-1\right| $$

Step-by-Step Solution

Answer

The function is not differentiable at \( x = \pm \frac{\sqrt{2}}{2} \).

Step 1: Analyze the Function

The given function is \( y = |2x^2 - 1| \). This is an absolute value function, which means it may have a sharp corner or cusp at certain points, making it non-differentiable there. To find potential points of non-differentiability, we first consider the expression inside the absolute value, \( 2x^2 - 1 \).

Step 2: Determine Critical Points

Set the expression inside the absolute value equal to zero to find points where the function changes concavity or shape. Solve \( 2x^2 - 1 = 0 \):\[ 2x^2 = 1 \]\[ x^2 = \frac{1}{2} \]\[ x = \pm\sqrt{\frac{1}{2}} \]These are \( x = \pm\frac{\sqrt{2}}{2} \).

Step 3: Graph the Function

Plot the function \( y = |2x^2 - 1| \) using the critical points. The function is a parabola that opens upwards mirrored over the x-axis for portions that are negative inside the absolute bracket. It will have sharp corners or cusps at \( x = \pm \frac{\sqrt{2}}{2} \).

Step 4: Identify Non-differentiable Points

From the graph, observe the points where there is a sharp corner or cusp. The function is non-differentiable at the x-values found in Step 2: \( x = \pm\frac{\sqrt{2}}{2} \) due to the nature of the absolute value, which causes a 'kink' in the graph.

Key Concepts

Absolute Value Functions

An absolute value function, such as \( y = |2x^2 - 1| \), displays distinct behavior compared to standard linear or quadratic functions.
Generally, the absolute value function translates any negative output of the contained expression into a positive one, causing unique features like sharp corners or cusps in its graph.
These characteristics are due to the sudden change in direction when crossing the x-axis.

• An expression within the absolute value, such as \(2x^2 - 1\), influences the function's shape and position on the coordinate plane.
• The expression can be split at zero points, dividing the function into segments that mirror each other across the x-axis.
• Wherever the inside of the absolute value switches from negative to positive, the graph may demonstrate a cusp or sharp edge.

Absolute value functions are important in understanding non-differentiable points, as these sharp transitions cannot be smoothed out.

Critical Points

Critical points of a function are values of x where the behavior or shape of the graph changes notably.
For the function \( y = |2x^2 - 1| \), critical points are found by setting the expression inside the absolute value equal to zero: \( 2x^2 - 1 = 0 \).

• Solve \( 2x^2 = 1 \) to get \( x = \pm\sqrt{\frac{1}{2}} \), which simplifies to \( x = \pm\frac{\sqrt{2}}{2} \).
• These critical points divide the graph into regions, where the function displays different characteristics.
• Near these points, the absolute value function can cause a 'kink' or sharp turn, highlighting a location of non-differentiability.

Identifying critical points helps in graphing the function accurately and analyzing where and why the function may not be differentiable.

Differentiability in Calculus

Differentiability is a core concept in calculus.
A function is differentiable at a point if it has a well-defined tangent line there.
In simpler terms, the graph should not have any breaks, holes, or sharp turns at that point. When dealing with absolute value functions, we must pay special attention to points of interest identified by critical points.

• If the graph has a cusp or sharp corner, like at \( x = \pm\frac{\sqrt{2}}{2} \) for \( y = |2x^2 - 1| \), the function is not differentiable there.
• Differentiability concerns smoothness; any abrupt direction changes or 'kinks' invalidate a derivative.
• Analyzing differentiability involves examining the slopes on either side of the critical points to identify any discrepancies indicating non-differentiability.

Understanding differentiability allows students to determine where functions are smooth and well-behaved, which is crucial in many calculus applications.