Problem 3 - APEX Calculus

Question

T/F: A point \(P\) is a critical point of \(f\) if \(f_{x}\) or \(f_{y}\) are undefined at \(P\).

Step-by-Step Solution

Answer

True; a point is critical if derivatives are zero or undefined.

Step 1: Define a Critical Point

A critical point of a function of several variables, such as \(f(x, y)\), is a point \((a, b)\) in the domain of \(f\) where the derivative is zero or undefined. That is, at a critical point, \[ f_x(a, b) = 0, \quad f_y(a, b) = 0 \] or either \( f_x \) or \( f_y \) do not exist.

Step 2: Analyze Condition for Critical Points

According to the definition, for a point \((a, b)\) to be a critical point of \(f\), the partial derivatives \(f_x\) and \(f_y\) must either be zero or be undefined at the point \((a, b)\). Therefore, having \(f_x\) or \(f_y\) undefined at \((a, b)\) satisfies the condition for a critical point.

Step 3: Conclusion

A point \(P\) is indeed a critical point if \(f_x\) or \(f_y\) is undefined at \(P\), as this satisfies the definition of a critical point having either the derivative zero or undefiined. Hence, the statement is True.

Key Concepts

Partial Derivatives

In multivariable calculus, partial derivatives are quite like their single-variable counterparts, except they provide information about how a function changes as just one of its variables changes, keeping all others fixed. This concept is key when dealing with functions that depend on more than one variable.
Suppose we have a function \( f(x, y) \). The partial derivatives \( f_x \) and \( f_y \) represent the rates at which \( f \) changes with respect to \( x \) and \( y \), respectively.

• \( f_x(x, y) \) is the derivative of \( f \) with respect to \( x \), treating \( y \) as a constant.
• \( f_y(x, y) \) is the derivative of \( f \) with respect to \( y \), treating \( x \) as a constant.

Calculating partial derivatives often involves using similar rules as single-variable derivatives, such as the power rule, product rule, and chain rule, applied to the specified variable while treating other variables as constants. Understanding how to compute these derivatives is crucial for analyzing the behavior of multivariable functions.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions of several variables, not just one. It embraces concepts like partial derivatives and critical points to describe the behavior of complex systems.
When you think about functions of one variable, you consider how they change as their single input changes.

However, multivariable functions depend on two or more variables. For instance, if we consider a function \( f(x, y) \), the output is determined by varying two inputs, \( x \) and \( y \).
Topics in multivariable calculus include:

• Gradient vectors, which generalize the first derivative to higher dimensions.
• Double and triple integrals, which extend the concept of integration to areas and volumes.
• Critical points, which help identify maxima, minima, or saddle points of surfaces.

By analyzing these aspects, multivariable calculus allows us to deal with more complex, real-world problems that involve multiple influences or variables.

Undefined Derivatives

A derivative is undefined at a point where the function does not change smoothly. In terms of partial derivatives, if the surface represented by a function \( f(x, y) \) has an abrupt change, the derivatives \( f_x \) or \( f_y \) may become undefined.

When we say a derivative is undefined at a point \((a, b)\), it typically means one of the following:

• The derivative is approaching different values from different directions (a cusp).
• The function has a vertical tangent at the point.
• The function is not continuous at that point.

In multivariable calculus, encountering an undefined derivative at a point \( P \) signifies a critical point. Such points may correspond to important characteristics like peaks, troughs, or ridges on the graph of a function. These are similar to critical points found in single-variable calculus, where the derivative is either zero or fails to exist, but they involve partial derivatives instead. Understanding these aspects is vital for fully grasping the behavior of surfaces and finding optimal points in a given range.