Problem 1 - APEX Calculus

Question

Let a level curve of \(z=f(x, y)\) be described by \(x=g(t)\), \(y=h(t) .\) Explain why \(\frac{d z}{d t}=0\)

Step-by-Step Solution

Answer

The total derivative \(\frac{dz}{dt}\) is zero because level curves have constant function values and their change with respect to a parameter \(t\) is zero.

Step 1: Understand the Meaning of Level Curves

A level curve of a function \(z = f(x, y)\) is a curve along which the function has a constant value, say \(c\). This means that as you move along the curve, the value of \(z\) does not change. So, for level curves, \(f(x, y) = c\) is constant.

Step 2: Differentiate Both Sides of the Level Curve Equation

Given that \(x = g(t)\) and \(y = h(t)\), substituting these into the level curve equation gives \(f(x, y) = f(g(t), h(t)) = c\). Differentiate both sides with respect to \(t\): \(\frac{d}{dt}[f(g(t), h(t))] = \frac{d}{dt}[c]\).

Step 3: Apply the Chain Rule

Using the chain rule, we differentiate \(f(g(t), h(t))\) as follows: \(\frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}\).

Step 4: Set the Derivatives Equal

Since \(c\) is a constant, its derivative with respect to \(t\) is zero, \(\frac{d}{dt}[c] = 0\). So, the equation becomes \(\frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt} = 0\).

Step 5: Relate to \\frac{dz}{dt}

This means that the total derivative of \(z\) with respect to \(t\) is zero because it captures the change in the function value along the level curve, relating \(\frac{dz}{dt} = \frac{df}{dt} = 0\).

Key Concepts

Partial Derivatives

When dealing with functions of multiple variables such as \(z = f(x, y)\), it's essential to understand partial derivatives. A partial derivative measures how much the function changes as you change one of the variables while keeping the other variables constant. Think of it as observing how the height of a hill changes if you walk only east or only north, but not both simultaneously.

• For the function \(f(x, y)\), the partial derivative with respect to \(x\) is denoted as \(\frac{\partial f}{\partial x}\). It represents the rate of change of \(z\) as \(x\) changes, holding \(y\) constant.
• Similarly, \(\frac{\partial f}{\partial y}\) is the rate of change of \(z\) with respect to \(y\), holding \(x\) constant.

These partial derivatives are crucial for applying the chain rule, especially when dealing with functions represented parametrically as \(x = g(t)\) and \(y = h(t)\). Knowing partial derivatives helps us decipher how each variable alone affects the function.

Chain Rule

The chain rule is a technique in calculus used to differentiate composite functions. It allows us to find the derivative of a function based on the derivatives of its component parts. Imagine you are examining how one variable indirectly influences another through an intermediate variable.
In our scenario where \(f(x, y) = f(g(t), h(t))\), the chain rule helps establish how changes in \(t\) affect \(f\). By differentiating \(f\) with respect to \(t\), using the chain rule, we can express the result as:

• \(\frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}\)

This equation is a blend of the rates of change \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) weighted by how much \(f\) depends on changes in \(x\) and \(y\) individually. In the context of level curves, employing the chain rule ensures we peel back how each parameter influences the entire function's rate of change.

Total Derivative

The total derivative with respect to a parameter, such as \(t\), captures the overall rate of change of a function. It accounts for all paths along which a change might occur. For a level curve where \(c\) is constant,we derive the total derivative of \(z = f(x, y)\) with respect to \(t\) by combining all partial effects out of necessity.

• In our example, we found \(\frac{dz}{dt} = \frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}\).
• Since \(c\) remains constant, this derivative turns into zero, indicating that while \(x\) or \(y\) might individually change, these changes exactly balance each other to keep \(z\) unchanged on the level curve.

The concept of the total derivative is pivotal when analyzing level curves, as it highlights the conditions for balance and constancy necessary for maintaining \(z\)'s value as the parameter \(t\) varies.