Problem 411 - Calculus Volume 3

Question

Find the differential \(d z\) of \(h(x, y)=4 x^{2}+2 x y-3 y\) and approximate \(\Delta z\) at the point \((1,-2) .\) Let \(\Delta x=0.1\) and \(\Delta y=0.01\).

Step-by-Step Solution

Answer

\(\Delta z \approx 0.39\)

Step 1: Find the partial derivatives

To find the differential \(dz\), we first need to calculate the partial derivatives of \(h(x, y)\) with respect to \(x\) and \(y\). The partial derivative of \(h\) with respect to \(x\) is \(\frac{\partial h}{\partial x} = 8x + 2y\). The partial derivative with respect to \(y\) is \(\frac{\partial h}{\partial y} = 2x - 3\).

Step 2: Evaluate the partial derivatives at the point (1, -2)

Now, substitute \(x = 1\) and \(y = -2\) into the partial derivatives. For \(\frac{\partial h}{\partial x}\), evaluate \(8(1) + 2(-2) = 8 - 4 = 4\). For \(\frac{\partial h}{\partial y}\), evaluate \(2(1) - 3 = 2 - 3 = -1\).

Step 3: Write the expression for the differential dz

The differential \(dz\) is given by the formula \(dz = \frac{\partial h}{\partial x} dx + \frac{\partial h}{\partial y} dy\). Since \(\frac{\partial h}{\partial x} = 4\) and \(\frac{\partial h}{\partial y} = -1\), the expression becomes \(dz = 4dx - dy\).

Step 4: Calculate dz using given increments

Now, substitute \(\Delta x = 0.1\) and \(\Delta y = 0.01\) into the differential expression \(dz = 4dx - dy\). Thus, \(dz = 4(0.1) - 1(0.01) = 0.4 - 0.01\).

Step 5: Compute the approximate change in z, \(\Delta z\)

Using the computed value of \(dz\), we approximate \(\Delta z \approx dz = 0.4 - 0.01 = 0.39\).

Key Concepts

Partial Derivatives

When dealing with multivariable functions, a partial derivative measures how a function changes as one of its inputs is varied, while keeping the others constant. For the function \( h(x, y)=4x^{2}+2xy-3y \), we have two variables, \( x \) and \( y \). Taking the partial derivative with respect to \( x \) means letting \( y \) be constant and finding out how much \( h \) changes as \( x \) changes:

• \( \frac{\partial h}{\partial x} = 8x + 2y \)

Similarly, when we take the partial derivative concerning \( y \), we treat \( x \) as a constant. This gives us:

• \( \frac{\partial h}{\partial y} = 2x - 3 \)

Understanding partial derivatives is crucial in multivariable calculus because they allow us to analyze how multi-dimensional figures behave in specific directions. By calculating \( \frac{\partial h}{\partial x} \) and \( \frac{\partial h}{\partial y} \) at a specific point like \((1, -2)\), we can better understand the local behavior and the rate of change of the function in different directions at that point.

Differential Approximation

Differential approximation is a way to estimate how a function changes when its variables change slightly. In other words, it's a method to approximate the change in a function's output based on small changes in its inputs. For a function \( h(x, y) \), the differential is given by:

• \( dz = \frac{\partial h}{\partial x} dx + \frac{\partial h}{\partial y} dy \)

This formula combines the effects of the small changes \( dx \) and \( dy \) on the total change in \( z \), denoted as \( dz \). By substituting the known increments \( \Delta x = 0.1 \) and \( \Delta y = 0.01 \), we compute \( dz \) to approximate \( \Delta z \), the real change in the function's value. With the partial derivatives already evaluated at the point \((1, -2)\), the formula becomes:

• \( dz = 4\times0.1 - 1\times0.01 = 0.39 \)

This means that a small change in \( x \) by 0.1 units and \( y \) by 0.01 units approximates a change in \( z \) by about 0.39 units. This technique is useful for predicting responses of systems described by multivariable functions in real-world applications.

Multivariable Functions

A multivariable function is a type of mathematical function that depends on more than one input variable. In our example, \( h(x, y) \) is a function of two variables \( x \) and \( y \). These functions can describe scenarios like temperature changes across a landscape or the pressure distribution on a surface.
The concept of multivariable functions extends to multiple dimensions, which means each input can affect the output in complex ways. That’s why using tools like partial derivatives and differentials is important when handling these functions to analyze changes and predict behaviors.

• They provide detailed insights by letting us focus on one variable at a time.
• They help break down complex systems into more manageable regional analyses.

Understanding how each variable impacts the function individually through partial derivatives gives us an invaluable look at the dynamics of multivariable systems. Because each input can independently or simultaneously affect the outcome, it equips us to handle complex practical problems effectively, ranging from engineering to economics.