Problem 25 - Advanced Engineering Mathematics

Question

Find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. $$ F(x, y, z)=x^{2}+4 x z+2 y z^{2} ;(1,2,-1) $$

Step-by-Step Solution

Answer

The vector is (-2, 2, -4); maximum rate is \(2\sqrt{6}\).

Step 1: Identify the Gradient Function

The direction in which a function increases most rapidly at a point is given by the gradient of the function, denoted as \( abla F(x, y, z) \). First, we need to compute the partial derivatives of the function.\[ \frac{\partial F}{\partial x} = 2x + 4z, \quad \frac{\partial F}{\partial y} = 2z^2, \quad \frac{\partial F}{\partial z} = 4x + 4yz \]

Step 2: Compute the Gradient at the Given Point

Substitute the coordinates of the given point \((1, 2, -1)\) into the partial derivatives to find the gradient at that point.\[ abla F(1, 2, -1) = \left( 2(1) + 4(-1), 2(-1)^2, 4(1) + 4(2)(-1) \right) \]Simplifying these, we get:\[ abla F(1, 2, -1) = (2 - 4, 2, 4 - 8) = (-2, 2, -4) \]

Step 3: Determine the Direction of Most Rapid Increase

The vector \((-2, 2, -4)\) is the direction in which the function \(F(x, y, z)\) increases most rapidly at the point \((1, 2, -1)\). This is because the gradient vector points in the direction of the greatest increase of the function.

Step 4: Find the Maximum Rate of Increase

The maximum rate of increase of the function at a point is the magnitude of the gradient vector at that point. Compute the magnitude of \((-2, 2, -4)\) using the formula for the magnitude of a vector:\[ \| abla F(1, 2, -1) \| = \sqrt{(-2)^2 + 2^2 + (-4)^2} = \sqrt{4 + 4 + 16} = \sqrt{24} = 2\sqrt{6} \]

Key Concepts

Partial Derivatives

Partial derivatives play a crucial role in understanding how functions change with respect to each individual variable, holding the others constant. This is particularly important when dealing with functions of more than one variable, such as the function given in this exercise, which involves three variables: \(x\), \(y\), and \(z\). To understand the behavior of the function \(F(x, y, z) = x^2 + 4xz + 2yz^2\), we compute the partial derivatives.

• \(\frac{\partial F}{\partial x} = 2x + 4z\)
• \(\frac{\partial F}{\partial y} = 2z^2\)
• \(\frac{\partial F}{\partial z} = 4x + 4yz\)

Each of these derivatives tells us how the function changes along a small increment of \(x\), \(y\), or \(z\) individually, with the other variables held constant. These calculations form the foundation for determining how a function behaves in a multi-dimensional space.

Direction of Increase

Once we have partial derivatives, we can use them to construct the gradient vector. The gradient vector signifies the direction in which the function increases most rapidly. This vector is found by assembling the partial derivatives into a single vector:\[abla F(x, y, z) = \left( \frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z} \right)\]In this exercise, the gradient at the given point \((1, 2, -1)\) is found by substituting these coordinates into our partial derivatives. The resultant gradient vector is \(-2\hat{i} + 2\hat{j} - 4\hat{k}\), providing the path of steepest ascent. Essentially, this vector acts like a compass, pointing in the direction where the function experiences the most rapid increase.

Maximum Rate of Increase

The maximum rate of increase of a function at a given point is crucial as it signifies how intensely the function is changing in the direction of the gradient. Mathematically, it's represented by the magnitude (or length) of the gradient vector.For the vector \((-2, 2, -4)\), we calculate the magnitude using:\[\|abla F(1, 2, -1)\| = \sqrt{(-2)^2 + (2)^2 + (-4)^2}\]Calculating gives \(\sqrt{24} = 2\sqrt{6}\). This value represents the maximum rate at which the function \(F(x, y, z)\) is increasing from the point \((1, 2, -1)\). Understanding this concept not only provides insights into the behavior of the function but also aids in optimization tasks where maximizing the increase is desired.