Question

Find the gradient of the function and the maximum value of the directional derivative at the given point. $$ \frac{\text { Function }}{w=x y^{2} z^{2}} \frac{\text { Point }}{(2,1,1)} $$

Short answer

The gradient of the function at the point (2,1,1) is \(\<1, 4, 4>\) and the maximum value of the directional derivative at this point is \(\sqrt{33}\).

Step-by-step solution

Calculate Partial Derivatives

The first step involves calculating the partial derivatives of \(w\) with respect to \(x\), \(y\), and \(z\). The partial derivative with respect to each variable is calculated by treating all other variables as constants. So, \(w_x = y^2 z^2\), \(w_y = 2xyz^2\), and \(w_z = 2xy^2z\).

Evaluate at the Given Point

In the next step, evaluate the partial derivatives calculated in Step 1 at the given point (2,1,1). Substituting \(x=2\), \(y=1\), and \(z=1\) into each of the partial derivatives gives the following values: \(w_x(2,1,1) = 1^2 * 1^2 = 1\), \(w_y(2,1,1) = 2 * 2 * 1^2 = 4\), \(w_z(2,1,1) = 2 * 2 * 1^2 * 1 = 4\).

Construct the Gradient Vector

Construct the gradient vector, which comprises the values of the partial derivatives evaluated in Step 2. Thus, the gradient of \(w\) at the point (2,1,1) is \(\<1, 4, 4>\).

Determine the Maximum Value of the Directional Derivative

The maximum value of the directional derivative is achieved in the direction of the gradient vector and is equal to the magnitude of the gradient. The magnitude is calculated as the square root of the sum of squares of the vector components. Therefore, \(\|\nabla w\| = \sqrt{1^2 + 4^2 + 4^2} = \sqrt{33}\).

Key concepts

Partial Derivatives

Partial derivatives are at the heart of multivariable calculus, representing how a function changes as each variable is altered individually while all others remain constant. For instance, imagine you're baking a cake and adjusting ingredients one by one to see how each affects the overall flavor. In the same manner, with a function like \(w = x y^2 z^2\), you would compute partial derivatives for each variable — \(x\), \(y\), and \(z\) — to discern their isolated impact on \(w\).

The calculations are quite straightforward: to find \(w_x\), the partial derivative of \(w\) with regard to \(x\), treat \(y\) and \(z\) as constants and differentiate. Repeat this to find \(w_y\) and \(w_z\) by treating the other variables as constants during each differentiation. This step is crucial to understand the function's behavior and sets the stage for constructing the gradient vector, which is an amalgam of these partial derivatives and points in the direction where the function increases most rapidly from a given point.

Gradient Vector

The gradient vector is a pivotal concept in multivariable calculus, symbolizing the direction and rate of the steepest ascent of a function from a point. Consider it as a compass pointing towards the peak of a hill. After finding the partial derivatives of \(w\), the gradient vector at a point \((x, y, z)\) is composed of these derivative values, represented as \(abla w = \langle w_x, w_y, w_z \rangle\).

In our cake-baking analogy, if you've gauged the individual ingredient effects (partial derivatives), then the gradient vector would tell you the most effective combination of changes for the best flavor increase from your current recipe. In the problem involving the gradient of \(w\) at the point \((2,1,1)\), after inserting the variable values into the gradients, we combine them to form the gradient vector \(\langle 1, 4, 4 \rangle\). This vector points to the function's steepest increase near the point of interest.

Multivariable Calculus

Multivariable calculus extends the principles of single-variable calculus to functions of several variables. It's like a conductor managing an orchestra — it coordinates the effects of all variables to understand a function’s overall dynamics. In this field, concepts such as partial derivatives, gradient vectors, and directional derivatives become integral tools for analyzing the function's topography.

In multivariable calculus, the directional derivative in the direction of the gradient vector typically denotes the maximum rate of increase of the function. It's as if you're hiking and seeking the path that ascends the fastest from your current position. The magnitude of the gradient vector \(\| abla w \|\) gives this steepest ascent's rate. For a function like \(w = x y^2 z^2\) at \((2,1,1)\), evaluating the gradient and then its magnitude provides this maximum directional derivative value, which identifies the swiftest route to the function's summit, similar to the steepest slope on our imagined hill.