Question

Change on a line Suppose \(w=f(x, y, z)\) and \(\ell\) is the line \(\mathbf{r}(t)=\langle a t, b t, c t\rangle,\) for \(-\infty< t <\infty\). a. Find \(\left.w^{\prime}(t) \text { on } \ell \text { (in terms of } a, b, c, w_{x}, w_{y}, \text { and } w_{z}\right)\) b. Apply part (a) to find \(w^{\prime}(t)\) when \(f(x, y, z)=x y z\) c. Apply part (a) to find \(w^{\prime}(t)\) when \(f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}\) d. For a general twice differentiable function \(w=f(x, y, z),\) find \(w^{\prime \prime}(t)\)

Short answer

In summary, we found the derivative and the second derivative of a function w=f(x, y, z) with respect to t along the given line by following these steps: 1. Find the derivatives of the components of the parametrized line with respect to t. 2. Apply the chain rule to find the first derivative of w with respect to t. 3. Apply the result to specific given functions. 4. Apply the chain rule again to find the second derivative of w with respect to t for general twice differentiable functions. The general expressions we derived for the first and second derivatives are: $$ w^{\prime}(t) = a w_{x} + b w_{y} + c w_{z} $$ $$ w^{\prime\prime}(t) = a^2 w_{xx} + abw_{xy} + acw_{xz} + abw_{yx} + b^2w_{yy} + bcw_{yz} + acw_{zx} + bcw_{zy} + c^2w_{zz} $$

Step-by-step solution

Find the derivatives of \(\mathbf{r}(t)\) with respect to \(t\)

To find \(w^{\prime}(t)\) along the line, we will need to find the derivatives of the components of \(\mathbf{r}(t)\) with respect to \(t\). So, we have: $$ \frac{d x}{d t}=a \\ \frac{d y}{d t}=b \\ \frac{d z}{d t}=c $$

Apply the chain rule to find \(w^{\prime}(t)\)

Now we apply the chain rule to find the derivative of \(w\) with respect to \(t\) along the line. The chain rule states that: $$ w^{\prime}(t) = \frac{\partial w}{\partial x}\frac{d x}{d t}+\frac{\partial w}{\partial y}\frac{d y}{d t}+\frac{\partial w}{\partial z}\frac{d z}{d t} $$ Plugging in the derivatives of \(x, y,\) and \(z\) with respect to \(t\), we get: $$ w^{\prime}(t) = a w_{x} + b w_{y} + c w_{z} $$

Apply the result to the given functions

Now we apply the above result to find \(w^{\prime}(t)\) for the specific functions given in parts (b) and (c). a) For \(f(x, y, z) = xyz\), we have: $$ w_{x} = yz \\ w_{y} = xz \\ w_{z} = xy $$ Plugging these into the expression for \(w^{\prime}(t)\), we get: $$ w^{\prime}(t) = a(yz) + b(xz) + c(xy) $$ b) For \(f(x, y, z) = \sqrt{x^2 + y^2 + z^2}\), we have: $$ w_{x} = \frac{x}{\sqrt{x^2 + y^2 + z^2}} \\ w_{y} = \frac{y}{\sqrt{x^2 + y^2 + z^2}} \\ w_{z} = \frac{z}{\sqrt{x^2 + y^2 + z^2}} $$ Plugging these into the expression for \(w^{\prime}(t)\), we get: $$ w^{\prime}(t) = a\frac{x}{\sqrt{x^2 + y^2 + z^2}} + b\frac{y}{\sqrt{x^2 + y^2 + z^2}} + c\frac{z}{\sqrt{x^2 + y^2 + z^2}} $$

Find the second derivative \(w^{\prime\prime}(t)\) for general twice differentiable functions

To find the second derivative \(w^{\prime\prime}(t)\), we need to apply the chain rule once again, taking the derivative of \(w^{\prime}(t)\) with respect to \(t\). We have: $$ w^{\prime\prime}(t) = \frac{d}{dt}\left(aw_{x} + bw_{y} + cw_{z}\right) $$ Applying the chain rule, we end up with the following expression for the second derivative of \(w\) with respect to \(t\): $$ w^{\prime\prime}(t) = a^2 w_{xx} + abw_{xy} + acw_{xz} + abw_{yx} + b^2w_{yy} + bcw_{yz} + acw_{zx} + bcw_{zy} + c^2w_{zz} $$

Key concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to find the derivative of composite functions. It is essential when dealing with functions of multiple variables, especially in scenarios like our exercise where we want to differentiate a function along a line.
To apply the chain rule, you need the derivatives of each component function with respect to an intermediary variable, typically denoted as 't' in parameterized curves.
Here's a breakdown of how it works:

• Let a function be described by several variables, each a function of another variable 't'.
• The chain rule states that the derivative of the function is the sum of products of partial derivatives with respect to each variable and the derivative of each variable with respect to 't'.

In this exercise, the chain rule is used to compute the derivative of the function with respect to a parameter 't' along a line. By calculating the derivatives of each positional component with respect to 't', and combining them with the partial derivatives of the function, we can find how the function changes along that path.

Directional Derivative

A directional derivative represents the rate at which a function changes as you move in a specified direction from a given point. It extends the concept of a derivative to functions of multiple variables, allowing us to understand how changes occur not just in one dimension, but in any direction in a multidimensional space.
To compute a directional derivative, the gradient of the function (a vector of partial derivatives) is used. This gradient is then dotted with the directional vector.

• The gradient, \(abla f\), indicates the direction of maximum increase of function \(f\).
• The directional derivative is computed by dotting \(abla f\) with a unit vector \(\mathbf{u}\) pointing in the desired direction.

In our problem, the line parameter \(\mathbf{r}(t)\) gives us a direction. Taking partial derivatives of the function \(f(x, y, z)\), and combining them with the direction components from \(\mathbf{r}(t)\), gives a directional look at the function's rate of change along the line \(\ell\).

Multivariable Calculus

Multivariable calculus is the field of mathematics that extends calculus to functions of several variables.

• Unlike single-variable calculus, where functions depend on only one variable, multivariable calculus considers functions that depend on two or more variables, such as \(f(x, y, z)\).
• It involves computing partial derivatives, gradients, and optimizing functions in higher-dimensional spaces.

In this context, our exercise is a practical application of multivariable calculus, where we study the behavior of functions along paths defined in multi-dimensional spaces. By using derivatives, we assess how functions transform as parameters change, which is crucial in various fields like physics, engineering, and economics. Multivariable calculus tools, such as the chain rule and gradient, are integral to understanding and manipulating these functions.

Second Derivative

The second derivative is an extension of the first derivative and provides information about the curvature or concavity of a function. In the context of functions of several variables, the second derivative tells us how the rate of change of a function's rate of change itself varies.

• In practical terms, the second derivative gives insight into acceleration if the context involves motion along a line.
• It helps determine concavity in optimization problems, crucial for identifying local maximums and minimums.

In our exercise, finding the second derivative \(w''(t)\) entailed differentiating \(w'(t)\) with respect to \(t\) again, using second-order partial derivatives. This process highlights how the influence of each coordinate direction affects the rate of change along our specified line, providing more in-depth analysis than the first derivative alone.