Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

Give definitions in terms of limits for the following partial derivatives: (a) \(f_{y}(x, y, z)\) (b) \(f_{z}(x, y, z)\) (c) \(G_{x}(w, x, y, z)\) (d) \(\frac{\partial}{\partial z} \lambda(x, y, z, t)\) (e) \(\frac{\partial}{\partial b_{2}} S\left(b_{0}, b_{1}, b_{2}, \ldots, b_{n}\right)\)

Step-by-Step Solution

Answer

Partial derivatives are defined using limits, focusing on the variable of interest while keeping others constant.

Step 1: Definition of Partial Derivative in terms of Limit for (a)

The partial derivative of a function \( f(x, y, z) \) with respect to \( y \), denoted as \( f_{y}(x, y, z) \), is defined as the limit of the difference quotient as \( h \) approaches 0:\[ f_{y}(x, y, z) = \lim_{{h \to 0}} \frac{f(x, y + h, z) - f(x, y, z)}{h} \] This represents how \( f \) changes as \( y \) changes while keeping \( x \) and \( z \) constant.

Step 2: Definition of Partial Derivative in terms of Limit for (b)

For the partial derivative of \( f(x, y, z) \) with respect to \( z \), denoted as \( f_{z}(x, y, z) \), it is defined in terms of limits as follows:\[ f_{z}(x, y, z) = \lim_{{h \to 0}} \frac{f(x, y, z + h) - f(x, y, z)}{h} \] This expression measures the rate of change of \( f \) in the \( z \) direction while \( x \) and \( y \) are held constant.

Step 3: Definition of Partial Derivative in terms of Limit for (c)

Consider the partial derivative of \( G(w, x, y, z) \) with respect to \( x \), denoted \( G_{x}(w, x, y, z) \). The definition using limits is:\[ G_{x}(w, x, y, z) = \lim_{{h \to 0}} \frac{G(w, x + h, y, z) - G(w, x, y, z)}{h} \] Thus, it quantifies how \( G \) changes as \( x \) changes while other variables \( w, y, \) and \( z \) are constant.

Step 4: Definition of Partial Derivative in terms of Limit for (d)

For the derivative \( \frac{\partial}{\partial z} \lambda(x, y, z, t) \), the partial derivative is defined using limits as follows:\[ \frac{\partial}{\partial z} \lambda(x, y, z, t) = \lim_{{h \to 0}} \frac{\lambda(x, y, z + h, t) - \lambda(x, y, z, t)}{h} \] This derivative shows the change in \( \lambda \) with respect to \( z \), with \( x, y, t \) constant.

Step 5: Definition of Partial Derivative in terms of Limit for (e)

The partial derivative with respect to \( b_{2} \) of the function \( S(b_{0}, b_{1}, b_{2}, \ldots, b_{n}) \) is defined using limits as:\[ \frac{\partial}{\partial b_{2}} S\left(b_{0}, b_{1}, b_{2}, \ldots, b_{n}\right) = \lim_{{h \to 0}} \frac{S(b_{0}, b_{1}, b_{2} + h, \ldots, b_{n}) - S(b_{0}, b_{1}, b_{2}, \ldots, b_{n})}{h} \] This expression finds how \( S \) changes in the direction of \( b_{2} \) with other \( b's \) constant.

Key Concepts

Limits

The concept of limits is crucial in calculus, especially when dealing with derivatives. Essentially, a limit describes the behavior of a function as the input approaches a certain point. This is not only about reaching that point but understanding how the function behaves around it.

When you see a derivative defined in terms of limits, like \[ \lim_{{h \to 0}} \frac{f(x + h, y, z) - f(x, y, z)}{h} \] this expression is telling you how the function changes as we make tiny changes in one specific direction (by altering one variable while keeping others constant).

Limits are fundamental in defining other advanced concepts, such as continuity and differentiability, which help describe the smoothness and behavior of functions both in 2D and 3D spaces. By understanding limits, you set the foundation for all of calculus.

Functions of Several Variables

In multivariable calculus, we often deal with functions that depend on more than one variable. Consider a function described as \( f(x, y, z) \).

This means the output of the function depends on the values chosen for each of \( x, y, \) and \( z \). Imagine the surface of a mountain where the height depends on both longitude and latitude.

To better understand these functions, we can explore their behavior in each variable independently by using partial derivatives. Partial derivatives examine the rate of change of the function with respect to one variable at a time, which gives us invaluable insights into how each variable influences the overall function.

• For example, \( f_x \) represents how the function changes as only the \( x \) variable changes.
• Similarly, \( f_y \) and \( f_z \) show changes concerning \( y \) and \( z \) respectively.

Grasping this concept lets us model complex real-world phenomena.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to functions with more than one variable. It allows us to explore how these functions behave in multi-dimensional spaces. Think of it as expanding our view from lines and curves to surfaces and volumes.

The core ideas include differentiation and integration of functions that may depend on two or more variables, such as \( f(x, y) \) or \( f(x, y, z) \).

• Partial derivatives and multiple integrals are key tools in this domain.
• They help in analyzing rates of change and calculating areas or volumes under surfaces.

By understanding multivariable calculus, we can study and optimize systems characterized by multiple influencing factors across diverse fields—physics, engineering, economics, and beyond.

Directional Derivative

While partial derivatives provide insight into a function's behavior along a specific axis, directional derivatives let us determine the rate of change of a function in any given direction.

The directional derivative in a direction given by a vector \( \mathbf{u} \) is defined as:
\[ D_{\mathbf{u}}f(x, y) = abla f(x, y) \cdot \mathbf{u} \]
where \( abla f(x, y) \) is the gradient vector, which includes all partial derivatives, representing the direction and rate of the steepest ascent.

• It effectively tells us how the function changes as we move along a vector \( \mathbf{u} \) from a point.
• Understanding directional derivatives is crucial when optimizing functions or finding maxima and minima within multivariable functions.

This concept proves particularly useful in scenarios where movement along arbitrary directions needs to be evaluated, like in physics and computational simulations.