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Chapter 12: Problem 3

Interpret the direction of the gradient vector at a point.

Short Answer

Expert verified
Answer: The direction of the gradient vector at a given point indicates the direction of the steepest ascent or maximum rate of increase for the function at that point. It represents the direction in the \(x\) and \(y\) plane where the function increases most rapidly.

Step by step solution

01

Understand what a gradient vector is

The gradient vector is a vector normal to the level curves or surface of a given continuous scalar function and points in the direction of the maximum rate of increase. The gradient vector is denoted as \(\nabla f(x,y)\), or sometimes as \(\text{grad}\,f(x,y)\). If the given scalar function is \(f(x,y)\), its gradient vector is: $$ \nabla f(x,y) = \frac{\partial f(x,y)}{\partial x}\mathbf{i} + \frac{\partial f(x,y)}{\partial y}\mathbf{j} $$ Here, \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the \(x\) and \(y\) directions, and the partial derivatives with respect to \(x\) and \(y\) represent the rate of change of the function in those directions.
02

Derive the gradient vector for a given function

Consider a scalar function \(f(x, y)\). To find the gradient vector, we need to determine: 1. The partial derivative of \(f\) with respect to \(x\), which we denote as \(\frac{\partial f}{\partial x}\), and 2. The partial derivative of \(f\) with respect to \(y\), which we denote as \(\frac{\partial f}{\partial y}\). Once we find these two derivatives, we can determine the gradient vector of the function by using the above formula.
03

Evaluate the gradient vector at a specified point

Given a point \((x_0, y_0)\), evaluate the gradient vector at this point by substituting the \(x\) and \(y\) coordinates into the gradient vector expression: $$ \nabla f(x_0, y_0) = \frac{\partial f(x_0,y_0)}{\partial x}\mathbf{i} + \frac{\partial f(x_0,y_0)}{\partial y}\mathbf{j} $$
04

Interpret the direction of the gradient vector

The direction of the gradient vector will be in the direction of the steepest ascent or maximum rate of increase at the given point \((x_0, y_0)\). The magnitude of the gradient vector indicates the rate of change while its direction shows the way in the \(x\) and \(y\) plane wherein the function increases the most rapidly. To interpret the direction of the gradient vector, we can look at the components of the vector. If the gradient vector points predominantly in the direction of the \(x\)-axis, it means the maximum rate of increase is along the \(x\)-axis. Similarly, if the gradient vector points predominantly in the direction of the \(y\)-axis, it means the maximum rate of increase is along the \(y\)-axis.

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Most popular questions from this chapter

Find the absolute maximum and minimum values of the following functions over the given regions \(R\). Use Lagrange multipliers to check for extreme points on the boundary. $$f(x, y)=x^{2}-4 y^{2}+x y ; R=\left\\{(x, y): 4 x^{2}+9 y^{2} \leq 36\right\\}$$

Extending Exercise \(62,\) when three electrical resistors with resistance \(R_{1}>0, R_{2}>0,\) and \(R_{3}>0\) are wired in parallel in a circuit (see figure), the combined resistance \(R,\) measured in ohms \((\Omega),\) is given by \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\) Estimate the change in \(R\) if \(R_{1}\) increases from \(2 \Omega\) to \(2.05 \Omega, R_{2}\) decreases from \(3 \Omega\) to \(2.95 \Omega,\) and \(R_{3}\) increases from \(1.5 \Omega\) to \(1.55 \Omega.\)

Suppose \(P\) is a point in the plane \(a x+b y+c z=d .\) Then the least distance from any point \(Q\) to the plane equals the length of the orthogonal projection of \(\overrightarrow{P Q}\) onto the normal vector \(\mathbf{n}=\langle a, b, c\rangle\) a. Use this information to show that the least distance from \(Q\) to the plane is \(\frac{|\overrightarrow{P Q} \cdot \mathbf{n}|}{|\mathbf{n}|}\) b. Find the least distance from the point (1,2,-4) to the plane \(2 x-y+3 z=1\)

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