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Chapter 13: Problem 32

In what directions is the derivative of \(f(x, y)=\left(x^{2}-y^{2}\right) /\left(x^{2}+y^{2}\right)\) at \(P(1,1)\) equal to zero?

Short Answer

Expert verified
The derivative is zero in the directions \((1, 1)\) and \((-1, -1)\).

Step by step solution

01

Compute Partial Derivatives

First, we need to compute the partial derivatives of the given function \(f(x, y)\). Use quotient rule for \(f(x, y) = \frac{x^2 - y^2}{x^2 + y^2}\).The partial derivative with respect to \(x\) is:\( f_x(x, y) = \frac{(2x)(x^2 + y^2) - (x^2 - y^2)(2x)}{(x^2 + y^2)^2} = \frac{2x(x^2 + y^2) - 2x(x^2 - y^2)}{(x^2 + y^2)^2} \).Simplifying gives:\( f_x(x, y) = \frac{4xy^2}{(x^2 + y^2)^2} \).Similarly, the partial derivative with respect to \(y\) is:\( f_y(x, y) = \frac{-2y(x^2 + y^2) - (x^2 - y^2)(2y)}{(x^2 + y^2)^2} \).Simplifying gives:\( f_y(x, y) = \frac{-4yx^2}{(x^2 + y^2)^2} \).
02

Evaluate Partial Derivatives at P(1,1)

Substitute \(P(1,1)\) into the partial derivatives:- \(f_x(1, 1) = \frac{4(1)(1)^2}{(1^2 + 1^2)^2} = \frac{4}{4} = 1\)- \(f_y(1, 1) = \frac{-4(1)^2(1)}{(1^2 + 1^2)^2} = \frac{-4}{4} = -1\)
03

Use the Gradient Vector

The gradient \(abla f(x, y)\) at a point is \((f_x(x, y), f_y(x, y))\). Thus, at \(P(1, 1)\), the gradient vector is \((1, -1)\).
04

Determine Directions of Zero Derivative

The directional derivative in the direction of \(\mathbf{v} = (a, b)\) is \( (a, b) \cdot abla f(x, y) = 0\).We have:\[ a \cdot 1 + b \cdot (-1) = 0 \]This simplifies to \(a = b\).
05

Parameterize Direction Vector

Since \(a = b\), choose \(\mathbf{v} = (a, a)\). Normalize \(\mathbf{v}\) to make it a unit vector:\[ \frac{1}{\sqrt{2}}(1, 1) \text{ or } \frac{1}{\sqrt{2}}(-1, -1) \].Thus, the directions are along \((1, 1)\) and \((-1, -1)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Derivatives
Partial derivatives are like regular derivatives, but they are used in functions with more than one variable. They show how a function changes as each variable changes individually, while keeping the other variables constant. This is particularly useful in multivariable calculus because it lets you look at the effect of just one variable on the function at a time.
  • To find the partial derivative with respect to a given variable, treat the other variables as constants.
  • It's similar to taking the regular derivative of a single-variable function.
  • Notation includes using a subscript, like \(f_x\), to denote the partial derivative with respect to \(x\).
In our exercise, the function \(f(x, y) = \frac{x^2 - y^2}{x^2 + y^2}\) involves two variables, \(x\) and \(y\). The partial derivatives \(f_x(x, y)\) and \(f_y(x, y)\) were calculated using the Quotient Rule to handle the ratio in the function.
Gradient Vector
The gradient vector of a multivariable function is a vector that points in the direction of the greatest rate of increase of the function. It is like a map that shows the direction of the steepest hill you might climb.
  • The gradient is represented by \(abla f(x, y)\) for a function \(f(x, y)\).
  • It is composed of partial derivatives, specifically \(f_x\) and \(f_y\) for functions of two variables.
  • In vector notation, it is expressed as \(abla f = (f_x, f_y)\).
In our problem, the gradient at point \(P(1,1)\) was calculated to be \( (1, -1)\). This means the function increases most steeply in the direction of this vector.
Quotient Rule
When you have a function that is the ratio of two other functions, the Quotient Rule is a handy tool to find its derivative. It describes how to take the derivative of a quotient of two functions, which is more complex than simple addition or multiplication of functions.
  • The Quotient Rule states that if you have a function \(g(x)/h(x)\), its derivative is \[\frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}\].\
  • It's like the product rule for multiplication, but with a subtraction.
  • The denominator is squared in the final result, so it's important to ensure it doesn't become zero in the domain of the function.
In the exercise, applying the Quotient Rule allowed us to carefully handle the partial derivatives \(f_x(x, y)\) and \(f_y(x, y)\) for the given function \(f(x, y) = \frac{x^2 - y^2}{x^2 + y^2}\), taking into account the numerator and denominator separately during differentiation.

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Most popular questions from this chapter

Find the point closest to the origin on the line of intersection of the planes \(y+2 z=12\) and \(x+y=6\)

Just as you describe curves in the plane parametrically with a pair of equations \(x=f(t), y=g(t)\) defined on some parameter interval \(I\), you can sometimes describe surfaces in space with a triple of equations \(x=f(u, v), y=g(u, v), z=h(u, v)\) defined on some parameter rectangle \(a \leq u \leq b, c \leq v \leq d .\) Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section 16.5.) Use a CAS to plot the surfaces in Exercises \(77-80 .\) Also plot several level curves in the \(x y\) -plane. $$\begin{aligned} &x=u \cos v, \quad y=u \sin v, \quad z=v, \quad 0 \leq u \leq 2\\\ &0 \leq v \leq 2 \pi \end{aligned}$$

The condition \(\nabla \boldsymbol{f}=\lambda \nabla g\) is not sufficient Although \(\nabla f=\lambda \nabla g\) is a necessary condition for the occurrence of an extreme value of \(f(x, y)\) subject to the conditions \(g(x, y)=0\) and \(\nabla g \neq \mathbf{0},\) it does not in itself guarantee that one exists. As a case in point, try using the method of Lagrange multipliers to find a maximum value of \(f(x, y)=x+y\) subject to the constraint that \(x y=16 .\) The method will identify the two points (4,4) and (-4,-4) as candidates for the location of extreme values. Yet the \(\operatorname{sum}(x+y)\) has no maximum value on the hyperbola \(x y=16\) The farther you go from the origin on this hyperbola in the first quadrant, the larger the sum \(f(x, y)=x+y\) becomes.

During the 1920 s, Charles Cobb and Paul Douglas modeled total production output \(P\) (of a firm, industry, or entire economy) as a function of labor hours involved \(x\) and capital invested \(y\) (which includes the monetary worth of all buildings and equipment). The Cobb-Douglas production function is given by $$P(x, y)=k x^{\alpha} y^{1-\alpha}$$ where \(k\) and \(\alpha\) are constants representative of a particular firm or economy. a. Show that a doubling of both labor and capital results in a doubling of production \(P\) b. Suppose a particular firm has the production function for \(k=\) 120 and \(\alpha=3 / 4 .\) Assume that each unit of labor costs $$ 250$ and each unit of capital costs $$ 400, and that the total expenses for all costs cannot exceed $$ 100,000 . Find the maximum production level for the firm.

Use a CAS to plot the implicitly defined level surfaces. $$\sin \left(\frac{x}{2}\right)-(\cos y) \sqrt{x^{2}+z^{2}}=2$$
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