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Chapter 13: Problem 15

Computing gradients Compute the gradient of the following functions and evaluate it at the given point \(P\). $$F(x, y)=e^{-x^{2}-2 y^{2}} ; P(-1,2)$$

Short Answer

Expert verified
Answer: The gradient of the function at point \(P(-1,2)\) is $$\nabla F(-1, 2) = \left\langle 2e^{-17}, -8e^{-17} \right\rangle$$.

Step by step solution

01

Compute the partial derivative with respect to 'x'

To find the partial derivative of the function \(F(x, y)=e^{-x^{2}-2 y^{2}}\) with respect to 'x', treat 'y' as a constant and differentiate the function with respect to 'x'. Using the chain rule, we get: $$\frac{\partial F}{\partial x} = e^{-x^{2}-2y^{2}} \cdot (-2x) = -2xe^{-x^{2}-2y^{2}}$$
02

Compute the partial derivative with respect to 'y'

Similarly, to find the partial derivative of the function \(F(x, y)=e^{-x^{2}-2 y^{2}}\) with respect to 'y', treat 'x' as a constant and differentiate the function with respect to 'y'. Again, using the chain rule, we get: $$\frac{\partial F}{\partial y} = e^{-x^{2}-2y^{2}} \cdot (-4y) = -4ye^{-x^{2}-2y^{2}}$$
03

Write the gradient vector

Now that we have both partial derivatives, we can write the gradient vector by combining these two derivatives as components of the vector: $$\nabla F(x, y) = \left\langle \frac{\partial F}{\partial x}, \frac{\partial F}{\partial y} \right\rangle = \left\langle -2xe^{-x^{2}-2y^{2}}, -4ye^{-x^{2}-2y^{2}} \right\rangle$$
04

Evaluate the gradient vector at the given point

Now, we will evaluate the gradient vector at the given point \(P(-1, 2)\). Substitute the coordinates of the point into the gradient vector: $$\nabla F(-1, 2) = \left\langle -2(-1)e^{-(-1)^{2}-2(2)^{2}}, -4(2)e^{-(-1)^{2}-2(2)^{2}} \right\rangle$$ Evaluate the exponential function and simplify the expression: $$\nabla F(-1, 2) = \left\langle 2e^{-17}, -8e^{-17} \right\rangle$$ So, the gradient at point P is: $$\nabla F(-1, 2) = \left\langle 2e^{-17}, -8e^{-17} \right\rangle$$.

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Most popular questions from this chapter

Suppose \(n\) houses are located at the distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right) .\) A power substation must be located at a point such that the sum of the squares of the distances between the houses and the substation is minimized. a. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at \((0,0),(2,0),\) and (1,1) b. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at distinct points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) c. Find the optimal location of the substation in the general case of \(n\) houses located at distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots\) \(\left(x_{n}, y_{n}\right)\) d. You might argue that the locations found in parts (a), (b), and (c) are not optimal because they result from minimizing the sum of the squares of the distances, not the sum of the distances themselves. Use the locations in part (a) and write the function that gives the sum of the distances. Note that minimizing this function is much more difficult than in part (a). Then use a graphing utility to determine whether the optimal location is the same in the two cases. (Also see Exercise 75 about Steiner's problem.)

Let \(w=f(x, y, z)=2 x+3 y+4 z\), which is defined for all \((x, y, z)\) in \(\mathbb{R}^{3}\). Suppose that we are interested in the partial derivative \(w_{x}\) on a subset of \(\mathbb{R}^{3}\), such as the plane \(P\) given by \(z=4 x-2 y .\) The point to be made is that the result is not unique unless we specify which variables are considered independent. a. We could proceed as follows. On the plane \(P\), consider \(x\) and \(y\) as the independent variables, which means \(z\) depends on \(x\) and \(y,\) so we write \(w=f(x, y, z(x, y)) .\) Differentiate with respect to \(x\) holding \(y\) fixed to show that \(\left(\frac{\partial w}{\partial x}\right)_{y}=18,\) where the subscript \(y\) indicates that \(y\) is held fixed. b. Alternatively, on the plane \(P,\) we could consider \(x\) and \(z\) as the independent variables, which means \(y\) depends on \(x\) and \(z,\) so we write \(w=f(x, y(x, z), z)\) and differentiate with respect to \(x\) holding \(z\) fixed. Show that \(\left(\frac{\partial w}{\partial x}\right)_{z}=8,\) where the subscript \(z\) indicates that \(z\) is held fixed. c. Make a sketch of the plane \(z=4 x-2 y\) and interpret the results of parts (a) and (b) geometrically. d. Repeat the arguments of parts (a) and (b) to find \(\left(\frac{\partial w}{\partial y}\right)_{x}\), \(\left(\frac{\partial w}{\partial y}\right)_{z},\left(\frac{\partial w}{\partial z}\right)_{x},\) and \(\left(\frac{\partial w}{\partial z}\right)_{y}\).

Problems with two constraints Given a differentiable function \(w=f(x, y, z),\) the goal is to find its maximum and minimum values subject to the constraints \(g(x, y, z)=0\) and \(h(x, y, z)=0\) where \(g\) and \(h\) are also differentiable. a. Imagine a level surface of the function \(f\) and the constraint surfaces \(g(x, y, z)=0\) and \(h(x, y, z)=0 .\) Note that \(g\) and \(h\) intersect (in general) in a curve \(C\) on which maximum and minimum values of \(f\) must be found. Explain why \(\nabla g\) and \(\nabla h\) are orthogonal to their respective surfaces. b. Explain why \(\nabla f\) lies in the plane formed by \(\nabla g\) and \(\nabla h\) at a point of \(C\) where \(f\) has a maximum or minimum value. c. Explain why part (b) implies that \(\nabla f=\lambda \nabla g+\mu \nabla h\) at a point of \(C\) where \(f\) has a maximum or minimum value, where \(\lambda\) and \(\mu\) (the Lagrange multipliers) are real numbers. d. Conclude from part (c) that the equations that must be solved for maximum or minimum values of \(f\) subject to two constraints are \(\nabla f=\lambda \nabla g+\mu \nabla h, g(x, y, z)=0,\) and \(h(x, y, z)=0\)

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(0,0)} \frac{|x-y|}{|x+y|}$$

Temperature of an elliptical plate The temperature of points on an elliptical plate \(x^{2}+y^{2}+x y \leq 1\) is given by \(T(x,y)=25\left(x^{2}+y^{2}\right) .\) Find the hottest and coldest temperatures on the edge of the elliptical plate.
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