Problem 39 - Calculus Early Transcendentals

Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Suppose you are standing at the center of a sphere looking at a point \(P\) on the surface of the sphere. Your line of sight to \(P\) is orthogonal to the plane tangent to the sphere at \(P\). b. At a point that maximizes \(f\) on the curve \(g(x, y)=0,\) the dot product \(\nabla f \cdot \nabla g\) is zero.

Step-by-Step Solution

Answer

Question: True or False, explain your answer or provide a counterexample: When standing at the center of a sphere and looking at a point P on the surface of the sphere, the line of sight to P is orthogonal to the plane tangent to the sphere at P. Moreover, at a point that maximizes a function f on the curve g(x, y) = 0, the dot product between the gradients ∇f and ∇g is zero. Answer: True. Both statements hold true. The line of sight to P on a sphere's surface is orthogonal or perpendicular to the tangent plane at P. For the constrained optimization problem, the dot product between the gradients ∇f and ∇g can indeed be zero at a point that maximizes the function f on the curve g(x, y) = 0.

Step 1: Visualize the sphere and point P

Imagine standing at the center of a sphere and looking at a point P on the surface of the sphere. The line that connects the center of the sphere to this point P forms a radius of the sphere. The plane tangent to the sphere at point P is a flat plane that touches the sphere only at that point.

Step 2: Observe the properties of the tangent plane

Now, visualize the line of sight from the center of the sphere to point P and the tangent plane at P. Since this plane is tangent to the sphere at P, it means it will not intersect the sphere anywhere else. The radius connecting the center and point P is perpendicular to the tangent plane, being the shortest distance from the center to the tangent plane.

Step 3: Conclude the statement's validity

Thus, we can conclude that the statement is true: When standing at the center of a sphere and looking at a point P on the surface of the sphere, the line of sight to P is orthogonal (perpendicular) to the plane tangent to the sphere at P. #b. Analyzing the dot product of gradient vectors in constrained optimization#

Step 1: Understand the given conditions

The conditions state that we have a function f(x, y) and a constraint function g(x, y) = 0. The point in question maximizes f(x, y) on the curve given by the constraint function.

Step 2: Relate the gradient vectors to the constrained maximum

When a point maximizes or minimizes a function with a constraint, there exists a relationship between the gradient vectors of the function and the constraint. Specifically, the gradient vectors will be proportional to each other, i.e., \(\nabla f = \lambda \nabla g\), where \(\lambda\) is a constant called the Lagrange Multiplier.

Step 3: Apply the dot product property

Now, since \(\nabla f = \lambda \nabla g\), taking the dot product between \(\nabla f\) and \(\nabla g\) gives: \((\nabla f) \cdot (\nabla g) = (\lambda \nabla g) \cdot (\nabla g)\). Now, since the dot product is commutative and associative, we can write \((\lambda \nabla g) \cdot (\nabla g) = \lambda(\nabla g) \cdot (\nabla g)\).

Step 4: Observe the results of the dot product

Since \(\lambda\) is a constant, its value will not change the fact that the dot product of the gradient vectors will either be positive, negative, or zero. Therefore, based on the geometry of the curve and the position of the constraint, it is possible that the dot product \(\nabla f \cdot \nabla g\) is zero at a point that maximizes f on the curve g(x, y) = 0.

Step 5: Conclude the statement's validity

Hence, we can conclude that the statement is true: At a point that maximizes f on the curve g(x, y) = 0, the dot product \(\nabla f \cdot \nabla g\) is indeed zero.

Key Concepts

Gradient Vectors

In calculus, gradients are fundamental as they give us the direction of steepest ascent of a function. The gradient vector, denoted as \(abla f\) for a function \(f(x, y)\), contains the partial derivatives with respect to every variable: \((\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})\).
This vector is perpendicular to the level curves of the function, which are the curves along which the function has a constant value.
When dealing with a constrained optimization problem, such as maximizing \(f(x, y)\) subject to \(g(x, y) = 0\), the gradient plays a vital role. Here, the gradients of the function \(abla f\) and the constraint \(abla g\) are related through the Lagrange multiplier \(\lambda\) by the equation \(abla f = \lambda abla g\). This relation is key in finding points of optimization on the constraint curve or surface.
This means at the point of optimization, the direction of change in \(f\) and the constraint are not independent, but are aligned in a proportional manner. This provides a perfect geometric insight to the constraint optimization approach using Lagrange multipliers.

Tangent Planes

A tangent plane is an essential concept in multivariable calculus, especially when visualizing surfaces. Imagine a small sphere, like a globe, and the infinite flat plane that just lightly "kisses" it at one point: that is the tangent plane.
When looking at a sphere, such a tangent plane at a point \(P\) will only touch the sphere at \(P\) without cutting across it. The shortest line connecting the center of the sphere to this plane is the sphere's radius, thus it is perpendicular or orthogonal to the tangent plane.If you visualize this, it makes sense why, from the center of the sphere, your line of sight to point \(P\) is orthogonal to its tangent plane.
Tangent planes in more general terms help us approximate the surface locally with these "flatter" geometric surfaces. They allow us to generate linear approximations for complex-shaped surfaces, thus simplifying many complex calculus operations.

Dot Product

The dot product is a fundamental operation in vector calculus and physics. It helps us determine the alignment or interaction between two vectors. For two vectors \(\mathbf{a} = (a_1, a_2)\) and \(\mathbf{b} = (b_1, b_2)\), the dot product is calculated as: \[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2. \]
The result is a scalar value and can tell you how "aligned" the vectors are. Typically,

• If the dot product is zero, the vectors are perpendicular (or orthogonal).
• If it's positive, they form an acute angle.
• If negative, they form an obtuse angle.

In our context of constrained optimization using Lagrange multipliers, the dot product \(abla f \cdot abla g = 0\) at a maximal or minimal point indicates that these gradient vectors are orthogonal. This orthogonality is significant as it reflects the precise angle relationship necessary for the conditions of optimization to be satisfied.