Problem 80 - Calculus Early Transcendentals

Question

Prove that for the plane described by \(f(x, y)=A x+B y,\) where \(A\) and \(B\) are nonzero constants, the gradient is constant (independent of \((x, y)\) ). Interpret this result.

Step-by-Step Solution

Answer

Question: Prove that the gradient of the plane \(f(x, y) = Ax + By\) is constant and independent of \((x, y)\). Answer: To prove this, we computed the partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\), which are \(A\) and \(B\), respectively. Both partial derivatives do not depend on \((x, y)\). The gradient of the function \(f(x, y)\) is given by \(\nabla f = \langle A, B \rangle\), which is also independent of \((x, y)\). Therefore, the gradient of the plane \(f(x, y) = Ax + By\) is constant and independent of \((x, y)\).

Step 1: Calculate the partial derivatives of \(f(x, y)\)

To find the gradient of the function \(f(x, y) = Ax + By\), we need to compute the partial derivatives with respect to \(x\) and \(y\). The partial derivative of \(f\) with respect to \(x\) is denoted as \(\frac{\partial f}{\partial x}\), and the partial derivative of \(f\) with respect to \(y\) is denoted as \(\frac{\partial f}{\partial y}\). Let's compute these partial derivatives: $$\frac{\partial f}{\partial x} = \frac{\partial (Ax + By)}{\partial x} = A$$ $$\frac{\partial f}{\partial y} = \frac{\partial (Ax + By)}{\partial y} = B$$

Step 2: Check if the partial derivatives depend on \((x, y)\)

Now that we have computed the partial derivatives of \(f(x, y)\), we can check if they depend on the variables \((x, y)\). Looking at the expressions above, we can see that the partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) are constants (\(A\) and \(B\), respectively) that do not depend on \((x, y)\).

Step 3: Calculate the gradient of \(f(x, y)\)

The gradient of a function \(f(x, y)\) is a vector that combines the partial derivatives. The gradient is denoted as \(\nabla f\), and is given by the following formula: $$\nabla f = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \rangle$$ Using the partial derivatives we calculated in Step 1, the gradient of the function \(f(x, y) = Ax + By\) is: $$\nabla f = \langle A, B \rangle$$

Step 4: Interpret the result

Since the gradient of the function \(f(x, y) = Ax + By\) is given by \(\nabla f = \langle A, B \rangle\), which does not depend on \((x, y)\), we have proven that the gradient of this plane is constant and independent of \((x, y)\). This result tells us that the slope of the plane remains the same regardless of the point \((x, y)\) on the plane. In other words, the plane has a constant slope in all directions.