Problem 294 - Calculus Volume 3

Question

For the following exercises, find the derivative of the function. \(f(x, y)=x^{2}+x y+y^{2}\) at point (-5,-4) in the direction the function increases most rapidly.

Step-by-Step Solution

Answer

The rate of increase is \( \sqrt{365} \).

Step 1: Find the Gradient

The rate at which a function increases most rapidly is given by its gradient. The gradient \( abla f(x, y) \) for a function \( f(x, y) = x^2 + xy + y^2 \) is found by calculating the partial derivatives with respect to \( x \) and \( y \). The partial derivative with respect to \( x \) is:\[ \frac{\partial f}{\partial x} = \frac{d}{dx}(x^2 + xy + y^2) = 2x + y \] The partial derivative with respect to \( y \) is:\[ \frac{\partial f}{\partial y} = \frac{d}{dy}(x^2 + xy + y^2) = x + 2y \] Thus, the gradient is:\[ abla f(x, y) = (2x + y, x + 2y) \]

Step 2: Evaluate the Gradient at the Given Point

Substitute the point \((-5, -4)\) into the gradient \( abla f(x, y) \) to find the direction of maximum increase at that point.\[ abla f(-5, -4) = (2(-5) + (-4), (-5) + 2(-4)) \] Calculate each component:For the \( x \) component:\[ 2(-5) + (-4) = -10 - 4 = -14 \]For the \( y \) component:\[ (-5) + 2(-4) = -5 - 8 = -13 \]Thus, the gradient at point \((-5, -4)\) is:\[ abla f(-5, -4) = (-14, -13) \]

Step 3: Calculate the Magnitude of the Gradient

The rate of increase of the function in the direction of maximum increase is the magnitude of the gradient. Calculate the magnitude of \( abla f(-5, -4) = (-14, -13) \).The magnitude is given by:\[ \| abla f(-5, -4) \| = \sqrt{(-14)^2 + (-13)^2} \]Calculating further:\[ (-14)^2 = 196, \quad (-13)^2 = 169 \]Add these values:\[ 196 + 169 = 365 \]So, the magnitude is:\[ \| abla f(-5, -4) \| = \sqrt{365} \]

Step 4: Conclusion

The rate at which the function \( f(x, y) = x^2 + xy + y^2 \) increases most rapidly at the point \((-5, -4)\) is given by the magnitude of the gradient, which is \( \sqrt{365} \).

Key Concepts

Gradient

In multivariable calculus, the gradient is a vital tool representing the direction and rate of the steepest increase of a function. It's essentially a vector composed of partial derivatives. For a function with multiple variables, such as two variables \(x\) and \(y\), the gradient \( abla f(x, y) \) is a vector that points in the direction where the function increases the fastest.

To find the gradient of a function, we compute the partial derivatives with respect to each variable involved. Consider a function \(f(x, y) = x^2 + xy + y^2\). The gradient is:

• The partial derivative with respect to \(x\) gives \(2x + y\).
• The partial derivative with respect to \(y\) gives \(x + 2y\).

Thus, the gradient \( abla f(x, y) \) is \((2x + y, x + 2y)\). This result is crucial as it indicates the direction to move from any given point in order to increase the function's value most quickly. Calculating the gradient at a specific point, like \((-5, -4)\), determines the precise direction of steepest ascent at that location.

Partial Derivatives

Partial derivatives are fundamental in understanding functions of several variables. They provide insight into how a function changes as one of the variables changes, while keeping the other variables constant. This is particularly useful when dealing with real-world phenomena where multiple factors influence the outcome.

Imagine a function \( f(x, y) = x^2 + xy + y^2 \). To find how this function changes with respect to \(x\), while \(y\) remains constant, we compute the partial derivative with respect to \(x\). The result, \(2x + y\), shows the change in the function for a small increment in \(x\).

Similarly, the partial derivative with respect to \(y\), \(x + 2y\), shows the effect of changing \(y\) while holding \(x\) constant. These derivatives are building blocks of the gradient, providing a complete picture of how the function behaves locally. Partial derivatives allow us to analyze each variable's individual impact on the function's output, making them indispensable for deeper insights into complex multivariable problems.

Rate of Increase

The rate at which a function increases at a specific point in its domain tells us how rapidly the function's value grows. In the context of multivariable functions, the gradient's magnitude is directly related to this rate of increase.

Once we have the gradient, its length or magnitude is computed to understand the steepness of the maximum ascent at a given point. For example, if \( abla f(-5, -4) = (-14, -13) \), calculating its magnitude involves:

• Squaring each component: \((-14)^2\) gives 196 and \((-13)^2\) gives 169.
• Adding these squares: \(196 + 169 = 365\).
• Finally, taking the square root: \(\sqrt{365}\).

The result, \(\sqrt{365}\), is the maximum rate of increase of the function at point \((-5, -4)\). It quantitatively describes how fast the function climbs in the direction of the gradient, making it a vital aspect of optimization and analysis in multivariable calculus.