Problem 301 - Calculus Volume 3

Question

For the following exercises, find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. $$\quad f(x, y)=\cos (3 x+2 y),\left(\frac{\pi}{6},-\frac{\pi}{8}\right)$$

Step-by-Step Solution

Answer

The maximum rate of change is \( \sqrt{\frac{13}{2}} \), occurring in the direction \( \left(\frac{-3\sqrt{2}}{4}, -\frac{\sqrt{2}}{\sqrt{13}}\right) \).

Step 1: Find the Gradient of f

The gradient of a function \( f(x, y) = \cos(3x + 2y) \) is found by taking the partial derivatives with respect to \( x \) and \( y\). First, compute the partial derivative with respect to \( x \): \[ \frac{\partial f}{\partial x} = -3 \sin(3x + 2y) \]Next, compute the partial derivative with respect to \( y \): \[ \frac{\partial f}{\partial y} = -2 \sin(3x + 2y) \] The gradient of \( f \) is given by: \[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = \left(-3 \sin(3x + 2y), -2 \sin(3x + 2y)\right) \]

Step 2: Evaluate the Gradient at the Given Point

Substitute \( x = \frac{\pi}{6} \) and \( y = -\frac{\pi}{8} \) into the gradient:\[ abla f\left(\frac{\pi}{6}, -\frac{\pi}{8}\right) = \left(-3 \sin\left(3\frac{\pi}{6} + 2\left(-\frac{\pi}{8}\right)\right), -2 \sin\left(3\frac{\pi}{6} + 2\left(-\frac{\pi}{8}\right)\right)\right) \]Simplify the terms inside the sine function:\[ 3\frac{\pi}{6} + 2(-\frac{\pi}{8}) = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} \]Thus, \[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]Substitute back:\[ abla f\left(\frac{\pi}{6}, -\frac{\pi}{8}\right) = \left(-3 \frac{\sqrt{2}}{2}, -2 \frac{\sqrt{2}}{2}\right) = \left(-\frac{3\sqrt{2}}{2}, -\sqrt{2}\right) \]

Step 3: Find the Magnitude of the Gradient

The magnitude of the gradient vector represents the maximum rate of change. Calculate it using:\[ |abla f| = \sqrt{\left(-\frac{3\sqrt{2}}{2}\right)^2 + \left(-\sqrt{2}\right)^2} \]Squaring the components:\[ \left(-\frac{3\sqrt{2}}{2}\right)^2 = \frac{9 \cdot 2}{4} = \frac{18}{4} = \frac{9}{2} \]\[ \left(-\sqrt{2}\right)^2 = 2 \]Sum inside the square root:\[ \sqrt{\frac{9}{2} + 2} = \sqrt{\frac{9}{2} + \frac{4}{2}} = \sqrt{\frac{13}{2}} \]This gives:\[ |abla f| = \sqrt{\frac{13}{2}} \]

Step 4: Determine the Direction of Maximum Change

The maximum rate of change occurs in the direction of the gradient vector. Normalize it to get the unit direction vector. The gradient is \( \left(-\frac{3\sqrt{2}}{2}, -\sqrt{2}\right) \) and its magnitude is \( \sqrt{\frac{13}{2}} \). Thus, the unit vector is: \[ \frac{1}{\sqrt{\frac{13}{2}}} \left(-\frac{3\sqrt{2}}{2}, -\sqrt{2}\right) \]This simplifies directionwise to: \[ \left(\frac{-\frac{3\sqrt{2}}{2}}{\sqrt{\frac{13}{2}}}, \frac{-\sqrt{2}}{\sqrt{\frac{13}{2}}}\right) \]

Key Concepts

Rate of Change

Understanding the rate of change is fundamental when dealing with functions. The rate of change refers to how much a function's output changes with a change in input. For functions of multiple variables, like our example function \( f(x, y) = \cos(3x + 2y) \), the rate of change at a given point is best described by the gradient vector.

The magnitude of this gradient vector tells us the maximum rate of change. This simply means it tells us how steeply the function's value increases or decreases at that point. To find this maximum rate of change, we first calculate the gradient, which involves finding the partial derivatives with respect to each variable.

• The greater the magnitude of the gradient, the faster the values of the function are changing in that direction.

By evaluating this at the specific point \( \left(\frac{\pi}{6}, -\frac{\pi}{8}\right) \), we discover the rate at which our function is changing right there, revealing important insights into the behavior of the function around that location.

Partial Derivatives

Partial derivatives play a critical role in multivariable calculus. They measure the rate at which a function changes as we change one of the variables, while keeping the others constant.

For the function \( f(x, y) = \cos(3x + 2y) \), the partial derivative with respect to \( x \), \( \frac{\partial f}{\partial x} \), captures how the function changes as \( x \) varies, fixing \( y \). Similarly, \( \frac{\partial f}{\partial y} \) indicates how \( f \) changes with changes in \( y \), keeping \( x \) steady.

• \( \frac{\partial f}{\partial x} = -3 \sin(3x + 2y) \)
• \( \frac{\partial f}{\partial y} = -2 \sin(3x + 2y) \)

These derivatives are essential for building the gradient vector, which combines them into a single vector representing the direction and rate of greatest increase of the function. Partial derivatives thus form the backbone of understanding how changes in individual variables impact the overall outcome of the function.

Direction of Maximum Change

When dealing with functions of multiple variables, knowing which direction to move in to achieve the maximum increase can be very important. The _direction of maximum change_ is determined by the gradient of the function.

The gradient vector \( abla f \) points in the direction where the function increases most quickly. In our problem, the gradient \( \left(-\frac{3\sqrt{2}}{2}, -\sqrt{2}\right) \) tells us where, at the point \( \left(\frac{\pi}{6}, -\frac{\pi}{8}\right) \), the function's value changes fastest.

• The unit vector in the direction of the gradient gives us a normalized direction vector, effectively standardizing the direction while removing the scale.
  
• This direction is found by dividing each component of the gradient by its magnitude, resulting in a clear path of fastest ascent.

This concept is crucial, especially in optimization problems, where one might seek a maximum increase in a system dependent on multiple inputs.