Problem 16 - Calculus with Concepts in Calculus

Question

Find the direction in which \(f\) increases most rapidly at the given point, and find the maximal directional derivative at that point. $$ f(x, y)=e^{2 x}(\cos y-\sin y) ;\left(\frac{1}{6},-\pi / 2\right) $$

Step-by-Step Solution

Answer

The function increases most rapidly in the direction \((-2e^{\frac{1}{3}}, 0)\), with a maximal directional derivative of \(2e^{\frac{1}{3}}\).

Step 1: Compute the gradient of the function

The first step is to determine the gradient \( abla f \) of the function \( f(x, y) = e^{2x}(\cos y - \sin y) \). The gradient is a vector of partial derivatives. \[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \].

Step 2: Calculate partial derivatives

First, find the partial derivative with respect to \( x \):\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( e^{2x}(\cos y - \sin y) \right) = 2e^{2x}(\cos y - \sin y) \].Next, the partial derivative with respect to \( y \):\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left( e^{2x}(\cos y - \sin y) \right) = -e^{2x}(\sin y + \cos y) \].

Step 3: Evaluate the gradient at the given point

Substitute \( \left(\frac{1}{6}, -\frac{\pi}{2}\right) \) into the gradient to find \( abla f \) at that point:\[ abla f\left( \frac{1}{6}, -\frac{\pi}{2} \right) = \left( 2e^{\frac{1}{3}}(-1), -e^{\frac{1}{3}}(0) \right) = \left( -2e^{\frac{1}{3}}, 0 \right) \].

Step 4: Determine the direction of maximum increase

The function \( f \) increases most rapidly in the direction of the gradient vector, \( abla f \) evaluated at the given point. The direction is \( \left( -2e^{\frac{1}{3}}, 0 \right) \).

Step 5: Calculate the maximal directional derivative

The magnitude of the gradient vector \( abla f \left( \frac{1}{6}, -\frac{\pi}{2} \right) \) gives the maximal rate of increase of the function at the point. The maximal directional derivative is:\( \left| abla f \left( \frac{1}{6}, -\frac{\pi}{2} \right) \right| = \sqrt{(-2e^{\frac{1}{3}})^2 + 0^2} = 2e^{\frac{1}{3}} \).

Key Concepts

Directional Derivative

A directional derivative measures how a function changes as you move in a specific direction from a particular point. It's a generalization of a partial derivative, which considers change in only one variable at a time. To find the directional derivative of a function \( f(x, y) \) at a point \( (a, b) \), we compute the dot product between the gradient of \( f \) at that point and a unit vector representing the direction of interest.

• The directional derivative helps us understand the rate of change of a function not just vertically, but in any direction we choose.
• It is particularly useful for optimizing problems where you need to know how an increase or decrease in a direction affects the function value.

Say you have a function \( f \) that represents the height of a hill. The directional derivative at a point on this hill in the direction of \( v \) will tell you how steeply the hill rises or falls in that specific direction.

The directional derivative formula is:\[ D_v f(a, b) = abla f(a, b) \cdot \vec{v} \]where \( abla f(a, b) \) is the gradient at point \((a, b)\), and \( \vec{v} \) is a unit vector in your chosen direction.

Partial Derivatives

Partial derivatives are essential components of calculus which measure the rate at which a function changes as we change one variable, keeping the other variables fixed. In mathematical terms, for a function \( f(x, y) \), the partial derivative with respect to \( x \) is calculated by differentiating \( f \) while treating \( y \) as a constant, and vice versa.

• The partial derivative \( \frac{\partial f}{\partial x} \) tells you how \( f \) changes as \( x \) changes, while keeping \( y \) constant.
• Similarly, \( \frac{\partial f}{\partial y} \) examines the change in \( f \) due to a variation in \( y \), keeping \( x \) unchanged.
• This is crucial when dealing with multivariable functions, allowing us to assess the isolated impact of each variable on the function's output.

The importance of partial derivatives becomes more apparent in many fields, such as physics, engineering, and economics, where systems depend on several input variables. By understanding each variable's individual effect, solutions can be tailored to meet specific objectives.

Maximal Rate of Increase

The maximal rate of increase for a function at a given point is determined through the gradient. The gradient vector \( abla f \) points in the direction where the function increases most sharply. Its magnitude elucidates the steepness of the incline.

• The direction of the gradient is the path along which the function rapidly ascends.
• The size of the gradient vector—calculated using the Euclidean norm—represents the maximal rate of increase at that specific location.
• This is useful for optimization, assisting in identifying paths of greatest uplift within a field.

In the context of our exercise, after evaluating the gradient at the point \( \left(\frac{1}{6}, -\frac{\pi}{2}\right) \), we discovered the direction where the function rises most quickly. The magnitude, \( 2e^{\frac{1}{3}} \), offers a quantifiable measure of this rapid increase, essential for optimization strategies across various disciplines.