Problem 286

Question

For the following exercises, find the derivative of the function at $P$ in the direction of $\mathbf{u} .$ $$ f(x, y)=-7 x+2 y, P(2,-4), \quad \mathbf{u}=4 \mathbf{i}-3 \mathbf{j} $$

Step-by-Step Solution

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Answer

The directional derivative is $-\frac{34}{5}$.

1Step 1: Find the Gradient of the Function

To find the derivative in a specific direction, first find the gradient of the function, $f(x, y) = -7x + 2y$. The gradient is the vector of partial derivatives: $abla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)$.The partial derivatives are:\[\frac{\partial f}{\partial x} = -7\]\[\frac{\partial f}{\partial y} = 2\]Thus, the gradient is $abla f = (-7, 2)$.

2Step 2: Normalize the Direction Vector

The direction vector given is $\mathbf{u} = 4\mathbf{i} - 3\mathbf{j}$. Normalize this vector to ensure it has a magnitude of 1 before using it to find the directional derivative.Calculate the magnitude of $\mathbf{u}$:\[|\mathbf{u}| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = 5\]Normalize $\mathbf{u}$ by dividing each component by its magnitude:\[\mathbf{u}_{unit} = \left(\frac{4}{5}, \frac{-3}{5}\right)\].

3Step 3: Compute the Directional Derivative

The directional derivative of $f$ at point $P$ in the direction of $\mathbf{u}$ is calculated by the dot product of the gradient and the normalized direction vector:\[D_{\mathbf{u}}f = abla f \cdot \mathbf{u}_{unit}\]Plug in the values:\[D_{\mathbf{u}}f = (-7, 2) \cdot \left(\frac{4}{5}, \frac{-3}{5}\right) = (-7)\left(\frac{4}{5}\right) + (2)\left(\frac{-3}{5}\right)\]Calculating this:\[D_{\mathbf{u}}f = \left(-\frac{28}{5}\right) + \left(-\frac{6}{5}\right) = -\frac{34}{5}\]So, the directional derivative of $f$ at $P$ in the direction of $\mathbf{u}$ is $-\frac{34}{5}$.

Key Concepts

GradientPartial DerivativesVector Normalization

Gradient

When exploring directional derivatives, the concept of a gradient is central. In simple terms, the gradient of a function gives us a vector that points in the direction of the greatest rate of increase of the function. For a two-variable function like $f(x, y) = -7x + 2y$, the gradient is a vector made up of the partial derivatives with respect to each variable.

To find the gradient $abla f$ of our function, we calculate the partial derivative with respect to $x$, which is $-7$, and the partial derivative with respect to $y$, which is $2$.

This gives us the gradient as a vector:

Gradient: $abla f = (-7, 2)$

This vector can help us understand how the function changes as we move along different directions in the $xy$-plane.

Partial Derivatives

Partial derivatives are the building blocks of the gradient. They measure how a function changes as one variable changes while keeping the others constant. Consider the function $f(x, y) = -7x + 2y$.

For partial derivatives:

The partial derivative with respect to $x$ $(\frac{\partial f}{\partial x})$ indicates how much $f$ changes as $x$ changes, keeping $y$ constant. Here, it results in $-7$. This means that for every unit increase in $x$, $f$ decreases by 7.
The partial derivative with respect to $y$ $(\frac{\partial f}{\partial y})$ measures how much $f$ changes as $y$ changes, with $x$ constant. For our function, it's $2$, indicating $f$ increases by 2 for every unit increase in $y$.

These derivatives form the components of the gradient vector, letting you see which variable affects the function the most at any point.

Vector Normalization

Vector normalization is crucial when working with directions in contexts like directional derivatives. The directional vector we use must have a magnitude of 1 to ensure that it merely represents a direction, not a scaled distance.

Our direction vector in the exercise is $ \mathbf{u} = 4 \mathbf{i} - 3 \mathbf{j} $. To normalize it, we first need its magnitude: