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The gradient of a function is a vector that consists of all its partial derivatives. It tells you the direction of steepest ascent of the function. For a function of two variables, like \( g(p, q) = p^4 - p^2q^3 \), the gradient \( abla g \) is given by the vector:

• \( \frac{\partial g}{\partial p} \)
• \( \frac{\partial g}{\partial q} \)

This means you take the derivative of the function with respect to each independent variable. In this case, \( abla g = \left( 4p^3 - 2pq^3, -3p^2q^2 \right) \). The gradient is crucial because when we compute the directional derivative, we need to know how the function changes in space, and the gradient provides that information.

Partial derivatives are used to understand how a function changes as one of its variables changes while holding the other constant. For a function \( f(x, y) \), the partial derivative with respect to \( x \) is denoted \( \frac{\partial f}{\partial x} \). In the example, you compute:

• \( \frac{\partial g}{\partial p} = 4p^3 - 2pq^3 \)
• \( \frac{\partial g}{\partial q} = -3p^2q^2 \)

These derivatives show how \( g(p, q) \) changes as \( p \) or \( q \) changes individually. Evaluating these at the point \((2, 1)\) provides the rates of change precisely at that point in each direction.

A unit vector is a vector with a magnitude of one. It keeps the direction of an original vector but without changing the length. For a given vector \( \mathbf{v} = [1, 3] \), the magnitude is \( \sqrt{1^2 + 3^2} = \sqrt{10} \). To convert \( \mathbf{v} \) to a unit vector \( \mathbf{u} \), each component of \( \mathbf{v} \) is divided by \( \sqrt{10} \):

• \( \mathbf{u} = \left( \frac{1}{\sqrt{10}}, \frac{3}{\sqrt{10}} \right) \)

Unit vectors are especially important in directional derivatives, as they ensure you're measuring change in exactly the intended direction without changing the scale by which it is measured.

The dot product is an algebraic operation taking two equal-length sequences of numbers (usually coordinate vectors) and returning a single number. It's calculated as the sum of the products of the corresponding entries of the two sequences of numbers. The formula for vectors \( \mathbf{a} \) and \( \mathbf{b} \) is:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \)

In the given exercise, the dot product between the gradient \( (28, -12) \) and the unit vector \( \left( \frac{1}{\sqrt{10}}, \frac{3}{\sqrt{10}} \right) \) yields the directional derivative. It effectively tells you how much of the gradient is in the direction of the unit vector. This dot product and particularly its result, the directional derivative, give insight into how the function changes as you move in that specified direction.