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The gradient, often denoted by \( abla f \), is a vector that points in the direction of the greatest rate of increase of a function. In multivariable calculus, the gradient of a function \( f(x, y) = x + y^2 \) is calculated by taking the partial derivatives with respect to each variable. For the given function:
\[ \frac{\partial f}{\partial x} = 1 \] and \[ \frac{\partial f}{\partial y} = 2y \].
The gradient vector is thus \( abla f = (1, 2y) \). This vector field shows how the function \( f \) changes at every point in the \( xy \)-plane.

Some key points to remember about gradients are:

• The gradient vector is always perpendicular to contour lines of the function.
• It indicates the direction in which the function increases the fastest.

Understanding the gradient is crucial for tackling problems involving directional derivatives, as the directional derivative leverages this concept.

Partial derivatives represent the rate of change of a function with respect to one variable, keeping the other variables constant. It is a fundamental tool in calculus, specifically for functions of multiple variables. For the function \( f(x, y) = x + y^2 \), the partial derivatives are:
\[ \frac{\partial f}{\partial x} = 1 \]
and \[ \frac{\partial f}{\partial y} = 2y \].

This means that:

• When \( y \) is held constant, the change in \( f \) with respect to \( x \) is consistent and linear, given by 1.
• When \( x \) is held constant, the change in \( f \) with respect to \( y \) is quadratic, meaning it varies with the value of \( y \).

Understanding partial derivatives allows one to analyze how changes in each variable affect the overall function, which is critical in calculating gradients and directional derivatives.

A tangent vector refers to a vector that touches a curve or surface at a given point without crossing it. In the context of this exercise, the tangent vector is derived from the constraint \( 2x^2 + y^2 = 9 \). To find a tangent vector, we first consider the gradient of the constraint, which is perpendicular to the tangent vector.

The gradient of the constraint \( g(x, y) = 2x^2 + y^2 \) results in \( abla g = (4x, 2y) \). At the point \((2, 1)\), this gives \( abla g (2, 1) = (8, 2) \).
To find the tangent vector, we take a vector perpendicular to \( abla g \), like swapping components and reversing the sign of one, resulting in \( (2, -8) \).

Tangent vectors are crucial in directional derivative problems because they provide the direction required for the derivative calculation.
Additionally, tangent vectors:

• Indicate the linear approximation direction of the curve at a point.
• Define an instantaneous path the function follows at the surface.

The constraint gradient pertains to the gradient vector derived from a constraint function, such as \( g(x, y) = 2x^2 + y^2 - 9 \). This constraint defines a curve on which we must focus. The constraint gradient \( abla g \) is \( (4x, 2y) \), found by taking the partial derivatives with respect to \( x \) and \( y \).

Evaluating this at the point \((2, 1)\) results in \( abla g (2, 1) = (8, 2) \). The constraint gradient reveals the direction perpendicular to the constraint surface, and thus, provides essential information for calculating tangent vectors.

Important aspects of constraint gradients include:

• They are used to find directions perpendicular to the constraint.
• The tangent vector can be derived as it is perpendicular to this gradient.
• The constraint gradient is crucial for identifying feasible regions and optimizing under certain conditions, particularly with Lagrange multipliers.

Understanding constraint gradients is indispensable when solving problems involving conditions and restrictions at specified points.