Calculus of Several Variables: Tangent Lines, Planes, and Linearization

Flashcards covering directional derivatives, gradient properties, tangent line and plane equations, linearization, and multivariable Taylor series based on MATH 252 lecture notes.

Directional Derivative

The rate of change of a function $f(x, y)$ in the direction of a unit vector $v = \langle v_x, v_y \rangle$ at a point $(x, y)$, defined by the limit $D_v(f)(x, y) = \lim_{h \to 0} \frac{f(x + h v_x, y + h v_y) - f(x, y)}{h}$.

Gradient ($\nabla f$)

For a function $f(x, y)$, the vector-valued function defined by $\nabla f(x, y) = \langle f_x(x, y), f_y(x, y) \rangle$. For a function $g(x, y, z)$, it is $\nabla g(x, y, z) = \langle g_x(x, y, z), g_y(x, y, z), g_z(x, y, z) \rangle$.

Gradient-Directional Derivative Theorem

If $v$ is any unit vector and $f$ has continuous partial derivatives, the directional derivative satisfies $D_v f = \nabla f \cdot v$.

Maximum Value of $D_v f$

The maximum rate of increase occurs when the unit vector $v$ is in the same direction as the gradient $\nabla f$, and its value is equal to the magnitude of the gradient, $\|\nabla f\|$.

Minimum Value of $D_v f$

The minimum rate of change (maximum decrease) occurs when the unit vector $v$ is in the opposite direction of the gradient $\nabla f$, and its value is $-\|\nabla f\|$.

Orthogonality Property of the Gradient

The value of the directional derivative $D_v f$ is zero if and only if the direction vector $v$ is orthogonal ($v \cdot \nabla f = 0$) to the gradient vector.

Normal Vector to Level Curves

For a level curve $f(x, y) = c$, the gradient vector $\nabla f(a, b)$ acts as a normal vector to the graph of the curve at that specific point.

Tangent Line of Implicit Curve

The line tangent to the curve $f(x, y) = d$ at point $(a, b)$ with equation $f_x(a, b) \cdot (x - a) + f_y(a, b) \cdot (y - b) = 0$.

Tangent Plane of Implicit Surface

The plane tangent to the surface $f(x, y, z) = d$ at point $(a, b, c)$ with equation $f_x(a, b, c) \cdot (x - a) + f_y(a, b, c) \cdot (y - b) + f_z(a, b, c) \cdot (z - c) = 0$.

Linearization of $f(x, y)$

The best linear approximation to a differentiable function $f$ near point $(a, b)$, given by $L(x, y) = f(a, b) + f_x(a, b) \cdot (x - a) + f_y(a, b) \cdot (y - b)$, which coincides with the tangent plane equation.

Linearization of $f(x, y, z)$

The best linear approximation to a function of three variables at $(a, b, c)$, defined as $L(x, y, z) = f(a, b, c) + f_x(a, b, c) \cdot (x - a) + f_y(a, b, c) \cdot (y - b) + f_z(a, b, c) \cdot (z - c)$.

Taylor Series (Multivariable)

An infinite sum representing a function through its partial derivatives at a point, expressed for two variables at $(a, b)$ as $T(x, y) = \sum_{n=0}^{\infty} \sum_{k=0}^{n} \frac{(x - a)^k (y - b)^{n-k}}{k! (n - k)!} \frac{\partial^n f}{(\partial x)^k (\partial y)^{n-k}}(a, b)$.

Taylor's Remainder Theorem (Multivariable)

A theorem providing an upper bound on the error between a function and its degree-$d$ Taylor polynomial: $|T_k(x, y) - f(x, y)| \le \frac{M \cdot (|x - a| + |y - b|)^{k+1}}{(k + 1)!}$, where $M$ bounds all $(d+1)$-order partial derivatives.