Study Guide: Multivariable Calculus Exam Topics (Critical Points, Gradients, Multiple Integrals, and Tangent Planes)

Critical Points and Extrema of Multivariable Functions

Identifying and Classifying Critical Points

Critical points of a function of several variables are points where all first partial derivatives vanish or are undefined. These points are candidates for local maxima, minima, or saddle points.

• Critical Point: A point $(x_0, y_0)$ where $f_x(x_0, y_0) = 0$ and $f_y(x_0, y_0) = 0$.

• Classification: Use the second derivative test for functions $f(x, y)$:

$D = f_{xx}(x_0, y_0) f_{yy}(x_0, y_0) - [f_{xy}(x_0, y_0)]^2$

• If $D > 0$ and $f_{xx}(x_0, y_0) > 0$, local minimum.

• If $D > 0$ and $f_{xx}(x_0, y_0) < 0$, local maximum.

• If $D < 0$, saddle point.

• If $D = 0$, the test is inconclusive.

Example: For $f(x, y) = x^4 + y^4 - 4xy + 1$, find and classify all critical points.

• Find $f_x$ and $f_y$, set to zero, solve for $(x, y)$.

• Compute $f_{xx}$, $f_{yy}$, $f_{xy}$ at each critical point and evaluate $D$.

Gradients and Tangent Planes

The Gradient Vector

The gradient of a function $f(x, y, z)$ is a vector of its first partial derivatives:

$\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle$

• The gradient points in the direction of greatest increase of $f$.

• The gradient is perpendicular (normal) to the level surface $f(x, y, z) = c$ at any point.

Equation of the Tangent Plane

The tangent plane to the surface $f(x, y, z) = 0$ at point $(x_0, y_0, z_0)$ is:

$\frac{\partial f}{\partial x}(x_0, y_0, z_0)(x - x_0) + \frac{\partial f}{\partial y}(x_0, y_0, z_0)(y - y_0) + \frac{\partial f}{\partial z}(x_0, y_0, z_0)(z - z_0) = 0$

Example: Find the tangent plane to $x^2 - 2y - 2z = 1$ at $(-1, -1, -2)$.

• Compute $\nabla f(-1, -1, -2) = \langle -2, -2, -2 \rangle$.

• Plug into the tangent plane formula.

Multiple Integrals

Double Integrals Over Regions

Double integrals are used to compute volumes under surfaces or to integrate over regions in the plane.

$\iint_R f(x, y)\, dA$

• Limits of integration are determined by the region $R$.

• Order of integration (dy dx or dx dy) depends on the region's description.

Example: Evaluate $\int_0^1 \int_0^x x^2 y\, dy\, dx$.

• Integrate with respect to $y$ first, then $x$.

Vector-Valued Functions and Tangent Vectors

Vector Functions and Their Derivatives

A vector-valued function $\vec{r}(t)$ describes a curve in space. Its derivative gives the tangent vector at each point.

$\vec{r}'(t) = \frac{d\vec{r}}{dt}$

• The unit tangent vector is $\vec{T}(t) = \frac{\vec{r}'(t)}{\|\vec{r}'(t)\|}$.

Example: For $\vec{r}(t) = \langle t, t^2, t^3 \rangle$, $\vec{r}'(t) = \langle 1, 2t, 3t^2 \rangle$.

Domain of Multivariable Functions

Determining the Domain

The domain of a function $f(x, y)$ is the set of all $(x, y)$ for which the formula makes sense (e.g., no division by zero, no square roots of negative numbers).

• For $f(x, y) = \frac{1}{x^2 + y^2 - 1}$, the domain is all $(x, y)$ such that $x^2 + y^2 \neq 1$.

Antiderivatives and Integration

Finding Antiderivatives

An antiderivative of a function $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$.

• For $f(x) = 3x^2$, an antiderivative is $F(x) = x^3$.

Summary Table: Key Concepts

Concept	Definition	Key Formula
Critical Point	Where all first partial derivatives vanish	$f_x = 0,\ f_y = 0$
Second Derivative Test	Classifies critical points	$D = f_{xx} f_{yy} - (f_{xy})^2$
Gradient	Vector of partial derivatives	$\nabla f = \langle f_x, f_y, f_z \rangle$
Tangent Plane	Plane tangent to a surface at a point	$f_x(x_0, y_0)(x-x_0) + f_y(x_0, y_0)(y-y_0) + f_z(x_0, y_0)(z-z_0) = 0$
Double Integral	Integral over a region in the plane	$\iint_R f(x, y)\, dA$

Additional info:

• This exam covers topics from multivariable calculus, including critical points, gradients, tangent planes, and multiple integrals, which correspond to Chapters 4, 5, and 6 in a standard Calculus sequence.

• Some questions also involve vector-valued functions and their derivatives, relevant to parametric curves (Chapter 12).