Calculus III Study Guide: Multivariable Functions, Partial Derivatives, and Optimization

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Graphs and Level Curves

Level Curves of Multivariable Functions

Level curves, also known as contours, are a fundamental concept in multivariable calculus. They represent the set of points where a function of two variables, z = f(x, y), takes on a constant value. These curves help visualize surfaces and understand their behavior.

Definition: A level curve of z = f(x, y) is the set of points (x, y) where f(x, y) = c for some constant c.
Graphing: To graph level curves, set z to various constant values (e.g., z = 0, 2, 3) and solve for (x, y).
Matching Functions to Level Curves: Understanding the shape and spacing of level curves helps identify the corresponding function.
Example: For f(x, y) = x^2 + y^2, level curves are circles centered at the origin with radius \sqrt{c}.

Limits and Continuity in Multivariable Functions

Evaluating Limits

Limits in functions of two or more variables extend the concept from single-variable calculus. The limit \lim_{(x, y) \to (a, b)} f(x, y) exists if the function approaches the same value along every path to (a, b).

Direct Substitution: If f(x, y) is continuous at (a, b), substitute directly.
Factoring and Canceling: Factor expressions to simplify and remove indeterminate forms.
Conjugate Factors: Use conjugates for functions involving square roots.
Path Checking: If a limit does not exist, show it approaches different values along different paths (e.g., y = 0 vs. x = 0).
Example: For f(x, y) = \frac{2xy}{x^2 + y^2} as (x, y) \to (0, 0), check along y = mx to see if the limit depends on m.

Partial Derivatives and the Chain Rule

Partial Derivatives

Partial derivatives measure how a function changes as one variable changes, keeping others constant. They are essential for analyzing multivariable functions.

Notation: f_x and f_y denote partial derivatives with respect to x and y.
Second Partial Derivatives: f_{xx}, f_{yy}, f_{xy} represent second derivatives.
Derivative Rules: Product, quotient, and chain rules apply to partial derivatives.
Example: For f(x, y) = x^2y + y^3, f_x = 2xy, f_y = x^2 + 3y^2.

Chain Rule for Multivariable Functions

The chain rule extends to functions of several variables, allowing differentiation of composite functions.

Formula: If z = f(x, y) and x, y depend on t, then
Example: If f(x, y) = x^2y, x = t, y = t^2, then

Directional Derivatives and the Gradient

Gradient Vector

The gradient is a vector of partial derivatives, indicating the direction and rate of fastest increase of a function.

Definition: For f(x, y), the gradient is For g(x, y, z),
Interpretation: \nabla f(a, b) points in the direction of steepest ascent at (a, b).
Magnitude: |\nabla f(a, b)| gives the maximum rate of increase.
Direction of No Change: Vectors perpendicular to \nabla f(a, b) indicate directions of no change.

Directional Derivative

The directional derivative measures the rate of change of a function in a specified direction.

Formula: For unit vector \vec{u},
Example: If \nabla f = \langle 3, 4 \rangle and \vec{u} = \langle \frac{3}{5}, \frac{4}{5} \rangle, then

Maximum and Minimum Problems

Critical Points and Classification

Critical points are where the partial derivatives of a function are zero or undefined. The second derivative test helps classify these points as local maxima, minima, or saddle points.

Critical Point: (a, b) is a critical point if f_x(a, b) = 0 and f_y(a, b) = 0, or one or both do not exist.
Discriminant:
Second Derivative Test:
- If D(a, b) > 0 and f_{xx}(a, b) > 0, local minimum.
- If D(a, b) > 0 and f_{xx}(a, b) < 0, local maximum.
- If D(a, b) < 0, saddle point.
- If D(a, b) = 0, test is inconclusive.
Example: For f(x, y) = x^2 - y^2, critical point at (0, 0), D = -4 (saddle point).

Basic Partial Derivative Rules

Common Derivative Formulas

Partial derivatives follow similar rules as single-variable derivatives. Below is a summary of key formulas:

Function	Partial Derivative with Respect to x
u^p
e^u
b^u
\ln(u)
\log_b(u)
\sqrt{u}
\sin(u)
\cos(u)
\tan(u)
\sec(u)
\csc(u)
\cot(u)
\sin^{-1}(u)
\tan^{-1}(u)

Note: In all cases, u_x denotes the partial derivative of u with respect to x.

Additional info: These notes cover key topics from Calculus III, including multivariable functions, partial derivatives, gradients, and optimization. The content is expanded for clarity and completeness, suitable for exam preparation.