Calculus Volume 3

Key Concepts

Calculus Volume 3Key Concepts

4.1Functions of Several Variables

• The graph of a function of two variables is a surface in $ℝ3ℝ3$ and can be studied using level curves and vertical traces.
• A set of level curves is called a contour map.

4.2Limits and Continuity

• To study limits and continuity for functions of two variables, we use a $δδ$ disk centered around a given point.
• A function of several variables has a limit if for any point in a $δδ$ ball centered at a point $P,P,$ the value of the function at that point is arbitrarily close to a fixed value (the limit value).
• The limit laws established for a function of one variable have natural extensions to functions of more than one variable.
• A function of two variables is continuous at a point if the limit exists at that point, the function exists at that point, and the limit and function are equal at that point.

4.3Partial Derivatives

• A partial derivative is a derivative involving a function of more than one independent variable.
• To calculate a partial derivative with respect to a given variable, treat all the other variables as constants and use the usual differentiation rules.
• Higher-order partial derivatives can be calculated in the same way as higher-order derivatives.

4.4Tangent Planes and Linear Approximations

• The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables.
• Tangent planes can be used to approximate values of functions near known values.
• A function is differentiable at a point if it is ”smooth” at that point (i.e., no corners or discontinuities exist at that point).
• The total differential can be used to approximate the change in a function $z=f(x0,y0)z=f(x0,y0)$ at the point $(x0,y0)(x0,y0)$ for given values of $ΔxΔx$ and $Δy.Δy.$

4.5The Chain Rule

• The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables.
• Tree diagrams are useful for deriving formulas for the chain rule for functions of more than one variable, where each independent variable also depends on other variables.

• A directional derivative represents a rate of change of a function in any given direction.
• The gradient can be used in a formula to calculate the directional derivative.
• The gradient indicates the direction of greatest change of a function of more than one variable.

4.7Maxima/Minima Problems

• A critical point of the function $f(x,y)f(x,y)$ is any point $(x0,y0)(x0,y0)$ where either $fx(x0,y0)=fy(x0,y0)=0,fx(x0,y0)=fy(x0,y0)=0,$ or at least one of $fx(x0,y0)fx(x0,y0)$ and $fy(x0,y0)fy(x0,y0)$ do not exist.
• A saddle point is a point $(x0,y0)(x0,y0)$ where $fx(x0,y0)=fy(x0,y0)=0,fx(x0,y0)=fy(x0,y0)=0,$ but $(x0,y0)(x0,y0)$ is neither a maximum nor a minimum at that point.
• To find extrema of functions of two variables, first find the critical points, then calculate the discriminant and apply the second derivative test.

4.8Lagrange Multipliers

• An objective function combined with one or more constraints is an example of an optimization problem.
• To solve optimization problems, we apply the method of Lagrange multipliers using a four-step problem-solving strategy.
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