Chapter Review

Printed Page 899

• The directional derivative of a differentiable function $z=f( x,y)$ at the point $( x_{0},y_{0})$ in the direction of the unit vector $\mathbf{u=\cos }\theta \mathbf{i+\sin }\theta \mathbf{j}$ is $D_{\mathbf{u}}f(x_{0},y_{0})=f_{x}( x_{0},y_{0}) \cos \theta + f_{y}( x_{0},y_{0}) \sin \theta$. (p. 864)
• The directional derivative $D_{\mathbf{u}}f(x_{0},y_{0})$ equals the slope of the tangent line to the curve $C$ at the point $\left( x_{0},y_{0},f(x_{0},y_{0})\right)$ on the surface of $z=f(x,y)$, where $C$ is the intersection of the surface with the plane perpendicular to the $xy$-plane and containing the line through $( x_{0},y_{0})$ in the direction $\mathbf{u}$. (p. 865)
• The gradient of a differentiable function $f$

• of two variables :${\boldsymbol\nabla\! }f(x,y)=f_{x}(x,y)\mathbf{i}+f_{y}(x,y)\mathbf{j}$ (p. 866)
• of three variables: ${\boldsymbol\nabla \!}f(x,y,z)=f_{x}(x,y,z)\mathbf{i}+ f_{y}(x,y,z)\mathbf{j}+ f_{z}(x,y,z)\mathbf{k}$ (p. 871)

Properties of the Gradient: (p. 867)

• If ${\boldsymbol\nabla\! }f(x_{0},y_{0})=\mathbf{0}$, then $D_{\mathbf{u} }f(x_{0},y_{0})=0$ for all directions $\mathbf{u}$.
• If ${\boldsymbol\nabla\! }f(x_{0},y_{0})\neq \mathbf{0}$, then the directional derivative of $f$ at $(x_{0},y_{0})$ is a maximum when $\mathbf{u}$ is in the direction of ${\boldsymbol\nabla\! }f(x_{0},y_{0})$. The maximum value of $D_{\mathbf{u}}f(x_{0},y_{0})$ is $\left\Vert {\boldsymbol\nabla\! }f(x_{0},y_{0})\right\Vert$.
• If ${\boldsymbol\nabla\! }f(x_{0},y_{0})\neq \mathbf{0}$, then the directional derivative of $f$ at $(x_{0},y_{0})$ is a minimum when $\mathbf{u}$ is in the direction of $-{\boldsymbol\nabla\! }f(x_{0},y_{0})$. The minimum value of $D_{\mathbf{u}}f(x_{0},y_{0})$ is $-\left\Vert {\boldsymbol\nabla\! }f(x_{0},y_{0})\right\Vert$.
• $z=f(x,y)$ increases most rapidly in the direction of $\boldsymbol\nabla\! f(x_{0},y_{0})$.
• $z=f(x,y)$ decreases most rapidly in the direction of $-\boldsymbol\nabla\! f(x_{0},y_{0})$.
• The value of $z=f(x,y)$ remains the same for directions orthogonal to $\boldsymbol\nabla\! f(x_{0},y_{0})$. (p. 868)

The gradient ${\boldsymbol\nabla\! }f(x_{0},y_{0})$ is normal to the level curve of $f$ at $P_{0}=(x_{0},y_{0}).$(p. 870)

$D_{\mathbf{u}}f(x,y) =$$\boldsymbol\nabla\!$ $f(x,y)\,{\boldsymbol\cdot}\, \mathbf{u}$, where $\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}$. (p. 866)

$D_{\mathbf{u}}f(x,y,z)=$$\boldsymbol\nabla\!$$f(x,y,z)\,{\boldsymbol\cdot}\, \mathbf{u}$, where $\mathbf{u}=\cos \alpha \mathbf{i}+\cos \beta \mathbf{j} +\cos \gamma \mathbf{k}$. (p. 871)

• Equation of a tangent plane to a surface (p. 876)
• The normal line to the tangent plane to $F(x,y,z)=0$ at the point $P_{0}=(x_{0},y_{0},z_{0})$:

• vector equation: $\mathbf{r}(t)=\mathbf{r}_{0}+t{\boldsymbol\nabla\! }F(x_{0},y_{0},z_{0}),$ where $\mathbf{r}_{0}=x_{0}\mathbf{i}+y_{0}\mathbf{j}+z_{0}\mathbf{k}$ is the position vector of $P_{0}$ and $\mathbf{r}$ is the position vector of any point $P$ on the normal line. (p. 876)
• parametric equations: $x=x_{0}+at$, $y=y_{0}+bt,$ and $z=z_{0}+ct,$ where $a=F_{x}(x_{0},y_{0},z_{0})$, $b=F_{y}(x_{0},y_{0},z_{0})$, and $c=F_{z}(x_{0},y_{0},z_{0})$. (p. 876)
• symmetric equations: $\dfrac{x-x_{0}}{a}=\dfrac{y-y_{0}}{b}=\dfrac{z-z_{0}}{c}$ if $abc\neq 0.$(p. 877)

• Local maximum; local minimum (p. 879)
• Absolute maximum; absolute minimum (p. 879)
• Critical point (p. 880)
• Saddle point (p. 881)

Second Partial Derivative Test: (p. 882)

Let $z=f(x,y)$ be a function of two variables for which the first- and second-order partial derivatives are continuous in some disk containing the point $(x_{0},y_{0})$. Suppose that $f_{x}(x_{0},y_{0})=0$ and $f_{y}(x_{0},y_{0})=0$. Let $$A=f_{xx}(x_{0},y_{0})\qquad B=f_{xy}(x_{0},y_{0})\qquad C=f_{yy}(x_{0},y_{0})$$

• If $AC-B^{2}>0$ and $f_{xx}(x_{0},y_{0})>0$, then $f$ has a local minimum at $(x_{0},y_{0})$.
• If $AC-B^{2}>0$ and $f_{xx}(x_{0},y_{0})<0$, then $f$ has a local maximum at $(x_{0},y_{0})$.
• If $AC-B^{2}<0$, then $(x_{0},y_{0},f(x_{0},y_{0}))$ is a saddle point of $f.$
• If $AC-B^{2}=0$, then the test gives no information.

Extreme Value Theorem for Functions of Two Variables: (p. 883)

Let $z=f(x,y)$ be a function of two variables. If $f$ is continuous on a closed, bounded set $D$, then $f$ has an absolute maximum and an absolute minimum on $D$.

Test for Absolute Maximum and Absolute Minimum: (p. 884)

Let $z=f(x,y)$ be a function of two variables defined on a closed, bounded set $D.$ If $f$ is continuous on $D$, then the absolute maximum and the absolute minimum of $f$ are, respectively, the largest and smallest values found among the following:

• The values of $f$ at the critical points of $f$ in $D$
• The values of $f$ on the boundary of $D$

Lagrange multiplier: (p. 892)

The Method of Lagrange Multipliers: (pp. 891-892)

The extreme values of $z=f(x,y)$ subject to the condition $g(x,y)=0$, if they exist, occur at the solutions $(x,y)$ of the system equations. $$\left\{\begin{array}{rcl} {\boldsymbol\nabla\! }f(x,y) &=&\lambda {\boldsymbol\nabla }g(x,y) \\[2pt] g(x,y) &=&0 \end{array}\right.$$

Steps for Using Lagrange Multipliers: (p. 893)