Chapter Review

• 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)