Chapter 12

Let a level curve of \(z=f(x, y)\) be described by \(x=g(t)\), \(y=h(t) .\) Explain why \(\frac{d z}{d t}=0\)

What is the difference between a directional derivative and a partial derivative?

Describe in your own words the difference between boundary and interior points of a set.

T/F: If \(f(x, y)\) is differentiable on \(S\), the \(f\) is continuous on \(S\).

What is the difference between a constant and a coefficient?

T/F: A point \(P\) is a critical point of \(f\) if \(f_{x}\) and \(f_{y}\) are both 0 at \(P\).

Explain how the vector \(\vec{v}=\langle 0.6,0.8,-2\rangle\) can be thought of as having a "slope" of -2 .

Use your own words to describe (informally) what \(\lim _{(x, y) \rightarrow(1,2)} f(x, y)=17\) means.

The graph of a function of two variables is a __________.

T/F: A point \(P\) is a critical point of \(f\) if \(f_{x}\) or \(f_{y}\) are undefined at \(P\).

\(\mathrm{T} / \mathrm{F}\) : Let \(z=f(x, y)\) be differentiable at \(P\). If \(\vec{n}\) is a normal vector to the tangent plane of \(f\) at \(P\), then \(\vec{n}\) is orthogonal to \(\ell_{x}\) and \(\ell_{y}\) at \(P\).

Give an example of a closed, bounded set.

\(\mathrm{T} / \mathrm{F}:\) If \(z=f(x, y)\) is differentiable, then the change in \(z\) over small changes \(d x\) and \(d y\) in \(x\) and \(y\) is approximately \(d z .\)

Explain what it means to "solve a constrained optimization" problem.

If \(z=f(x, y),\) where \(x=g(t)\) and \(y=h(t),\) we can substitute and write \(z\) as an explicit function of \(t\). T/F: Using the Multivariable Chain Rule to find \(\frac{d z}{d t}\) is sometimes easier than first substituting and then taking the derivative.

Explain in your own words why we do not refer to the tangent line to a surface at a point, but rather to directional tangent lines to a surface at a point.

Give an example of a closed, unbounded set.

In the mixed partial fraction \(\frac{\partial^{2} f}{\partial x \partial y},\) which is computed first, \(f_{x}\) or \(f_{y} ?\)

T/F: Along a level curve, the output of a function does not change.

Find the critical points of the given function. Use the Second Derivative Test to determine if each critical point corresponds to a relative maximum, minimum, or saddle point. $$ f(x, y)=\frac{1}{2} x^{2}+2 y^{2}-8 y+4 x $$

A function \(z=f(x, y),\) a vector \(\vec{v}\) and a point \(P\) are given. Give the parametric equations of the following directional tangent lines to \(f\) at \(P\) : (a) \(\ell_{x}(t)\) (b) \(\ell_{y}(t)\) (c) \(\ell_{\vec{u}}(t),\) where \(\vec{u}\) is the unit vector in the direction of \(\vec{v}\). $$ f(x, y)=2 x^{2} y-4 x y^{2}, \vec{v}=\langle 1,3\rangle, P=(2,3) $$

The gradient points in the direction of __________ increase.

In Exercises \(5-8,\) find the total differential \(d z\). $$ z=x \sin y+x^{2} $$

Give an example of a open, bounded set.

Evaluate \(f_{x}(x, y)\) and \(f_{y}(x, y)\) at the indicated point. $$ f(x, y)=x^{2} y-x+2 y+3 \text { at }(1,2) $$

Find the critical points of the given function. Use the Second Derivative Test to determine if each critical point corresponds to a relative maximum, minimum, or saddle point. $$ f(x, y)=x^{2}+4 x+y^{2}-9 y+3 x y $$

A function \(z=f(x, y),\) a vector \(\vec{v}\) and a point \(P\) are given. Give the parametric equations of the following directional tangent lines to \(f\) at \(P\) : (a) \(\ell_{x}(t)\) (b) \(\ell_{y}(t)\) (c) \(\ell_{\vec{u}}(t),\) where \(\vec{u}\) is the unit vector in the direction of \(\vec{v}\). $$ f(x, y)=3 \cos x \sin y, \vec{v}=\langle 1,2\rangle, P=(\pi / 3, \pi / 6) $$

It is generally more informative to view the directional derivative not as the result of a limit, but rather as the result of a _________ product.

In Exercises \(5-8,\) find the total differential \(d z\). $$ z=\left(2 x^{2}+3 y\right)^{2} $$

Give an example of a open, unbounded set.

Evaluate \(f_{x}(x, y)\) and \(f_{y}(x, y)\) at the indicated point. $$ f(x, y)=x^{3}-3 x+y^{2}-6 y \text { at }(-1,3) $$

What does it mean when level curves are close together? Far apart?

Find the critical points of the given function. Use the Second Derivative Test to determine if each critical point corresponds to a relative maximum, minimum, or saddle point. $$ f(x, y)=x^{2}+3 y^{2}-6 y+4 x y $$

In Exercises \(7-12,\) functions \(z=f(x, y), x=g(t)\) and \(y=h(t)\) are given. (a) Use the Multivariable Chain Rule to compute \(\frac{d z}{d t}\). (b) Evaluate \(\frac{d z}{d t}\) at the indicated \(t\) -value. $$ z=3 x+4 y, \quad x=t^{2}, \quad y=2 t ; \quad t=1 $$

A function \(z=f(x, y),\) a vector \(\vec{v}\) and a point \(P\) are given. Give the parametric equations of the following directional tangent lines to \(f\) at \(P\) : (a) \(\ell_{x}(t)\) (b) \(\ell_{y}(t)\) (c) \(\ell_{\vec{u}}(t),\) where \(\vec{u}\) is the unit vector in the direction of \(\vec{v}\). $$ f(x, y)=3 x-5 y, \vec{v}=\langle 1,1\rangle, P=(4,2) $$

In Exercises \(7-12,\) a function \(z=f(x, y)\) is given. Find \(\nabla f\). $$ f(x, y)=-x^{2} y+x y^{2}+x y $$

In Exercises \(5-8,\) find the total differential \(d z\). $$ z=5 x-7 y $$

Evaluate \(f_{x}(x, y)\) and \(f_{y}(x, y)\) at the indicated point. $$ f(x, y)=\sin y \cos x \text { at }(\pi / 3, \pi / 3) $$

In Exercises \(7-14,\) give the domain and range of the multivariable function. $$ f(x, y)=x^{2}+y^{2}+2 $$

Find the critical points of the given function. Use the Second Derivative Test to determine if each critical point corresponds to a relative maximum, minimum, or saddle point. $$ f(x, y)=\frac{1}{x^{2}+y^{2}+1} $$

In Exercises \(7-12,\) functions \(z=f(x, y), x=g(t)\) and \(y=h(t)\) are given. (a) Use the Multivariable Chain Rule to compute \(\frac{d z}{d t}\). (b) Evaluate \(\frac{d z}{d t}\) at the indicated \(t\) -value. $$ z=x^{2}-y^{2}, \quad x=t, \quad y=t^{2}-1 ; \quad t=1 $$

A function \(z=f(x, y)\) is given. Find \(\nabla f\). $$ f(x, y)=\sin x \cos y $$

A function \(z=f(x, y),\) a vector \(\vec{v}\) and a point \(P\) are given. Give the parametric equations of the following directional tangent lines to \(f\) at \(P\) : (a) \(\ell_{x}(t)\) (b) \(\ell_{y}(t)\) (c) \(\ell_{\vec{u}}(t),\) where \(\vec{u}\) is the unit vector in the direction of \(\vec{v}\). $$ f(x, y)=x^{2}-2 x-y^{2}+4 y, \vec{v}=\langle 1,1\rangle, P=(1,2) $$

A set \(S\) is given. (a) Give one boundary point and one interior point, when possible, of \(S\). (b) State whether \(S\) is open, closed, or neither. (c) State whether \(S\) is bounded or unbounded. $$ S=\left\\{(x, y) \mid y \neq x^{2}\right\\} $$

In Exercises \(5-8,\) find the total differential \(d z\). $$ z=x e^{x+y} $$

Evaluate \(f_{x}(x, y)\) and \(f_{y}(x, y)\) at the indicated point. $$ f(x, y)=\ln (x y) \text { at }(-2,-3) $$

Give the domain and range of the multivariable function. $$ f(x, y)=x+2 y $$

Find the critical points of the given function. Use the Second Derivative Test to determine if each critical point corresponds to a relative maximum, minimum, or saddle point. $$ f(x, y)=x^{2}+y^{3}-3 y+1 $$

In Exercises \(7-12,\) functions \(z=f(x, y), x=g(t)\) and \(y=h(t)\) are given. (a) Use the Multivariable Chain Rule to compute \(\frac{d z}{d t}\). (b) Evaluate \(\frac{d z}{d t}\) at the indicated \(t\) -value. $$ z=5 x+2 y, \quad x=2 \cos t+1, \quad y=\sin t-3 ; \quad t=\pi / 4 $$

A function \(z=f(x, y)\) is given. Find \(\nabla f\). $$ f(x, y)=\frac{1}{x^{2}+y^{2}+1} $$