Chapter 10

Suppose \(w=\frac{x}{y}+\frac{y}{z},\) where $$x=e^{5 t}, y=2+\sin (3 t), \text { and } z=2+\cos (6 t)$$ A) Use the chain rule to find \(\frac{d w}{d t}\) as a function of \(x, y, z,\) and \(t .\) Do not rewrite \(x, y,\) and \(z\) in terms of \(t,\) and do not rewrite \(e^{5 t}\) as \(x\). \(\frac{d w}{d t}=\) ____________. B) Use part A to evaluate \(\frac{d w}{\|}\) when \(t=0\).

A student was asked to find the equation of the tangent plane to the surface \(z=x^{4}-y^{5}\) at the point \((x, y)=(4,5) .\) The student's answer was \(z=-2869+4 x^{3}(x-4)-\left(5 y^{4}\right)(y-5)\) (a) At a glance, how do you know this is wrong. What mistakes did the student make? Select all that apply. \(\square\) The answer is not a linear function. \(\square\) The \((\mathrm{x}-4)\) and \((\mathrm{y}-\) 5) should be \(x\) and y. \(\quad \square\) The partial derivatives were not evaluated a the point. \(\square\) The -2869 should not be in the answer. All of the above (b) Find the correct equation for the tangent plane. \(z=\) ____________.

Find the partial derivatives of the function $$f(x, y)=x y e^{4 y}$$ \(f_{x}(x, y)=\) ________. \(f_{y}(x, y)=\) _________. \(f_{x y}(x, y)=\) _________. \(f_{y x}(x, y)=\) _________.

Find the partial derivatives of the function $$w=\sqrt{4 r^{2}+3 s^{2}+7 t^{2}}$$ \(\frac{\partial w}{\partial r}=\) ________. \(\frac{\partial w}{\partial s}=\) ________. \(\frac{\partial w}{\partial t}=\) ________.

Show that the function $$f(x, y)=\frac{x^{3} y}{x^{6}+y^{3}}$$ does not have a limit at (0,0) by examining the following limits. (a) Find the limit of \(f\) as \((x, y) \rightarrow(0,0)\) along the line \(y=x\). \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)=\) ________. (b) Find the limit of \(f\) as \((x, y) \rightarrow(0,0)\) along the curve \(y=x^{3}\). \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)=\) ___________.

Find the absolute maximum and minimum of the function \(f(x, y)=x^{2}-\) \(y^{2}\) subject to the constraint \(x^{2}+y^{2}=361 .\) As usual, ignore unneeded answer blanks, and list points in lexicographic order. Absolute minimum value: ____________________. attained at ( ____________, ______________) ( ____________, ) Absolute minimum value: ____________________. attained at ( ____________, ______________) ( ____________, )

Let \(f(x, y)=1 / x+2 / y+3 x+4 y\) in the region \(R\) where \(x, y>0\). Explain why \(f\) must have a global minimum at some point in \(R\) (note that \(R\) is unbounded- - how does this influence your explanation?). Then find the global minimum. minimum \(=\) ______________.

The temperature at any point in the plane is given by \(T(x, y)=\frac{100}{x^{2}+y^{2}+3}\). (a) What shape are the level curves of \(T ?\) \(\odot\) hyperbolas \(\odot\) circles \(\odot\) lines \(\odot\) ellipses \(\odot\) parabolas \(\odot\) none of the above (b) At what point on the plane is it hottest? __________ What is the maximum temperature? ___________. (c) Find the direction of the greatest increase in temperature at the point (-3,-3) ____________. What is the value of this maximum rate of change, that is, the maximum value of the directional derivative at (-3,-3)\(?\) ___________. (d) Find the direction of the greatest decrease in temperature at the point (-3,-3) ___________. What is the value of this most negative rate of change, that is, the minimum value of the directional derivative at (-3,-3)\(?\) __________.

If \(z=(x+y) e^{y}\) and \(x=u^{2}+v^{2}\) and \(y=u^{2}-v^{2},\) find the following partial derivatives using the chain rule. Enter your answers as functions of \(u\) and \(v\). \(\frac{\partial z}{\partial u}=\) __________. \(\frac{\partial z}{\partial v}=\) ___________.

(a) Check the local linearity of \(f(x, y)=e^{-x} \cos (y)\) near \(x=1, y=1.5\) by filling in the following table of values of \(f\) for \(x=0.9,1,1.1\) and \(y=1.4,1.5,1.6 .\) Express values of \(f\) with 4 digits after the decimal point. (b) Next, fill in the table for the values \(x=0.99,1,1.01\) and \(y=\) 1.49,1.5,1.51 , again showing 4 digits after the decimal point. Notice if the two tables look nearly linear, and whether the second looks more linear than the first (in particular, think about how you would decide if they were linear, or if the one were more closely linear than the other). (c) Give the local linearization of \(f(x, y)=e^{-x} \cos (y)\) at (1,1.5) : Using the second of your tables: \(f(x, y) \approx\) ____________. Using the fact that \(f_{x}(x, y)=-e^{-x} \cos (y)\) and \(f_{y}(x, y)=-e^{-x} \sin (y):\) \(f(x, y) \approx\) ___________.

Calculate all four second-order partial derivatives of \(f(x, y)=\sin \left(\frac{5 x}{y}\right)\). \(f_{x x}(x, y)=\) __________. \(f_{x y}(x, y)=\) __________. \(f_{y x}(x, y)=\) __________. \(f_{y y}(x, y)=\) __________.

Suppose that \(f(x, y)\) is a smooth function and that its partial derivatives have the values, \(f_{x}(0,9)=-4\) and \(f_{y}(0,9)=-2 .\) Given that \(f(0,9)=\) 1, use this information to estimate the value of \(f(1,10)\). Note this is analogous to finding the tangent line approximation to a function of one variable. In fancy terms, it is the first Taylor approximation. Estimate of (integer value) \(f(0,10)\) ______________. Estimate of (integer value) \(f(1,9)\) ______________. Estimate of (integer value) \(f(1,10)\) ______________.

Find the limit, if it exists, or type \(\mathrm{N}\) if it does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{3 x^{2}}{3 x^{2}+5 y^{2}}=$$ ___________.

Find the minimum distance from the point (1,1,11) to the paraboloid given by the equation \(z=x^{2}+y^{2}\). Minimum distance = ___________.

Each of the following functions has at most one critical point. Graph a few level curves and a few gradients and, on this basis alone, decide whether the critical point is a local maximum, a local minimum, a saddle point, or that there is no critical point. For \(f(x, y)=e^{-2 x^{2}-4 y^{2}},\) type of critical point: \(\quad(\square\) Local Maximum Local Minimum \(\square\) Saddle Point \(\square\) No Critical Point) For \(f(x, y)=e^{2 x^{2}-4 y^{2}},\) type of critical point: \(\quad\) (\square Local Maximum Local Minimum \(\square\) Saddle Point \(\square\) No Critical Point) For \(f(x, y)=2 x^{2}+4 y^{2}+4,\) type of critical point: \(\quad(\square\) Local Maximum Local Minimum \(\square\) Saddle Point \(\square\) No Critical Point) For \(f(x, y)=2 x^{2}+4 y+4,\) type of critical point: \(\quad(\square\) Local Maximum Local Minimum \(\square\) Saddle Point \(\square\) No Critical Point)

If $$z=\sin \left(x^{2}+y^{2}\right), \quad x=v \cos (u), \quad y=v \sin (u)$$ find \(\partial z / \partial u\) and \(\partial z / \partial v\). The variables are restricted to domains on which the functions are defined. \(\partial z / \partial u=\) _________. \(\partial z / \partial v=\) _________.

Suppose that \(z\) is a linear function of \(x\) and \(y\) with slope -5 in the \(x\) direction and slope 5 in the \(y\) direction. (a) A change of 0.2 in \(x\) and 0.5 in \(y\) produces what change in \(z ?\) change in \(z=\) __________. (b) If \(z=6\) when \(x=3\) and \(y=2\), what is the value of \(z\) when \(x=2.7\) and \(y=1.9 ?\) \(z=\) ___________.

Given \(F(r, s, t)=r\left(9 s^{4}-t^{5}\right),\) compute: $$F_{r s t}=$$ _________.

Find the limit, if it exists, or type \(\mathrm{N}\) if it does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{(x+13 y)^{2}}{x^{2}+13^{2} y^{2}}=$$ _____________.

For each value of \(\lambda\) the function \(h(x, y)=x^{2}+y^{2}-\lambda(2 x+8 y-20)\) has a minimum value \(m(\lambda)\). (a) Find \(m(\lambda)\) \(m(\lambda)=\) ___________. (Use the letter \(\boldsymbol{L}\) for \(\lambda\) in your expression. \()\) (b) For which value of \(\lambda\) is \(m(\lambda)\) the largest, and what is that maximum value? \(\lambda=\) \(\operatorname{maximum} m(\lambda)=\) _________. (c) Find the minimum value of \(f(x, y)=x^{2}+y^{2}\) subject to the constraint \(2 x+8 y=20\) using the method of Lagrange multipliers and evaluate \(\lambda\) \(\operatorname{minimum} f=\) ___________ \(\lambda=\) ___________

If \(f(x, y, z)=2 z y^{2},\) then the gradient at the point (2,2,4) is \(\nabla f(2,2,4)=\) _____________.

Let \(z=g(u, v)\) and \(u(r, s), v(r, s)\). How many terms are there in the expression for \(\partial z / \partial r ?\) __________ terms

Find the differential of the function \(w=x^{3} \sin \left(y^{5} z^{1}\right)\) \(d w=\) ____________ \(d x+\) ___________ \(d y+\) \(d z\).

Calculate all four second-order partial derivatives and check that \(f_{x y}=\) \(f_{y x}\). Assume the variables are restricted to a domain on which the function is defined. $$f(x, y)=e^{2 x y}$$ \(f_{x x}=\) ________. \(f_{y y}=\) ________. \(f_{x y}=\) ________. \(f_{y x}=\) ________.

Find the first partial derivatives of \(f(x, y, z)=z \arctan \left(\frac{y}{x}\right)\) at the point (4,4,-3) A. \(\frac{\partial f}{\partial x}(4,4,-3)=\) ____________. B. \(\frac{\partial f}{\partial y}(4,4,-3)=\) ____________. C. \(\frac{\partial f}{\partial z}(4,4,-3)=\) ____________.

Find the limit, if it exists, or type 'DNE' if it does not exist. $$\lim _{(x, y) \rightarrow(1,3)} e^{\sqrt{4 x^{2}+3 y^{2}}}=$$ ____________.

The plane \(x+y+2 z=6\) intersects the paraboloid \(z=x^{2}+y^{2}\) in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin. Point farthest away occurs at ( ____________, ______________) Point nearest occurs at ( ____________, ______________)

Find the maximum and minimum values of \(f(x, y)=x y\) on the ellipse \(5 x^{2}+y^{2}=3\) maximum value \(=\) __________. minimum value \(=\) __________.

The concentration of salt in a fluid at \((x, y, z)\) is given by \(F(x, y, z)=\) \(2 x^{2}+3 y^{4}+2 x^{2} z^{2} \mathrm{mg} / \mathrm{cm}^{3}\). You are at the point (-1,1,-1) . (a) In which direction should you move if you want the concentration to increase the fastest? direction: _________. (Give your answer as a vector.) (b) You start to move in the direction you found in part (a) at a speed of \(5 \mathrm{~cm} / \mathrm{sec} .\) How fast is the concentration changing? rate of change \(=\) ______________.

Let \(W(s, t)=F(u(s, t), v(s, t))\) where $$\begin{array}{r}u(1,0)=-4, u_{s}(1,0)=-7, u_{t}(1,0)=-4 \\ v(1,0)=8, v_{s}(1,0)=-8, v_{t}(1,0)=3 \\ F_{u}(-4,8)=1, F_{v(-4,8)=5\end{array}$$ \(W_{s}(1,0)= _______ W_{t}(1,0)=\) ____________.

The dimensions of a closed rectangular box are measured as 60 centimeters, 60 centimeters, and 80 centimeters, respectively, with the error in each measurement at most .2 centimeters. Use differentials to estimate the maximum error in calculating the surface area of the box. __________ square centimeters.