Exam 11: Partial Derivatives

A package in the shape of a cylindrical box can be mailed by the U.S. Postal Service if the sum of its height and girth (the circumference of the circular base) is at most 108 in. Find the dimension of the package with largest volume that can be mailed.

Use a linear approximation to estimate $\frac { 0.95 } { 5.05 }$ .

Find the critical points (if any) for $f ( x , y ) = \frac { 1 } { x } + \frac { 1 } { y } + x y$ and determine if each is a local extreme value or a saddle point.

The temperature at a point $( x , y )$ of a flat metal plate is $T ( x , y ) = 9 x ^ { 2 } + 16 y ^ { 2 }$ where $T ( x , y )$ is measured in degrees. Draw the isothermals for $T ( x , y ) =$ 0, 9, 16, and 144 degrees.

Let $f ( x , y ) = \tan ^ { - 1 } ( x / y )$ . Find the value of the partial derivative $f _ { y } ( 1,2 )$ .

Find the directional derivative of the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ at the point $( 1,1 )$ in the direction $\theta = \frac { \pi } { 4 }$ .

Determine how many critical points the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 2 x ^ { 2 } y + 3$ has.

Suppose that the quantity $Q$ produced of a certain good depends on the number of units of labor $L$ and the quantity of capital $K$ according to the function $Q = 900 L ^ { 1 / 3 } K ^ { 2 / 3 }$ . Suppose also that labor costs $100 per unit and that capital costs $200 per unit.(a) What combination of labor and capital should be used to produce 36,000 units of the goods at minimum cost? What is that minimum cost? (b) Draw the level curves of $C$ , the cost function, and the graph of the constraint on the same set of axes.(c) Complete the value of $\lambda$ . What does $\lambda$ represent?

Let $w = x ^ { 3 } + y ^ { 3 } + z ^ { 3 }$ , $x = s + t$ , $y = s ^ { 2 } - t ^ { 2 }$ and $z = s t$ . Use the chain rule to show that $s \left( \frac { \delta w } { \delta s } \right) + t \left( \frac { \delta w } { \delta t } \right) = 3 x ^ { 3 } + 6 y ^ { 3 } + 6 z ^ { 3 }$ .

Consider a function of three variables $P = f ( A , r , N )$ , where $P$ is the monthly mortgage payment in dollars, $A$ is the amount borrowed in dollars, $r$ is the annual interest rate, and $N$ is the number of years before the mortgage is paid off.(a) Suppose $f ( 180,000,6,30 ) = 1080$ . What does this tell you in financial terms? (b) Suppose $\frac { \delta f } { \delta r } ( 180,000,6,30 ) = 115.73$ . What does this tell you in financial terms? (c) Suppose $\frac { \delta f } { \delta N } ( 180,000,6,30 ) = - 12.86$ . What does this tell you in financial terms?

Find the range of the function $f ( x , y ) = e ^ { x - y ^ { 2 } }$ .

If $w = f ( x , y , z )$ has continuous partial derivatives, $x = s + t$ , $y = s ^ { 2 } - t ^ { 2 }$ , and $z = s t$ , show that $s \frac { \delta w } { \delta s } + t \frac { \delta w } { \delta t } = x \frac { \delta w } { \delta x } + 2 y \frac { \delta w } { \delta y } + 2 z \frac { \delta w } { \delta z }$

Let $x ^ { 3 } y + y ^ { 3 } z + z ^ { 3 } x = 4$ , find $\frac { \delta z } { \delta y }$ .

Find the point(s) on the surface $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 1$ such that the normal vector at the point is parallel to the vector $\{ 1,1 , - 1 \}$ .

Find all solutions to the partial differential equation $\frac { \delta ^ { 2 } f } { \delta y \delta x } = 0$ .

Let $f ( x , y ) = x ^ { 2 } e ^ { y }$ and $P ( - 2,0 )$ .(a) Find the rate of change in the direction of $\nabla f ( P )$ .(b) Calculate $D _ { u } f ( P )$ where $\mathbf { u }$ is a unit vector making an angle $\theta = 135 ^ { \circ }$ with $\nabla f ( P )$ .

Let $f ( x , y ) = e ^ { 2 x + y ^ { 2 } }$ . Find the value of the partial derivative $f _ { x yy } ( 0,0 )$ .

Let $f ( x , y ) = \sqrt { 4 - y ^ { 2 } }$ .(a) Evaluate $f ( 3,0 )$ .(b) Sketch the domain of $f$ .(c) What is the range of the function $f$ ?

Find $\delta z / \delta x$ and $\delta z/\delta y$ given that $\delta z / \delta x$ is defined implicitly as a function of $x$ and $y$ by the equation $\sin x y z + \ln \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) = 0$ .

The function $z = x y + ( x + y ) ( 120 - x - y )$ has a maximum. Find the values of $x$ and $y$ at which it occurs.