Exam 14: Differentiating Functions of Several Variables

Suppose that $T ( x , y ) = x ^ { 2 } + 3 y ^ { 2 } - x$ is the temperature (in degrees Celsius)at the point $( x , y )$ (where x and y are in meters).If you are standing at the point $( - 2,1 )$ and proceed in the direction of the point $( - 6 , - 3 )$ , will the temperature be increasing or decreasing at the moment you begin? At what rate? Give your answer to 4 decimal places.

Sally is on a day hike at Mt.Baker.From 9 to 11:00 a.m.she zig-zags up z = f(x, y)where x is the number of miles due east of her starting position, y is the number of miles due north of her starting position, and z is her elevation in miles above sea level.Feeling tired, she decides to continue walking, but in such a way that her altitude remains constant from 11 a.m.to noon to settle her stomach for lunch.At 11:30 a.m., she will be passing through (2, -1, 5)where f_x(2, -1)= 3 and f_y(2, -1)= -2. What is the slope of her "path" in the x, y plane at this instant? (This "path" is among the level curves in the plane.)

The ideal gas law states that $P V = R T$ for a fixed amount of gas, called a mole of gas, where P is the pressure (in atmospheres), V is the volume (in cubic meters), T is the temperature (in degrees Kelvin)and R is a positive constant. A mole of a certain gas is at a temperature of 290° K, a pressure of 1 atmosphere, and a volume of 0.04 m³. What is $\partial P / \partial T$ for this gas?

Calculate the following derivative: $\frac { \partial } { \partial P } \left( P \ln \left( V ^ { 5 } - P \right) \right)$ .

Let f be a differentiable function with local linearization L(x, y)= -1 + 4(x - 4)- 2(y - 2)at (4, 2).Evaluate f(4, 2).

In a neighborhood clinic, the number of patient visits can be described as a function of the number of doctors, x, and the number of nurses, y, by $f ( x , y ) = 1000 x ^ { 0.6 } y ^ { 03 }$ .With upcoming budget cuts, the clinic must reduce the number of doctors at the rate of 2 per month.Estimate the rate at which the number of nurses has to be increased in order to maintain the current patient load.Currently there are 30 doctors and 50 nurses.

Suppose that $\operatorname { grad } f ( 1 , - 1 ) = \vec { v }$ , with $\| \vec { v } \| = 4$ (a)What is the directional derivative of f at (1, -1)in the direction of $3 \vec { v }$ ? (b)What is the smallest value of the directional derivative of f at (1, -1)among all possible directions?

Let $f ( \theta , \phi ) = \sin ^ { 2 } 2 \theta \cos ^ { 3 } 2 \varphi$ .Find $\left. \frac { \partial f } { \partial \theta } \right| _ { ( \theta = \pi / 6 , \varphi = \pi / 8 ) }$ to 3 decimal places.

Below is a contour diagram for f(x, y)which is defined and continuous everywhere.The z-values have been omitted.Explain why it is true that $f _ { i } ( 1,1 ) = f _ { - 3 } ( 1,1 )$ .

Let $S ( x , y , z )$ represent the surface area of a rectangular solid of length x, width y, and height z.Find $\partial S / \partial z$ and explain its meaning in terms of how the solid changes with respect to z.

Let $V ( x , y , z )$ represent the volume of a rectangular solid of length x, width y, and height z.Find $\partial V / \partial x$ and explain its meaning in terms of how the solid changes with respect to x.

The level curves of a function z = f(x, y)are shown below.Assume that the scales along the x and y axes are the same. $f _ { y } ( \mathrm { Q } ) > f _ { y } ( \mathrm { P } )$

For the function $f ( x , y ) = x ^ { 2 } y ^ { 3 }$ find a unit vector in the direction of the steepest increase at the point (a, b)= (1, 1).

Suppose || $\nabla$ f(a, b, c)||=19.Is it possible to choose a direction from (a, b, c)so that $f _ { \vec { u } }$ in that direction is -19?

The monthly mortgage payment in dollars, P, for a house is a function of three variables P = f(A, r, N), where A is the amount borrowed in dollars, r is the interest rate, and N is the number of years before the mortgage is paid off.It is given that: f(100000,7,20)=775.29, f(100000,8,20)=836.44, f(100000,7,25)=706.77 f(120000,7,20)=930.35, f(120000,8,20)=1003.72, f(120000,7,25)=848.13 Estimate the value of $\left. \frac { \partial f } { \partial A } \right| _ { ( 10000,7,20 ) }$ and interpret your answer in terms of a mortgage payment.Select all answers that apply.

Let $z = \ln \left( x ^ { 2 } + 2 x y + 2 y \right) , x = t ^ { 2 } + 3 , y = t + \sqrt { t }$ Find dz/dt at t = 1 using the chain rule.Give your answer to 4 decimal places.

The equations of the tangent planes to the graph z = f(x, y)at the points (0, -2), (2, 1)are $z - 3 x + 2 y = 5$ and $1 = 2 x + 5 y - z$ , respectively.Determine the value of $f ( 0 , - 2 )$ State whether the value you find is exact or an approximation.

Suppose the graph of $z = f ( x , y )$ contains the point $P ( 2 , - 3,3 )$ , and that the tangent plane at P is given by $3 x + 2 y - 5 z = - 15$ .Find $f _ { x } ( 2 , - 3 )$ and $f _ { y } ( 2 , - 3 )$ .

Suppose that there exists a linear function $L ( x , y )$ near the point (0,0)such that the difference between $f ( x , y )$ and $L ( x , y )$ approaches 0 as $( x , y )$ approaches (0,0).Then f is differentiable at (0,0)with $L ( x , y )$ as its local linearization.

The volume $V = \pi r ^ { 2 } h$ of a right circular cylinder is to be calculated from measured values of r and h.Suppose r is measured with an error of no more than 2.5% and h with an error of no more than 1%.Using differentials, estimate the percentage error in the calculation of V. (In general, in measuring a quantity Q, the percentage error is dQ/Q.)