Exam 14: Differentiating Functions of Several Variables

If $g ( x , y ) = x ^ { 2 } \cos ( x y )$ , then find $\frac { \partial g } { \partial x } \left( 6 , \frac { \pi } { 6 } \right)$ .

What is the z-coordinate of the point $P ( 1,3 , z )$ if P lies on the plane which is tangent to the ellipsoid $4 x ^ { 2 } + y ^ { 2 } + 9 z ^ { 2 } = 17$ at the point $( - 1 , - 2 , - 1 )$ ? Give your answer to four decimal places.

Find the gradient of the function $f ( x , y ) = x ^ { 2 } y ^ { 3 }$ at the point $( 1,1 )$ and use the result to obtain a linear approximation for $1.04 ^ { 2 } 0.98 ^ { 3 }$

Find $\left.\frac{\partial H}{\partial w}\right|_{(\pi / 4, \pi / 2)}$ to 2 decimal places if $H ( w , t ) = e ^ { \cos ( 2 w + 2 t ) }$ .

Let $f ( x , y ) = 3 y ^ { 2 } + 2 x y$ Use the appropriate partial derivative to find the slope of the cross-section at the given point. (a)The cross-section f(x, 2)at the point (3, 2). (b)The cross-section f(1, y)at the point (1, -2).

Given the contour diagram shown below, state whether $f _ { x } ( 0.5,1 )$ is positive, negative or nearly zero.

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. Find $\frac { \partial P } { \partial V }$

Suppose that as you move away from the point (-1, -1, -1), the function $f ( x , y , z )$ increases most rapidly in the direction $0.6 \vec { i } + 0.8 \vec { j }$ and the rate of increase of f in this direction is 4.At what rate is f increasing as you move away from (-1, -1, -1)in the direction of $\vec { i } + \vec { j } + \vec { k }$ ? Give your answer to 4 decimal places.

If $f ( x , y ) = 2 x ^ { 2 } + 2 y ^ { 2 }$ and $x = e ^ { u } , y = u e ^ { w }$ find f_w(1, 1)using the chain rule.Give your answer to 4 decimal places.

An ant is walking along the surface which is the graph of the function $f ( x , y ) = x ^ { 2 } - \sin ( y )$ (a)When the ant is at the point (1, 0, 1), what direction should it move in order to be moving on the surface in the direction of greatest ascent? (b)If the ant moves in this direction at a speed of 6 units per second, what is the rate of change of height of the ant?

Suppose that the temperature at the point $( x , y , z )$ is given by $T ( x , y , z ) = e ^ { - z ^ { 2 } } \sin ( x y )$ .If you are at the point $( 1,2 \pi , 0 )$ , in which direction should you go to decrease your temperature the fastest?

Let $f(x, y)=y^{2}+3 y \int_{3}^{x} \sin \left(e^{t}\right) d t$ Find $f _ { x } ( 3,1 )$ and $f _ { y } ( 3,1 )$ to four decimal places.

The table of some values of f(x, y)is given below.Find a local linearization of f at (1 , 2).

Suppose that the function $f ( x , y )$ and the linear function $L ( x , y ) = 3 - 3 x + 4 y$ satisfy $| f ( x , y ) - L ( x , y ) | \leq 5 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 }$ for points $( x , y )$ close to $( 0,0 )$ .Is f differentiable at (0,0)?

If you know the directional derivative of f(x, y)in two distinct directions (i.e.not including opposite directions)at a point P then you can find $\frac { \partial f } { \partial x } ( P )$

The consumption of beef, C (in pounds per week per household)is given by the function C = f(I, p), where I is the household income in thousands of dollars per year, and p is the price of beef in dollars per pound.Explain the meaning of the statement: $f _ { \mathrm { p } } ( 80,3 ) = - 1.6$ , and include units in your answer.

A rectangular beam, supported at its two ends, will sag when subjected to a uniform load.The amount of sag is calculated from the formula: $S = C p x ^ { 4 } / w h ^ { 3 }$ , where p is the load (in Newtons per meter), x is the length between supports (in meters), w is the width of the beam (in meters), h is the height of the beam (in meters), and C is a constant (depending on material and units of measurement used).Determine $\partial S / \partial h$ for a beam 3 m long, 0.1 m wide, 0.2 m high subjected to a load of 80 N/m.

Find $f _ { H }$ if $f ( H , T ) = \frac { 5 H + T } { ( 3 - H ) ^ { 3 } }$ .

If $P is invested in a bank account earning r% interest a year, compounded continuously, the balance, $B, at the end of t years is given by $B=f(P, r, t)=P e^{n / 100}$ Find $\partial B / \partial P$

Let w = 3x cos 4y.If $x = e ^ { - t }$ and $y = \ln t _ { 1 }$ find $\frac { d w } { d t }$ at the point t = 3.Give your answer to 2 decimal places.