Exam 14: Differentiating Functions of Several Variables

Let $\vec { u } = \frac { 1 } { \sqrt { 2 } } ( \vec { i } + \vec { j } )$ and let $\vec { w } = \frac { 1 } { \sqrt { 13 } } ( 2 \vec { i } + 3 \vec { j } )$ .Suppose that $f _ { t } ( 1,1 ) = 4 \sqrt { 2 }$ and $f _ { t } ( 1,1 ) = 3$ .Find $f _ { \mathfrak { w } } ( 1,1 )$ .

Determine the tangent plane to $z = f ( x , y ) = 3 e ^ { x - 2 y }$ at (x, y)= (2, 1).

If f_x(0, 0)exists and f_y(0, 0)exists, then f is differentiable at (0, 0).

Consider the function $g ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ (a)Describe the level set g = 16. (b)Find a vector perpendicular to the tangent plane to the level set g = 16 at the point (-1, 2, 2).

If $z ( x , y ) = f \left( x ^ { 2 } - y ^ { 2 } \right)$ , then simplify $y \frac { \partial z } { \partial x } + x \frac { \partial z } { \partial y }$ .

$\operatorname { grad } f ( a , b )$ is perpendicular to the graph of $z = f ( x , y )$ at the point $( a , b , f ( a , b ) )$ .

If $\vec { u }$ is a unit vector and the level curves of f(x, y)are given below, then at point P we have $f_{u}(P)=\operatorname{grad} f .$

Consider the function $g ( x , y ) = x ^ { \frac { 1 } { 5 } } y ^ { \frac { 1 } { 5 } }$ (a)Find g_x(x, y)and g_y(x, y)for (x, y) $\neq$ (0, 0). (b)Use the limit definition of partial derivative to show that g_x(0, 0)= 0 and g_y(0, 0)= 0. (c)Are the functions g_x and g_y continuous at (0, 0)? Explain. (d)Is g differentiable at (0, 0)? Explain.

Let $z = f ( x , y ) = e ^ { ( x / y ) }$ .Find the tangent plane to f at the point $( 5,5 )$ .

If $\frac { \partial f } { \partial x } = \frac { \partial f } { \partial y }$ everywhere, then f(x, y)is a constant.

Given that f(2, 4)= 1.5 and f(2.1, 4.4)= 2.1, estimate the value of $f _ { u } ( 2,4 )$ , where $\vec { u }$ is the unit vector in the direction of $1 \vec { i } + 4 \vec { j }$ Give your answer to four decimal places.

The figure below shows the graph of z = f(x, y)and its intersection with various planes.(The x and y-axes have the same scale.) What is the sign of $f _ { y } ( 0,1 )$ ?

Using the contour diagram for f(x, y), find the sign of $f _ { y y } ( P )$ given that f_xx(P)< 0.

Let $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ .Then $\text { grad } f ( 0,0,0 ) = \overrightarrow { 0 }$ .

Let $f ( x , y ) = x ^ { 2 } + 2 x \cos ^ { 2 } y$ .Give a linear approximation of f at the point $( 2 , \pi / 4 )$ .

Find an equation for the tangent plane to the ellipsoid $( x - 1 ) ^ { 2 } + 4 ( y - 2 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 17$ at the point $( 3,1,0 )$ .

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

Consider the surface $z = x y$ and the point $P = ( - 4,3 , - 12 )$ .Find a vector normal to the surface at P.

There exists a function f(x, y)with f_x = 2y and f_y = 3x.

The table below gives values of a function f(x, y)near x = 1, y = 2. Give the equation of the tangent plane to the graph z = f(x, y)at x = 1, y = 2.