Exam 14: Differentiating Functions of Several Variables

Let w = 3x cos y.If $x = u ^ { 2 } + v ^ { 2 }$ $y=v / \mathcal u_{1}$ find $\partial$ w/ $\partial$ u and $\partial$ w/ $\partial$ v at the point $( u , v ) = ( 2,3 )$ .Give your answers to 2 decimal places.

Let $f ( x , y ) = x ^ { 2 } + 3 y ^ { 2 } - 3 x$ Find the gradient vector of f at the point (-1, 2).

Find the differential of the function $f ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ at the point (3, 4). A point is measured to be 3 units from the y-axis with an error of ±0.01 and 4 units from the x-axis with an error of ±0.02.Approximate the error in computing its distance from the origin.

Consider the surface given as the graph of $z = f ( x , y )$ .Let $\vec { u } = \frac { 1 } { \sqrt { 26 } } ( \vec { i } + 5 \vec { j } )$ and let $\vec { v } = \frac { 1 } { \sqrt { 20 } } ( 4 \vec { i } - 2 \vec { j } )$ .Suppose that $f_{0}(1,3)=\frac{21}{\sqrt{26}} ; f_{0}(1,3)=\frac{-4}{\sqrt{20}} ; f(1,3)=-7$ .Find the equation of the tangent plane to the surface at the point $( 1,3 )$ .

Consider the surface $z = x y$ and the point $P = ( - 4,3 , - 12 )$ .Find an equation for the plane tangent to the surface at P.

Find the following partial derivative: H_P(2, 1)if $H ( P , T ) = \frac { 2 P } { 3 P + T }$ Give your answer to 4 decimal places.

Suppose that as you move away from the point (2, 0, 2), the function 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 7.At what rate is f increasing as you move away from (2, 0, 2)in the direction of $\vec { i } + \vec { j } + \vec { k }$ ? Give your answer to 4 decimal places.

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

The quadratic Taylor approximations of $f ( x , y )$ at the points $( - 1 - 1 ) , ( 1,1 )$ , (2,0)and (0,2)are given respectively by: (x,y)=-3+3(x+1+4(y+1 (x,y)=1-3(x-1+4(y-1 (x,y)=-2-9(x-2)-6(x-2-2 (x,y)=8+3x+24(y-2)+22(y-2 Find an approximate value of $f ( 0.2,1.9 )$ .

Find $\partial$ z/ $\partial$ x if $z = - 3 \ln x + \sin \left( x y ^ { 5 } \right)$

Let $f ( x , y ) = 4 x ^ { 2 } - 9 y ^ { 2 }$ .What is the equation of the tangent plane to the graph of $z = f ( x , y )$ at the point $( 5,1 )$ ?

If $\text { grad } f = \operatorname { grad } g$ , then $f = g$ .

Suppose f(x, y)is a function of x and y and define $g ( u , v ) = f \left( e ^ { u } + \cos v , e ^ { u } + \sin ( v ) \right)$ Find $g _ { u } ( 0,0 )$ given that $f ( 0,0 ) = 5 , f ( 2,1 ) = - 1 , f _ { x } ( 0,0 ) = 2 , f _ { x } ( 1,2 ) = 1 , f _ { x } ( 2,1 ) = - 3 \text {, }$ and $g ( 0,0 ) = - 1 , g ( 2,1 ) = - 1 , f _ { y } ( 0,0 ) = 2 , f _ { y } ( 2,1 ) = - 3 , f ( 1,2 ) = 7$

The table below gives values of a function f(x, y)near x = 1, y = 2. Estimate $f _ { x } ( 0,1.5 )$ .

Suppose that $f ( x , y ) = x ^ { 2 } e ^ { y }$ Find an equation for the tangent plane to f at the point (3, 0).

Find the equation of the tangent plane to $x ^ { 2 } + 2 x y + 4 y + 6 = z ^ { 2 }$ at the point (-4, 1, 3).

Let $f ( x , y ) = x ^ { 2 } + 3 y ^ { 2 } - 4 x$ What is the maximum rate of change of f at (2, 1)?

Find an equation for the tangent plane to the graph of $f ( x , y ) = e ^ { x \sin y }$ at $( x , y ) = ( 2,6 \pi )$

Estimate the value of $f _ { y } ( 1,0 )$ from the given contour diagram of f.

Find the points on the ellipsoid $x ^ { 2 } + 2 y ^ { 2 } + z ^ { 2 } = 7$ where the tangent plane is parallel to the plane $- 2 x + 2 y - z = 0$ .