Exam 11: Partial Derivatives

If $z = f ( x , y )$ has continuous second partial derivatives, $x = 2 u + v$ , and $y = u - 3 v$ , express $\frac { \delta ^ { 2 } z } { \delta u \delta v }$ in terms of $\frac { \delta ^ { 2 } z } { \delta x ^ { 2 } }$ , $\frac { \delta ^ { 2 } z } { \delta x \delta y }$ , and $\frac { \delta ^ { 2 } z } { \delta y ^ { 2 } }$ .

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Question 1

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$\frac { \delta ^ { 2 } z } { \delta u \delta v } = 2 \frac { \delta ^ { 2 } z } { \delta x ^ { 2 } } - 5 \frac { \delta ^ { 2 } z } { \delta x \delta y } - 3 \frac { \delta ^ { 2 } z } { \delta y ^ { 2 } }$

Find $\frac { \delta z } { \delta x }$ for $x ^ { 3 } - y + x z ^ { 3 } = 0$

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Question 2

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$\frac { \delta z } { \delta x } = \frac { - \left( 3 x ^ { 2 } + z ^ { 3 } \right) } { 3 x z ^ { 2 } }$

Find the linear approximation to the function $f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 }$ at $( 1,1 )$ and use it to approximate $f ( 1.1,1.1 )$ .

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Question 3

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$z = 4 x + 2 y - 3 , f ( 1.1,1.1 ) \approx 3.6$

A cardboard box without a lid is to have volume 100 cubic inches, with total area of cardboard as small as possible. Find its height in inches.

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Question 4

Solve completely, using Lagrange multipliers: Find the dimension of a box with volume 1000 which minimizes the total length of the 12 edges.

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Question 5

Find an equation of the tangent plane to the surface $4 x ^ { 2 } - y ^ { 2 } + 3 z ^ { 2 } = 10$ at the point $( 2 , - 3,1 )$ .

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Question 6

Find the point(s) on the surface $x ^ { 2 } + y ^ { 2 } + 4 z ^ { 2 } = 6$ such that the tangent plane at the point is parallel to the plane $x + 2 y - 2 z = 4$

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Question 7

Let $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ , $x = s ^ { 2 } - t ^ { 2 }$ , and $y = 2 s t$ . Use the chain rule to find $\frac { \delta z } { \delta s }$ and $\frac { \delta z } { \delta t }$ .

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Question 8

The surface of a certain lake is represented by a region in the $x y$ -plane such that the depth under the point corresponding to $( x , y )$ is $f ( x , y ) = 300 - 2 x ^ { 2 } - 3 y ^ { 2 }$ . Zeke the dog is at the point $( 3,2 )$ .(a) In what direction should Zeke swim in order for the depth to decrease most rapidly? (b) In what direction would the depth remain the same?

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Question 9

Let $f ( x , y ) = \left\{ \begin{array} { l l } \frac { x ^ { 4 } - y ^ { 4 } } { x ^ { 2 } + y ^ { 2 } } & \text { if } ( x , y ) \neq ( 0,0 ) \\0 & \text { if } ( x , y ) = ( 0,0 )\end{array} \right.$ Where is $f$ continuous?

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Question 10

A rancher with 300 ft of fence intends to enclose a rectangular corral, dividing it in half by a fence parallel to the short sides of the corral. What is the maximum area he can enclose? Compute the value of $\lambda$ . What does $\lambda$ represent?

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Question 11

A company estimates that its annual profits for next year will be $z = 6 x y - y ^ { 3 } - x ^ { 2 }$ , where $x$ represents investment in research and $y$ represents investment in labor. All units are in tens of thousands of dollars. Determine the amount the company should spend on research and on labor to maximize its profit. What is maximum profit?

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Question 12

Find the differential of $w = x e ^ { y \sin x }$ at the point $( x , y , z ) = ( 1,1,0 )$ .

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Question 13

Describe the level curves of the function $f ( x , y ) = \frac { y } { x ^ { 2 } }$ .

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Question 14

Let $f ( x , y ) = x ^ { 2 } - 2 y$ .(a) Sketch the level curves for $f$ .(b) Find a formula for the level curve that passes through the point $( 3,1 )$ .

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Question 15

If $z = x ^ { 2 } \sin y + y e ^ { x }$ , find $\frac { \delta z } { \delta x } , \frac { \delta z } { \delta y } , \frac { \delta ^ { 2 } z } { \delta x ^ { 2 } } , \frac { \delta ^ { 2 } z } { \delta x \delta y } \text {, and } \frac { \delta ^ { 2 } z } { \delta y ^ { 2 } }$

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Question 16

Let $z = x y$ , and let $x$ and $y$ be functions of $t$ with $x ( 1 ) = 1 , y ( 1 ) = 2 , x ^ { \prime } ( 1 ) = 3 \text {, and } y ^ { \prime } ( 1 ) = 4$ . Find $d z / d t$ when $t = 1$ .

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Question 17

Let $z = \ln \sqrt { x ^ { 2 } + y ^ { 2 } }$ , $x = r \cos \theta$ , and $y = r \sin \theta$ . Use the chain rule to find $\frac { \delta z } { \delta r }$ and $\frac { d z } { \delta \theta }$ .

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Question 18

Let the temperature in a flat plate be given by the function $T ( x , y ) = 3 x ^ { 2 } + 2 x y$ . What is the value of the directional derivative of this function at the point $( 3 - 6 )$ in the direction $\mathbf { v } = 4 \mathbf { i } - 3 \mathbf { j }$ ? In what direction is the plate cooling most rapidly at $( 3 , - 6 )$ ?

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Question 19

Find the directional derivative of the function $f ( x , y , z ) = \sqrt { x y z }$ at the point $( 2,4,2 )$ in the direction of the vector $\{ 4,2 , - 4 \}$ .

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Question 20

Find $\frac { \delta z } { \delta x }$ for $x ^ { 3 } - y + x z ^ { 3 } = 0$

Find the linear approximation to the function $f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 }$ at $( 1,1 )$ and use it to approximate $f ( 1.1,1.1 )$ .

A cardboard box without a lid is to have volume 100 cubic inches, with total area of cardboard as small as possible. Find its height in inches.

Solve completely, using Lagrange multipliers: Find the dimension of a box with volume 1000 which minimizes the total length of the 12 edges.

Find an equation of the tangent plane to the surface $4 x ^ { 2 } - y ^ { 2 } + 3 z ^ { 2 } = 10$ at the point $( 2 , - 3,1 )$ .

Find the point(s) on the surface $x ^ { 2 } + y ^ { 2 } + 4 z ^ { 2 } = 6$ such that the tangent plane at the point is parallel to the plane $x + 2 y - 2 z = 4$

Let $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ , $x = s ^ { 2 } - t ^ { 2 }$ , and $y = 2 s t$ . Use the chain rule to find $\frac { \delta z } { \delta s }$ and $\frac { \delta z } { \delta t }$ .

Let $f ( x , y ) = \left\{ \begin{array} { l l } \frac { x ^ { 4 } - y ^ { 4 } } { x ^ { 2 } + y ^ { 2 } } & \text { if } ( x , y ) \neq ( 0,0 ) \\0 & \text { if } ( x , y ) = ( 0,0 )\end{array} \right.$ Where is $f$ continuous?

A rancher with 300 ft of fence intends to enclose a rectangular corral, dividing it in half by a fence parallel to the short sides of the corral. What is the maximum area he can enclose? Compute the value of $\lambda$ . What does $\lambda$ represent?

Find the differential of $w = x e ^ { y \sin x }$ at the point $( x , y , z ) = ( 1,1,0 )$ .

Describe the level curves of the function $f ( x , y ) = \frac { y } { x ^ { 2 } }$ .

Let $f ( x , y ) = x ^ { 2 } - 2 y$ .(a) Sketch the level curves for $f$ .(b) Find a formula for the level curve that passes through the point $( 3,1 )$ .

If $z = x ^ { 2 } \sin y + y e ^ { x }$ , find $\frac { \delta z } { \delta x } , \frac { \delta z } { \delta y } , \frac { \delta ^ { 2 } z } { \delta x ^ { 2 } } , \frac { \delta ^ { 2 } z } { \delta x \delta y } \text {, and } \frac { \delta ^ { 2 } z } { \delta y ^ { 2 } }$

Let $z = x y$ , and let $x$ and $y$ be functions of $t$ with $x ( 1 ) = 1 , y ( 1 ) = 2 , x ^ { \prime } ( 1 ) = 3 \text {, and } y ^ { \prime } ( 1 ) = 4$ . Find $d z / d t$ when $t = 1$ .

Let $z = \ln \sqrt { x ^ { 2 } + y ^ { 2 } }$ , $x = r \cos \theta$ , and $y = r \sin \theta$ . Use the chain rule to find $\frac { \delta z } { \delta r }$ and $\frac { d z } { \delta \theta }$ .

Find the directional derivative of the function $f ( x , y , z ) = \sqrt { x y z }$ at the point $( 2,4,2 )$ in the direction of the vector $\{ 4,2 , - 4 \}$ .

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Exam 11: Partial Derivatives

Find δzδx\frac { \delta z } { \delta x }δxδz​ for x3−y+xz3=0x ^ { 3 } - y + x z ^ { 3 } = 0x3−y+xz3=0

Find the linear approximation to the function f(x,y)=2x2+y2f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 }f(x,y)=2x2+y2 at (1,1)( 1,1 )(1,1) and use it to approximate f(1.1,1.1)f ( 1.1,1.1 )f(1.1,1.1) .

A cardboard box without a lid is to have volume 100 cubic inches, with total area of cardboard as small as possible. Find its height in inches.

Solve completely, using Lagrange multipliers: Find the dimension of a box with volume 1000 which minimizes the total length of the 12 edges.

Find an equation of the tangent plane to the surface 4x2−y2+3z2=104 x ^ { 2 } - y ^ { 2 } + 3 z ^ { 2 } = 104x2−y2+3z2=10 at the point (2,−3,1)( 2 , - 3,1 )(2,−3,1) .

Find the point(s) on the surface x2+y2+4z2=6x ^ { 2 } + y ^ { 2 } + 4 z ^ { 2 } = 6x2+y2+4z2=6 such that the tangent plane at the point is parallel to the plane x+2y−2z=4x + 2 y - 2 z = 4x+2y−2z=4

Let z=x2+y2z = \sqrt { x ^ { 2 } + y ^ { 2 } }z=x2+y2​ , x=s2−t2x = s ^ { 2 } - t ^ { 2 }x=s2−t2 , and y=2sty = 2 s ty=2st . Use the chain rule to find δzδs\frac { \delta z } { \delta s }δsδz​ and δzδt\frac { \delta z } { \delta t }δtδz​ .

A rancher with 300 ft of fence intends to enclose a rectangular corral, dividing it in half by a fence parallel to the short sides of the corral. What is the maximum area he can enclose? Compute the value of λ\lambdaλ . What does λ\lambdaλ represent?

Find the differential of w=xeysin⁡xw = x e ^ { y \sin x }w=xeysinx at the point (x,y,z)=(1,1,0)( x , y , z ) = ( 1,1,0 )(x,y,z)=(1,1,0) .

Describe the level curves of the function f(x,y)=yx2f ( x , y ) = \frac { y } { x ^ { 2 } }f(x,y)=x2y​ .

Let f(x,y)=x2−2yf ( x , y ) = x ^ { 2 } - 2 yf(x,y)=x2−2y .(a) Sketch the level curves for fff .(b) Find a formula for the level curve that passes through the point (3,1)( 3,1 )(3,1) .

Let z=xyz = x yz=xy , and let xxx and yyy be functions of ttt with x(1)=1,y(1)=2,x′(1)=3, and y′(1)=4x ( 1 ) = 1 , y ( 1 ) = 2 , x ^ { \prime } ( 1 ) = 3 \text {, and } y ^ { \prime } ( 1 ) = 4x(1)=1,y(1)=2,x′(1)=3, and y′(1)=4 . Find dz/dtd z / d tdz/dt when t=1t = 1t=1 .

Let z=ln⁡x2+y2z = \ln \sqrt { x ^ { 2 } + y ^ { 2 } }z=lnx2+y2​ , x=rcos⁡θx = r \cos \thetax=rcosθ , and y=rsin⁡θy = r \sin \thetay=rsinθ . Use the chain rule to find δzδr\frac { \delta z } { \delta r }δrδz​ and dzδθ\frac { d z } { \delta \theta }δθdz​ .

Find the directional derivative of the function f(x,y,z)=xyzf ( x , y , z ) = \sqrt { x y z }f(x,y,z)=xyz​ at the point (2,4,2)( 2,4,2 )(2,4,2) in the direction of the vector {4,2,−4}\{ 4,2 , - 4 \}{4,2,−4} .

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Multiple Integrals

Vector Calculus

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Find $\frac { \delta z } { \delta x }$ for $x ^ { 3 } - y + x z ^ { 3 } = 0$

Find the linear approximation to the function $f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 }$ at $( 1,1 )$ and use it to approximate $f ( 1.1,1.1 )$ .

Find an equation of the tangent plane to the surface $4 x ^ { 2 } - y ^ { 2 } + 3 z ^ { 2 } = 10$ at the point $( 2 , - 3,1 )$ .

Find the point(s) on the surface $x ^ { 2 } + y ^ { 2 } + 4 z ^ { 2 } = 6$ such that the tangent plane at the point is parallel to the plane $x + 2 y - 2 z = 4$

Let $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ , $x = s ^ { 2 } - t ^ { 2 }$ , and $y = 2 s t$ . Use the chain rule to find $\frac { \delta z } { \delta s }$ and $\frac { \delta z } { \delta t }$ .

A rancher with 300 ft of fence intends to enclose a rectangular corral, dividing it in half by a fence parallel to the short sides of the corral. What is the maximum area he can enclose? Compute the value of $\lambda$ . What does $\lambda$ represent?

Find the differential of $w = x e ^ { y \sin x }$ at the point $( x , y , z ) = ( 1,1,0 )$ .

Describe the level curves of the function $f ( x , y ) = \frac { y } { x ^ { 2 } }$ .

Let $f ( x , y ) = x ^ { 2 } - 2 y$ .(a) Sketch the level curves for $f$ .(b) Find a formula for the level curve that passes through the point $( 3,1 )$ .

Let $z = x y$ , and let $x$ and $y$ be functions of $t$ with $x ( 1 ) = 1 , y ( 1 ) = 2 , x ^ { \prime } ( 1 ) = 3 \text {, and } y ^ { \prime } ( 1 ) = 4$ . Find $d z / d t$ when $t = 1$ .

Let $z = \ln \sqrt { x ^ { 2 } + y ^ { 2 } }$ , $x = r \cos \theta$ , and $y = r \sin \theta$ . Use the chain rule to find $\frac { \delta z } { \delta r }$ and $\frac { d z } { \delta \theta }$ .

Find the directional derivative of the function $f ( x , y , z ) = \sqrt { x y z }$ at the point $( 2,4,2 )$ in the direction of the vector $\{ 4,2 , - 4 \}$ .