Question 1

Find the limit. $\lim _ { ( x , y ) \rightarrow ( 8,4 ) } x y \cos ( x - 2 y )$

Accepted Answer

The answer of Find the limit. \(\lim _ { (...

Question 2

If $f ( x , y ) = x ^ { 2 } + 7 y ^ { 2 }$ use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 136$ at the point $( 10,2 )$ .

Accepted Answer

A)  $10 x + 14 y = 12$ 
B)  $10 x + 14 y = 136$ 
C)  $100 x - 49 y = 12$ 
D)  $100 x + 49 y = 136$ 
E)  $100 x - 14 y = 12$ 
A)  $10 x + 14 y = 12$ 
B)  $10 x + 14 y = 136$ 
C)  $100 x - 49 y = 12$ 
D)  $100 x + 49 y = 136$ 
E)  $100 x - 14 y = 12$

Question 3

Find the differential of the function $z = 3 x ^ { 3 } y ^ { 6 }$

Accepted Answer

A)  $d z = 9 x ^ { 2 } y ^ { 5 } d x + 18 x ^ { 2 } y ^ { 5 } d y$ 
B)  $d z = 9 x ^ { 2 } y ^ { 6 } d x + 18 x ^ { 3 } y ^ { 5 } d y$ 
C)  $d z = 18 x ^ { 2 } y ^ { 5 } d x + 9 x ^ { 2 } y ^ { 5 } d y$ 
D)  $d z = 18 x ^ { 3 } y ^ { 5 } d x + 9 x ^ { 2 } y ^ { 6 } d y$ 
A)  $d z = 9 x ^ { 2 } y ^ { 5 } d x + 18 x ^ { 2 } y ^ { 5 } d y$ 
B)  $d z = 9 x ^ { 2 } y ^ { 6 } d x + 18 x ^ { 3 } y ^ { 5 } d y$ 
C)  $d z = 18 x ^ { 2 } y ^ { 5 } d x + 9 x ^ { 2 } y ^ { 5 } d y$ 
D)  $d z = 18 x ^ { 3 } y ^ { 5 } d x + 9 x ^ { 2 } y ^ { 6 } d y$

Question 4

Use partial derivatives to find the implicit partial derivatives $\frac { \partial z } { \partial x }$ and $\frac { \partial z } { \partial y }$ $2 x ^ { 2 } + 7 x y - 6 x ^ { 2 } z + 8 y z ^ { 2 } = 0$

Accepted Answer

The answer of Use partial derivatives to find the implicit...

Question 5

Sketch the graph of the function $g ( x , y ) = x ^ { 2 }$

Accepted Answer

The answer of Sketch the graph of the function \(g...

Question 6

Find and classify the relative extrema and saddle points of the function $f ( x , y ) = e ^ { - 2 x } \sin 4 y$ for $x \geq 0$ and $0 \leq y \leq \frac { \pi } { 2 }$ .

Accepted Answer

A)  None 
B)  Relative maximum $f \left( 0 , \frac { \pi } { 8 } \right) = 1$ 
C)  Saddle point $( 0,0,0 )$ 
D)  Relative minimum $f \left( 0 , \frac { \pi } { 4 } \right) = 0$ 
A)  None 
B)  Relative maximum $f \left( 0 , \frac { \pi } { 8 } \right) = 1$ 
C)  Saddle point $( 0,0,0 )$ 
D)  Relative minimum $f \left( 0 , \frac { \pi } { 4 } \right) = 0$

Question 7

Use spherical coordinates to find the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 6 x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$

Accepted Answer

The answer of Use spherical coordinates to find the limit....

Question 8

Find the domain and range of the function $h ( x , y ) = \sqrt { 2 x - 9 y }$ .

Accepted Answer

A)  $\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \geq \frac { 9 } { 2 } x \right\} \
R = \{ z \mid 0 < z \}
\end{array}$ 
B)  $\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \leq \frac { 2 } { 9 } x \right\} \
R = \{ z \mid 0 < z \}
\end{array}$ 
C)  $\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \leq \frac { 2 } { 9 } x \right\} \
R = \{ z \mid 0 \leq z \}
\end{array}$ 
D)  $\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \geq \frac { 9 } { 2 } x \right\} \
R = \{ z \mid 0 \leq z \}
\end{array}$ 
A)  $\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \geq \frac { 9 } { 2 } x \right\} \
R = \{ z \mid 0 < z \}
\end{array}$ 
B)  $\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \leq \frac { 2 } { 9 } x \right\} \
R = \{ z \mid 0 < z \}
\end{array}$ 
C)  $\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \leq \frac { 2 } { 9 } x \right\} \
R = \{ z \mid 0 \leq z \}
\end{array}$ 
D)  $\begin{array} { l } 
D = \left\{ ( x , y ) \mid y \geq \frac { 9 } { 2 } x \right\} \
R = \{ z \mid 0 \leq z \}
\end{array}$

Question 9

Find the absolute extrema of the function $f ( x , y ) = 2 x + 3 y - 5$ on the closed triangular region with vertices $( 0,0 )$ , $( 5,0 )$ and $( 5,4 )$ .

Accepted Answer

A)  Absolute minimum 0, Absolute maximum 5 
B)  Absolute minimum -5, Absolute maximum 5 
C)  Absolute minimum -5, Absolute maximum 17 
D)  Absolute minimum 5, Absolute maximum 17 
A)  Absolute minimum 0, Absolute maximum 5 
B)  Absolute minimum -5, Absolute maximum 5 
C)  Absolute minimum -5, Absolute maximum 17 
D)  Absolute minimum 5, Absolute maximum 17

Question 10

Find equations for the tangent plane and the normal line to the surface with equation $x y + y z + x z = 38$ at the point $P ( 2,4,5 )$

Accepted Answer

A)  $6 x + 7 y + 9 z = 76$ , $\frac { x - 2 } { 6 } = \frac { y - 4 } { 7 } = \frac { z - 5 } { 9 }$ 
B)  $9 x + 7 y + 6 z = 76$ , $\frac { x - 2 } { 9 } = \frac { y - 4 } { 7 } = \frac { z - 5 } { 6 }$ 
C)  $2 x + 4 y + 5 z = 76$ , $\frac { x - 6 } { 2 } = \frac { y - 7 } { 4 } = \frac { z - 9 } { 5 }$ 
D)  $2 x + 4 y + 5 z = 76$ , $\frac { x - 9 } { 2 } = \frac { y - 7 } { 4 } = \frac { z - 6 } { 5 }$ 
A)  $6 x + 7 y + 9 z = 76$ , $\frac { x - 2 } { 6 } = \frac { y - 4 } { 7 } = \frac { z - 5 } { 9 }$ 
B)  $9 x + 7 y + 6 z = 76$ , $\frac { x - 2 } { 9 } = \frac { y - 4 } { 7 } = \frac { z - 5 } { 6 }$ 
C)  $2 x + 4 y + 5 z = 76$ , $\frac { x - 6 } { 2 } = \frac { y - 7 } { 4 } = \frac { z - 9 } { 5 }$ 
D)  $2 x + 4 y + 5 z = 76$ , $\frac { x - 9 } { 2 } = \frac { y - 7 } { 4 } = \frac { z - 6 } { 5 }$

Question 11

Find the first partial derivatives of the function $z = x ^ { 6 } \sqrt { y + 4 }$ .

Accepted Answer

The answer of Find the first partial derivatives of the...

Question 12

Find an equation of the tangent plane to the given surface at the specified point. $z = \sqrt { 18 - x ^ { 2 } - 2 y ^ { 2 } } , ( 3,2 , - 4 )$

Accepted Answer

The answer of Find an equation of the tangent plane...

Question 13

Determine the largest set on which the function is continuous. $f ( x , y , z ) = \frac { x y z } { 6 x ^ { 2 } + 2 y ^ { 2 } - z }$

Accepted Answer

The answer of Determine the largest set on which the...

Question 14

Find the gradient of the function $f ( x , y , z ) = x ^ { 5 } e ^ { y } \sqrt { z }$ .

Accepted Answer

The answer of Find the gradient of the function \(f...

Question 15

If R is the total resistance of three resistors, connected in parallel, with resistances $R _ { 1 } , R _ { 2 } , R _ { 3 }$ , then $\frac { 1 } { R } = \frac { 1 } { R _ { 1 } } + \frac { 1 } { R _ { 2 } } + \frac { 1 } { R _ { 3 } }$ . If the resistances are measured in ohms as $R _ { 1 } = 30 \Omega , R _ { 2 } = 35 \Omega \text { and } R _ { 3 } = 50 \Omega$ , with a possible error of 0.8% in each case, estimate the maximum error in the calculated value of R.

Accepted Answer

A)  0.347 $ \Omega$ 
B)  0.476 $ \Omega$ 
C)  0.291 $ \Omega$ 
D)  0.098 $ \Omega$ 
E)  0.957 $ \Omega$ 
A)  0.347 $ \Omega$ 
B)  0.476 $ \Omega$ 
C)  0.291 $ \Omega$ 
D)  0.098 $ \Omega$ 
E)  0.957 $ \Omega$

Question 16

Use Lagrange multipliers to find the minimum value of the function subject to the given constraints. $f ( x , y , z , t ) = 5 x + 5 y + 5 z + 5 t , \quad 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + t ^ { 2 } \right) = 4$

Accepted Answer

The answer of Use Lagrange multipliers to find the minimum...

Question 17

Find the equation of the normal line to the given surface at the specified point. $z + 2 = x e ^ { y } \cos z , ( 4,0,0 )$

Accepted Answer

The answer of Find the equation of the normal line...

Question 18

Find the maximum rate of change of f at the given point. $f ( x , y , z ) = x ^ { 5 } y ^ { 3 } z ^ { 2 } , ( 1 , - 1,1 )$

Accepted Answer

The answer of Find the maximum rate of change of...

Question 19

Find the equation of the tangent plane to the given surface at the specified point. $z + 7 = x e ^ { y } \cos z , ( 7,0,0 )$

Accepted Answer

A)  $x + 7 y + z = 7$ 
B)  $x + 7 y - z = 7$ 
C)  $x + y - z = 7$ 
D)  $7 x + y - z = 7$ 
E)  $x + y - 7 z = 7$ 
A)  $x + 7 y + z = 7$ 
B)  $x + 7 y - z = 7$ 
C)  $x + y - z = 7$ 
D)  $7 x + y - z = 7$ 
E)  $x + y - 7 z = 7$

Question 20

Use partial derivatives to find the implicit partial derivatives $\frac { \partial z } { \partial x }$ and $\frac { \partial z } { \partial y }$ $2 x ^ { 2 } + 7 x y - 6 x ^ { 2 } z + 8 y z ^ { 2 } = 0$

Accepted Answer

The answer of Use partial derivatives to find the implicit...

Find the limit. $\lim _ { ( x , y ) \rightarrow ( 8,4 ) } x y \cos ( x - 2 y )$

If $f ( x , y ) = x ^ { 2 } + 7 y ^ { 2 }$ use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 136$ at the point $( 10,2 )$ .

Find the differential of the function $z = 3 x ^ { 3 } y ^ { 6 }$

Use partial derivatives to find the implicit partial derivatives $\frac { \partial z } { \partial x }$ and $\frac { \partial z } { \partial y }$ $2 x ^ { 2 } + 7 x y - 6 x ^ { 2 } z + 8 y z ^ { 2 } = 0$

Sketch the graph of the function $g ( x , y ) = x ^ { 2 }$

Find and classify the relative extrema and saddle points of the function $f ( x , y ) = e ^ { - 2 x } \sin 4 y$ for $x \geq 0$ and $0 \leq y \leq \frac { \pi } { 2 }$ .

Use spherical coordinates to find the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 6 x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$

Find the domain and range of the function $h ( x , y ) = \sqrt { 2 x - 9 y }$ .

Find the absolute extrema of the function $f ( x , y ) = 2 x + 3 y - 5$ on the closed triangular region with vertices $( 0,0 )$ , $( 5,0 )$ and $( 5,4 )$ .

Find equations for the tangent plane and the normal line to the surface with equation $x y + y z + x z = 38$ at the point $P ( 2,4,5 )$

Find the first partial derivatives of the function $z = x ^ { 6 } \sqrt { y + 4 }$ .

Find an equation of the tangent plane to the given surface at the specified point. $z = \sqrt { 18 - x ^ { 2 } - 2 y ^ { 2 } } , ( 3,2 , - 4 )$

Determine the largest set on which the function is continuous. $f ( x , y , z ) = \frac { x y z } { 6 x ^ { 2 } + 2 y ^ { 2 } - z }$

Find the gradient of the function $f ( x , y , z ) = x ^ { 5 } e ^ { y } \sqrt { z }$ .

Use Lagrange multipliers to find the minimum value of the function subject to the given constraints. $f ( x , y , z , t ) = 5 x + 5 y + 5 z + 5 t , \quad 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + t ^ { 2 } \right) = 4$

Find the equation of the normal line to the given surface at the specified point. $z + 2 = x e ^ { y } \cos z , ( 4,0,0 )$

Find the maximum rate of change of f at the given point. $f ( x , y , z ) = x ^ { 5 } y ^ { 3 } z ^ { 2 } , ( 1 , - 1,1 )$

Find the equation of the tangent plane to the given surface at the specified point. $z + 7 = x e ^ { y } \cos z , ( 7,0,0 )$

Use partial derivatives to find the implicit partial derivatives $\frac { \partial z } { \partial x }$ and $\frac { \partial z } { \partial y }$ $2 x ^ { 2 } + 7 x y - 6 x ^ { 2 } z + 8 y z ^ { 2 } = 0$

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 14: Partial Derivatives

Find the limit. lim⁡(x,y)→(8,4)xycos⁡(x−2y)\lim _ { ( x , y ) \rightarrow ( 8,4 ) } x y \cos ( x - 2 y )lim(x,y)→(8,4)​xycos(x−2y)

If f(x,y)=x2+7y2f ( x , y ) = x ^ { 2 } + 7 y ^ { 2 }f(x,y)=x2+7y2 use the gradient vector ∇f(10,2)\nabla f ( 10,2 )∇f(10,2) to find the tangent line to the level curve f(x,y)=136f ( x , y ) = 136f(x,y)=136 at the point (10,2)( 10,2 )(10,2) .

Find the differential of the function z=3x3y6z = 3 x ^ { 3 } y ^ { 6 }z=3x3y6

Use partial derivatives to find the implicit partial derivatives ∂z∂x\frac { \partial z } { \partial x }∂x∂z​ and ∂z∂y\frac { \partial z } { \partial y }∂y∂z​ 2x2+7xy−6x2z+8yz2=02 x ^ { 2 } + 7 x y - 6 x ^ { 2 } z + 8 y z ^ { 2 } = 02x2+7xy−6x2z+8yz2=0

Sketch the graph of the function g(x,y)=x2g ( x , y ) = x ^ { 2 }g(x,y)=x2

Find and classify the relative extrema and saddle points of the function f(x,y)=e−2xsin⁡4yf ( x , y ) = e ^ { - 2 x } \sin 4 yf(x,y)=e−2xsin4y for x≥0x \geq 0x≥0 and 0≤y≤π20 \leq y \leq \frac { \pi } { 2 }0≤y≤2π​ .

Use spherical coordinates to find the limit. lim⁡(x,y)→(0,0)6xyzx2+y2+z2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 6 x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }lim(x,y)→(0,0)​x2+y2+z26xyz​

Find the domain and range of the function h(x,y)=2x−9yh ( x , y ) = \sqrt { 2 x - 9 y }h(x,y)=2x−9y​ .

Find the absolute extrema of the function f(x,y)=2x+3y−5f ( x , y ) = 2 x + 3 y - 5f(x,y)=2x+3y−5 on the closed triangular region with vertices (0,0)( 0,0 )(0,0) , (5,0)( 5,0 )(5,0) and (5,4)( 5,4 )(5,4) .

Find equations for the tangent plane and the normal line to the surface with equation xy+yz+xz=38x y + y z + x z = 38xy+yz+xz=38 at the point P(2,4,5)P ( 2,4,5 )P(2,4,5)

Find the first partial derivatives of the function z=x6y+4z = x ^ { 6 } \sqrt { y + 4 }z=x6y+4​ .

Find an equation of the tangent plane to the given surface at the specified point. z=18−x2−2y2,(3,2,−4)z = \sqrt { 18 - x ^ { 2 } - 2 y ^ { 2 } } , ( 3,2 , - 4 )z=18−x2−2y2​,(3,2,−4)

Determine the largest set on which the function is continuous. f(x,y,z)=xyz6x2+2y2−zf ( x , y , z ) = \frac { x y z } { 6 x ^ { 2 } + 2 y ^ { 2 } - z }f(x,y,z)=6x2+2y2−zxyz​

Find the gradient of the function f(x,y,z)=x5eyzf ( x , y , z ) = x ^ { 5 } e ^ { y } \sqrt { z }f(x,y,z)=x5eyz​ .

Use Lagrange multipliers to find the minimum value of the function subject to the given constraints. f(x,y,z,t)=5x+5y+5z+5t,5(x2+y2+z2+t2)=4f ( x , y , z , t ) = 5 x + 5 y + 5 z + 5 t , \quad 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + t ^ { 2 } \right) = 4f(x,y,z,t)=5x+5y+5z+5t,5(x2+y2+z2+t2)=4

Find the equation of the normal line to the given surface at the specified point. z+2=xeycos⁡z,(4,0,0)z + 2 = x e ^ { y } \cos z , ( 4,0,0 )z+2=xeycosz,(4,0,0)

Find the maximum rate of change of f at the given point. f(x,y,z)=x5y3z2,(1,−1,1)f ( x , y , z ) = x ^ { 5 } y ^ { 3 } z ^ { 2 } , ( 1 , - 1,1 )f(x,y,z)=x5y3z2,(1,−1,1)

Find the equation of the tangent plane to the given surface at the specified point. z+7=xeycos⁡z,(7,0,0)z + 7 = x e ^ { y } \cos z , ( 7,0,0 )z+7=xeycosz,(7,0,0)

Use partial derivatives to find the implicit partial derivatives ∂z∂x\frac { \partial z } { \partial x }∂x∂z​ and ∂z∂y\frac { \partial z } { \partial y }∂y∂z​ 2x2+7xy−6x2z+8yz2=02 x ^ { 2 } + 7 x y - 6 x ^ { 2 } z + 8 y z ^ { 2 } = 02x2+7xy−6x2z+8yz2=0

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Find the limit. $\lim _ { ( x , y ) \rightarrow ( 8,4 ) } x y \cos ( x - 2 y )$

If $f ( x , y ) = x ^ { 2 } + 7 y ^ { 2 }$ use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 136$ at the point $( 10,2 )$ .

Find the differential of the function $z = 3 x ^ { 3 } y ^ { 6 }$

Use partial derivatives to find the implicit partial derivatives $\frac { \partial z } { \partial x }$ and $\frac { \partial z } { \partial y }$ $2 x ^ { 2 } + 7 x y - 6 x ^ { 2 } z + 8 y z ^ { 2 } = 0$

Sketch the graph of the function $g ( x , y ) = x ^ { 2 }$

Find and classify the relative extrema and saddle points of the function $f ( x , y ) = e ^ { - 2 x } \sin 4 y$ for $x \geq 0$ and $0 \leq y \leq \frac { \pi } { 2 }$ .

Use spherical coordinates to find the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 6 x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$

Find the domain and range of the function $h ( x , y ) = \sqrt { 2 x - 9 y }$ .

Find the absolute extrema of the function $f ( x , y ) = 2 x + 3 y - 5$ on the closed triangular region with vertices $( 0,0 )$ , $( 5,0 )$ and $( 5,4 )$ .

Find equations for the tangent plane and the normal line to the surface with equation $x y + y z + x z = 38$ at the point $P ( 2,4,5 )$

Find the first partial derivatives of the function $z = x ^ { 6 } \sqrt { y + 4 }$ .

Find an equation of the tangent plane to the given surface at the specified point. $z = \sqrt { 18 - x ^ { 2 } - 2 y ^ { 2 } } , ( 3,2 , - 4 )$

Determine the largest set on which the function is continuous. $f ( x , y , z ) = \frac { x y z } { 6 x ^ { 2 } + 2 y ^ { 2 } - z }$

Find the gradient of the function $f ( x , y , z ) = x ^ { 5 } e ^ { y } \sqrt { z }$ .

Use Lagrange multipliers to find the minimum value of the function subject to the given constraints. $f ( x , y , z , t ) = 5 x + 5 y + 5 z + 5 t , \quad 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + t ^ { 2 } \right) = 4$

Find the equation of the normal line to the given surface at the specified point. $z + 2 = x e ^ { y } \cos z , ( 4,0,0 )$

Find the maximum rate of change of f at the given point. $f ( x , y , z ) = x ^ { 5 } y ^ { 3 } z ^ { 2 } , ( 1 , - 1,1 )$

Find the equation of the tangent plane to the given surface at the specified point. $z + 7 = x e ^ { y } \cos z , ( 7,0,0 )$

Use partial derivatives to find the implicit partial derivatives $\frac { \partial z } { \partial x }$ and $\frac { \partial z } { \partial y }$ $2 x ^ { 2 } + 7 x y - 6 x ^ { 2 } z + 8 y z ^ { 2 } = 0$