Question 1

Given the contour diagram shown below
(a)Sketch a graph of f(1, y).
(b)Sketch a graph of f(x, 0).

Accepted Answer

The answer of Given the contour diagram shown below
(a)Sketch a...

Question 2

If $z = f ( x , y )$ and $x = s + t , y = s - t$ , then simplify $\left(\frac{\partial z}{\partial s}\right)\left(\frac{\partial z}{\partial t}\right)$ in terms of $f _ { x }$ and $f _ { y }$ .

Accepted Answer

The answer of If \(z = f ( x ,...

Question 3

In an electric circuit, two resistors (of resistance R1 and R2, respectively)are connected so that the combined resistance of the circuit, R, is given by $\frac { 1 } { R } = \frac { 1 } { R _ { 1 } } + \frac { 1 } { R _ { 2 } }$ Find $\frac { \partial R } { \partial R _ { 1 } }$ .

Accepted Answer

Question 4

Given that $f_{x}(x, y)=6 x y+y^{2} e^{xy^{2}}$ and $f _ { y } ( x , y ) = 3 x ^ { 2 } + 2 x y e ^ { x y ^ { 2 } }$ Suppose that f(1, 1)= 4.Find the quadratic Taylor polynomial of f(x, y)at (1, 1).

Accepted Answer

The answer of Given that $f_{x}(x, y)=6 x y+y^{2} e^{xy^{2}}$...

Question 5

Suppose that fx(2, 1)= 2.2, fx(2.5, 1)= 1, fx(2, 1.5)= 1.8, fy(2, 1)= -0.8, fy(2.5, 1)= -1.2 and fy(2, 1.5)= -1.4.
If $f ( 2,1 ) = 4$ estimate the value of f(1.85, 0.8)using a quadratic Taylor polynomial about (2,1).
Use difference quotients to approximate all second derivatives.

Accepted Answer

Question 6

For the function g(x, y)with contour diagram below:   Find the direction in which g is increasing the fastest at the point (1, 1).

Accepted Answer

The required direction is @#LAT-DLM& Thi

Question 7

The monthly mortgage payment in dollars, P, for a house is a function of three variables P = f(A, r, N), where A is the amount borrowed in dollars, r is the interest rate, and N is the number of years before the mortgage is paid off.It is given that: $$\begin{array} { l l l } 
f ( 100000,7,20 ) = 775.29 , & f ( 100000,8,20 ) = 836.44 , & f ( 100000,7,25 ) = 706.77 \
f ( 120000,7,20 ) = 930.35 , & f ( 120000,8,20 ) = 1003.72 , & f ( 120000,7,25 ) = 848.13
\end{array}$$ Estimate the value of $$\left. \frac { \partial f } { \partial r } \right| _ { ( 100000,7,20 ) }$$

Accepted Answer

@#LAT-DLM& is approx

Question 8

Suppose $f ( x , y ) = ( \ln ( x + 3 y ) ) ^ { y }$ .
Use a difference quotient to estimate $f _ { x } ( 2,1 )$ and $f _ { y } ( 2,1 )$ with h = 0.01.Give your answers to 3 decimal places.

Accepted Answer

The answer of Suppose \(f ( x , y )...

Question 9

Let $U ( x , t ) = e ^ { - ( x - c t ) ^ { 2 } }$ .Simplify $U _ { t t } - C ^ { 2 } U _ { x x }$ .

Accepted Answer

A)  $U _ { t t } - c ^ { 2 } U _ { x x } = 0$ . 
B)  $U _ { t t } - C ^ { 2 } U _ { x x } = U$ . 
C)  $U _ { t t } - c ^ { 2 } U _ { x x } = - U$ . 
D)  $U _ { t t } - c ^ { 2 } U _ { x x } = c ^ { 2 } U$ . 
A)  $U _ { t t } - c ^ { 2 } U _ { x x } = 0$ . 
B)  $U _ { t t } - C ^ { 2 } U _ { x x } = U$ . 
C)  $U _ { t t } - c ^ { 2 } U _ { x x } = - U$ . 
D)  $U _ { t t } - c ^ { 2 } U _ { x x } = c ^ { 2 } U$ .

Given the contour diagram shown below (a)Sketch a graph of f(1, y). (b)Sketch a graph of f(x, 0).

If $z = f ( x , y )$ and $x = s + t , y = s - t$ , then simplify $\left(\frac{\partial z}{\partial s}\right)\left(\frac{\partial z}{\partial t}\right)$ in terms of $f _ { x }$ and $f _ { y }$ .

In an electric circuit, two resistors (of resistance R₁ and R₂, respectively)are connected so that the combined resistance of the circuit, R, is given by $\frac { 1 } { R } = \frac { 1 } { R _ { 1 } } + \frac { 1 } { R _ { 2 } }$ Find $\frac { \partial R } { \partial R _ { 1 } }$ .

Given that $f_{x}(x, y)=6 x y+y^{2} e^{xy^{2}}$ and $f _ { y } ( x , y ) = 3 x ^ { 2 } + 2 x y e ^ { x y ^ { 2 } }$ Suppose that f(1, 1)= 4.Find the quadratic Taylor polynomial of f(x, y)at (1, 1).

Suppose that f_x(2, 1)= 2.2, f_x(2.5, 1)= 1, f_x(2, 1.5)= 1.8, f_y(2, 1)= -0.8, f_y(2.5, 1)= -1.2 and f_y(2, 1.5)= -1.4. If $f ( 2,1 ) = 4$ estimate the value of f(1.85, 0.8)using a quadratic Taylor polynomial about (2,1). Use difference quotients to approximate all second derivatives.

For the function g(x, y)with contour diagram below: Find the direction in which g is increasing the fastest at the point (1, 1).

Suppose $f ( x , y ) = ( \ln ( x + 3 y ) ) ^ { y }$ . Use a difference quotient to estimate $f _ { x } ( 2,1 )$ and $f _ { y } ( 2,1 )$ with h = 0.01.Give your answers to 3 decimal places.

Let $U ( x , t ) = e ^ { - ( x - c t ) ^ { 2 } }$ .Simplify $U _ { t t } - C ^ { 2 } U _ { x x }$ .

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Exam 14: Differentiating Functions of Several Variables

Given the contour diagram shown below (a)Sketch a graph of f(1, y). (b)Sketch a graph of f(x, 0).

If z=f(x,y)z = f ( x , y )z=f(x,y) and x=s+t,y=s−tx = s + t , y = s - tx=s+t,y=s−t , then simplify (∂z∂s)(∂z∂t)\left(\frac{\partial z}{\partial s}\right)\left(\frac{\partial z}{\partial t}\right)(∂s∂z​)(∂t∂z​) in terms of fxf _ { x }fx​ and fyf _ { y }fy​ .

Given that fx(x,y)=6xy+y2exy2f_{x}(x, y)=6 x y+y^{2} e^{xy^{2}}fx​(x,y)=6xy+y2exy2 and fy(x,y)=3x2+2xyexy2f _ { y } ( x , y ) = 3 x ^ { 2 } + 2 x y e ^ { x y ^ { 2 } }fy​(x,y)=3x2+2xyexy2 Suppose that f(1, 1)= 4.Find the quadratic Taylor polynomial of f(x, y)at (1, 1).

Suppose that fx(2, 1)= 2.2, fx(2.5, 1)= 1, fx(2, 1.5)= 1.8, fy(2, 1)= -0.8, fy(2.5, 1)= -1.2 and fy(2, 1.5)= -1.4. If f(2,1)=4f ( 2,1 ) = 4f(2,1)=4 estimate the value of f(1.85, 0.8)using a quadratic Taylor polynomial about (2,1). Use difference quotients to approximate all second derivatives.

For the function g(x, y)with contour diagram below: Find the direction in which g is increasing the fastest at the point (1, 1).

Suppose f(x,y)=(ln⁡(x+3y))yf ( x , y ) = ( \ln ( x + 3 y ) ) ^ { y }f(x,y)=(ln(x+3y))y . Use a difference quotient to estimate fx(2,1)f _ { x } ( 2,1 )fx​(2,1) and fy(2,1)f _ { y } ( 2,1 )fy​(2,1) with h = 0.01.Give your answers to 3 decimal places.

Let U(x,t)=e−(x−ct)2U ( x , t ) = e ^ { - ( x - c t ) ^ { 2 } }U(x,t)=e−(x−ct)2 .Simplify Utt−C2UxxU _ { t t } - C ^ { 2 } U _ { x x }Utt​−C2Uxx​ .

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

If $z = f ( x , y )$ and $x = s + t , y = s - t$ , then simplify $\left(\frac{\partial z}{\partial s}\right)\left(\frac{\partial z}{\partial t}\right)$ in terms of $f _ { x }$ and $f _ { y }$ .

Given that $f_{x}(x, y)=6 x y+y^{2} e^{xy^{2}}$ and $f _ { y } ( x , y ) = 3 x ^ { 2 } + 2 x y e ^ { x y ^ { 2 } }$ Suppose that f(1, 1)= 4.Find the quadratic Taylor polynomial of f(x, y)at (1, 1).

Suppose that f_x(2, 1)= 2.2, f_x(2.5, 1)= 1, f_x(2, 1.5)= 1.8, f_y(2, 1)= -0.8, f_y(2.5, 1)= -1.2 and f_y(2, 1.5)= -1.4. If $f ( 2,1 ) = 4$ estimate the value of f(1.85, 0.8)using a quadratic Taylor polynomial about (2,1). Use difference quotients to approximate all second derivatives.

Suppose $f ( x , y ) = ( \ln ( x + 3 y ) ) ^ { y }$ . Use a difference quotient to estimate $f _ { x } ( 2,1 )$ and $f _ { y } ( 2,1 )$ with h = 0.01.Give your answers to 3 decimal places.

Let $U ( x , t ) = e ^ { - ( x - c t ) ^ { 2 } }$ .Simplify $U _ { t t } - C ^ { 2 } U _ { x x }$ .