The partial derivatives of the function f ( x , y ) .

Question

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Answer 1

Question

Chapter 2, Problem 2.1P

a.

To determine

The partial derivatives of the function f(x,y) .

a.

Expert Solution

Answer to Problem 2.1P

The partial derivatives of the function f(x,y) are

∂f∂x=8x,∂f∂y=6y,∂2f∂x2=8,∂2f∂y2=6,∂2f∂x∂y=0

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

Since,

f(x,y)=4x2+3y2

Partially differentiate with respect to x and y ,

∂f∂x=∂∂x(4x2+3y2)=4×2x+0=8x

and

∂f∂y=∂∂y(4x2+3y2)=0+3×2y=6y

Now

∂2f∂x2=∂∂x(∂f∂x)=∂∂x(8x)=8×1=8

and

∂2f∂y2=∂∂y(∂f∂y)=∂∂y(6y)=6×1=6

Also,

∂2f∂x∂y=∂∂x(∂f∂y)=∂∂x(6y)=0

b.

To determine

The derivative dydx using the implicit function theorem.

b.

Expert Solution

Answer to Problem 2.1P

dydx=−8x6y

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

and

f(x,y)=16

Since,

f(x,y)=4x2+3y2

Then,

16=4x2+3y2

Differentiate with respect to x ,

0=4×2x+3×2ydydx8x+6ydydx=06ydydx=−8xdydx=−8x6y

c.

To determine

The value of dydx at x=1, y=2 .

c.

Expert Solution

Answer to Problem 2.1P

dydx=−23

Explanation of Solution

Since,

dydx=−8x6y

Substituting x=1, y=2 ,

dydx=−8×16×2

dydx=−812=−23

d.

To determine

To draw:The graph of dydx=−23 on the surface f(x,y)=4x2+3y2 .

d.

Expert Solution

Explanation of Solution

Since, dydx is the slope of the curve,

From the above part,

dydx=−83

So, draw the tangent plane on the surface

f(x,y)=4x2+3y2

Bundle: Microeconomic Theory: Basic Principles and Extensions, 12th + MindTap Economics, 1 term (6 months) Printed Access Card, Chapter 2, Problem 2.1P

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Answer 2

Question

Chapter 2, Problem 2.1P

a.

To determine

The partial derivatives of the function f(x,y) .

a.

Expert Solution

Answer to Problem 2.1P

The partial derivatives of the function f(x,y) are

∂f∂x=8x,∂f∂y=6y,∂2f∂x2=8,∂2f∂y2=6,∂2f∂x∂y=0

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

Since,

f(x,y)=4x2+3y2

Partially differentiate with respect to x and y ,

∂f∂x=∂∂x(4x2+3y2)=4×2x+0=8x

and

∂f∂y=∂∂y(4x2+3y2)=0+3×2y=6y

Now

∂2f∂x2=∂∂x(∂f∂x)=∂∂x(8x)=8×1=8

and

∂2f∂y2=∂∂y(∂f∂y)=∂∂y(6y)=6×1=6

Also,

∂2f∂x∂y=∂∂x(∂f∂y)=∂∂x(6y)=0

b.

To determine

The derivative dydx using the implicit function theorem.

b.

Expert Solution

Answer to Problem 2.1P

dydx=−8x6y

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

and

f(x,y)=16

Since,

f(x,y)=4x2+3y2

Then,

16=4x2+3y2

Differentiate with respect to x ,

0=4×2x+3×2ydydx8x+6ydydx=06ydydx=−8xdydx=−8x6y

c.

To determine

The value of dydx at x=1, y=2 .

c.

Expert Solution

Answer to Problem 2.1P

dydx=−23

Explanation of Solution

Since,

dydx=−8x6y

Substituting x=1, y=2 ,

dydx=−8×16×2

dydx=−812=−23

d.

To determine

To draw:The graph of dydx=−23 on the surface f(x,y)=4x2+3y2 .

d.

Expert Solution

Explanation of Solution

Since, dydx is the slope of the curve,

From the above part,

dydx=−83

So, draw the tangent plane on the surface

f(x,y)=4x2+3y2

Bundle: Microeconomic Theory: Basic Principles and Extensions, 12th + MindTap Economics, 1 term (6 months) Printed Access Card, Chapter 2, Problem 2.1P

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Answer 3

Question

Chapter 2, Problem 2.1P

a.

To determine

The partial derivatives of the function f(x,y) .

a.

Expert Solution

Answer to Problem 2.1P

The partial derivatives of the function f(x,y) are

∂f∂x=8x,∂f∂y=6y,∂2f∂x2=8,∂2f∂y2=6,∂2f∂x∂y=0

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

Since,

f(x,y)=4x2+3y2

Partially differentiate with respect to x and y ,

∂f∂x=∂∂x(4x2+3y2)=4×2x+0=8x

and

∂f∂y=∂∂y(4x2+3y2)=0+3×2y=6y

Now

∂2f∂x2=∂∂x(∂f∂x)=∂∂x(8x)=8×1=8

and

∂2f∂y2=∂∂y(∂f∂y)=∂∂y(6y)=6×1=6

Also,

∂2f∂x∂y=∂∂x(∂f∂y)=∂∂x(6y)=0

b.

To determine

The derivative dydx using the implicit function theorem.

b.

Expert Solution

Answer to Problem 2.1P

dydx=−8x6y

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

and

f(x,y)=16

Since,

f(x,y)=4x2+3y2

Then,

16=4x2+3y2

Differentiate with respect to x ,

0=4×2x+3×2ydydx8x+6ydydx=06ydydx=−8xdydx=−8x6y

c.

To determine

The value of dydx at x=1, y=2 .

c.

Expert Solution

Answer to Problem 2.1P

dydx=−23

Explanation of Solution

Since,

dydx=−8x6y

Substituting x=1, y=2 ,

dydx=−8×16×2

dydx=−812=−23

d.

To determine

To draw:The graph of dydx=−23 on the surface f(x,y)=4x2+3y2 .

d.

Expert Solution

Explanation of Solution

Since, dydx is the slope of the curve,

From the above part,

dydx=−83

So, draw the tangent plane on the surface

f(x,y)=4x2+3y2

Bundle: Microeconomic Theory: Basic Principles and Extensions, 12th + MindTap Economics, 1 term (6 months) Printed Access Card, Chapter 2, Problem 2.1P

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Answer 4

Question

Chapter 2, Problem 2.1P

a.

To determine

The partial derivatives of the function f(x,y) .

a.

Expert Solution

Answer to Problem 2.1P

The partial derivatives of the function f(x,y) are

∂f∂x=8x,∂f∂y=6y,∂2f∂x2=8,∂2f∂y2=6,∂2f∂x∂y=0

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

Since,

f(x,y)=4x2+3y2

Partially differentiate with respect to x and y ,

∂f∂x=∂∂x(4x2+3y2)=4×2x+0=8x

and

∂f∂y=∂∂y(4x2+3y2)=0+3×2y=6y

Now

∂2f∂x2=∂∂x(∂f∂x)=∂∂x(8x)=8×1=8

and

∂2f∂y2=∂∂y(∂f∂y)=∂∂y(6y)=6×1=6

Also,

∂2f∂x∂y=∂∂x(∂f∂y)=∂∂x(6y)=0

b.

To determine

The derivative dydx using the implicit function theorem.

b.

Expert Solution

Answer to Problem 2.1P

dydx=−8x6y

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

and

f(x,y)=16

Since,

f(x,y)=4x2+3y2

Then,

16=4x2+3y2

Differentiate with respect to x ,

0=4×2x+3×2ydydx8x+6ydydx=06ydydx=−8xdydx=−8x6y

c.

To determine

The value of dydx at x=1, y=2 .

c.

Expert Solution

Answer to Problem 2.1P

dydx=−23

Explanation of Solution

Since,

dydx=−8x6y

Substituting x=1, y=2 ,

dydx=−8×16×2

dydx=−812=−23

d.

To determine

To draw:The graph of dydx=−23 on the surface f(x,y)=4x2+3y2 .

d.

Expert Solution

Explanation of Solution

Since, dydx is the slope of the curve,

From the above part,

dydx=−83

So, draw the tangent plane on the surface

f(x,y)=4x2+3y2

Bundle: Microeconomic Theory: Basic Principles and Extensions, 12th + MindTap Economics, 1 term (6 months) Printed Access Card, Chapter 2, Problem 2.1P

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Answer 5

Chapter 2, Problem 2.1P

a.

To determine

The partial derivatives of the function f(x,y) .

a.

Expert Solution

Answer to Problem 2.1P

The partial derivatives of the function f(x,y) are

∂f∂x=8x,∂f∂y=6y,∂2f∂x2=8,∂2f∂y2=6,∂2f∂x∂y=0

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

Since,

f(x,y)=4x2+3y2

Partially differentiate with respect to x and y ,

∂f∂x=∂∂x(4x2+3y2)=4×2x+0=8x

and

∂f∂y=∂∂y(4x2+3y2)=0+3×2y=6y

Now

∂2f∂x2=∂∂x(∂f∂x)=∂∂x(8x)=8×1=8

and

∂2f∂y2=∂∂y(∂f∂y)=∂∂y(6y)=6×1=6

Also,

∂2f∂x∂y=∂∂x(∂f∂y)=∂∂x(6y)=0

b.

To determine

The derivative dydx using the implicit function theorem.

b.

Expert Solution

Answer to Problem 2.1P

dydx=−8x6y

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

and

f(x,y)=16

Since,

f(x,y)=4x2+3y2

Then,

16=4x2+3y2

Differentiate with respect to x ,

0=4×2x+3×2ydydx8x+6ydydx=06ydydx=−8xdydx=−8x6y

c.

To determine

The value of dydx at x=1, y=2 .

c.

Expert Solution

Answer to Problem 2.1P

dydx=−23

Explanation of Solution

Since,

dydx=−8x6y

Substituting x=1, y=2 ,

dydx=−8×16×2

dydx=−812=−23

d.

To determine

To draw:The graph of dydx=−23 on the surface f(x,y)=4x2+3y2 .

d.

Expert Solution

Explanation of Solution

Since, dydx is the slope of the curve,

From the above part,

dydx=−83

So, draw the tangent plane on the surface

f(x,y)=4x2+3y2

Bundle: Microeconomic Theory: Basic Principles and Extensions, 12th + MindTap Economics, 1 term (6 months) Printed Access Card, Chapter 2, Problem 2.1P

Answer 6

Chapter 2, Problem 2.1P

a.

To determine

The partial derivatives of the function f(x,y) .

a.

Expert Solution

Answer to Problem 2.1P

The partial derivatives of the function f(x,y) are

∂f∂x=8x,∂f∂y=6y,∂2f∂x2=8,∂2f∂y2=6,∂2f∂x∂y=0

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

Since,

f(x,y)=4x2+3y2

Partially differentiate with respect to x and y ,

∂f∂x=∂∂x(4x2+3y2)=4×2x+0=8x

and

∂f∂y=∂∂y(4x2+3y2)=0+3×2y=6y

Now

∂2f∂x2=∂∂x(∂f∂x)=∂∂x(8x)=8×1=8

and

∂2f∂y2=∂∂y(∂f∂y)=∂∂y(6y)=6×1=6

Also,

∂2f∂x∂y=∂∂x(∂f∂y)=∂∂x(6y)=0

Answer 7

Chapter 2, Problem 2.1P

a.

To determine

The partial derivatives of the function f(x,y) .

Answer 8

a.

Expert Solution

Answer to Problem 2.1P

The partial derivatives of the function f(x,y) are

∂f∂x=8x,∂f∂y=6y,∂2f∂x2=8,∂2f∂y2=6,∂2f∂x∂y=0

Answer 9

a.

Expert Solution

Answer 10

a.

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

Since,

f(x,y)=4x2+3y2

Partially differentiate with respect to x and y ,

∂f∂x=∂∂x(4x2+3y2)=4×2x+0=8x

and

∂f∂y=∂∂y(4x2+3y2)=0+3×2y=6y

Now

∂2f∂x2=∂∂x(∂f∂x)=∂∂x(8x)=8×1=8

and

∂2f∂y2=∂∂y(∂f∂y)=∂∂y(6y)=6×1=6

Also,

∂2f∂x∂y=∂∂x(∂f∂y)=∂∂x(6y)=0

Answer 13

b.

To determine

The derivative dydx using the implicit function theorem.

b.

Expert Solution

Answer to Problem 2.1P

dydx=−8x6y

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

and

f(x,y)=16

Since,

f(x,y)=4x2+3y2

Then,

16=4x2+3y2

Differentiate with respect to x ,

0=4×2x+3×2ydydx8x+6ydydx=06ydydx=−8xdydx=−8x6y

Answer 14

b.

To determine

The derivative dydx using the implicit function theorem.

Answer 15

b.

Expert Solution

Answer to Problem 2.1P

dydx=−8x6y

Answer 16

b.

Expert Solution

Answer 17

b.

Expert Solution

Answer 18

Expert Solution

Answer 19

Explanation of Solution

Given information:

The function,

f(x,y)=4x2+3y2

and

f(x,y)=16

Since,

f(x,y)=4x2+3y2

Then,

16=4x2+3y2

Differentiate with respect to x ,

0=4×2x+3×2ydydx8x+6ydydx=06ydydx=−8xdydx=−8x6y

Answer 20

c.

To determine

The value of dydx at x=1, y=2 .

c.

Expert Solution

Answer to Problem 2.1P

dydx=−23

Explanation of Solution

Since,

dydx=−8x6y

Substituting x=1, y=2 ,

dydx=−8×16×2

dydx=−812=−23

Answer 21

c.

To determine

The value of dydx at x=1, y=2 .

Answer 22

c.

Expert Solution

Answer to Problem 2.1P

dydx=−23

Answer 23

c.

Expert Solution

Answer 24

c.

Expert Solution

Answer 25

Expert Solution

Answer 26

Explanation of Solution

Since,

dydx=−8x6y

Substituting x=1, y=2 ,

dydx=−8×16×2

dydx=−812=−23

Answer 27

d.

To determine

To draw:The graph of dydx=−23 on the surface f(x,y)=4x2+3y2 .

d.

Expert Solution

Explanation of Solution

Since, dydx is the slope of the curve,

From the above part,

dydx=−83

So, draw the tangent plane on the surface

f(x,y)=4x2+3y2

Bundle: Microeconomic Theory: Basic Principles and Extensions, 12th + MindTap Economics, 1 term (6 months) Printed Access Card, Chapter 2, Problem 2.1P

Answer 28

d.

To determine

To draw:The graph of dydx=−23 on the surface f(x,y)=4x2+3y2 .

Answer 29

d.

Expert Solution

Explanation of Solution

Since, dydx is the slope of the curve,

From the above part,

dydx=−83

So, draw the tangent plane on the surface

f(x,y)=4x2+3y2

Bundle: Microeconomic Theory: Basic Principles and Extensions, 12th + MindTap Economics, 1 term (6 months) Printed Access Card, Chapter 2, Problem 2.1P