Find the Gradient Field of the Function $$f ( x , y , z ) = \ln \left( \frac { ( x + y ) ^ { 3 } } { z ^ { 10 } } \right) + z ^ { 9 }$$

question 230

Multiple Choice

Find the gradient field of the function.
- $$f ( x , y , z ) = \ln \left( \frac { ( x + y ) ^ { 3 } } { z ^ { 10 } } \right) + z ^ { 9 }$$

• A)\({\nabla} f = \frac { 3 } { \mathrm { x } } \mathrm { i } + \frac { 3 } { \mathrm { y } } \mathrm { j } + \left( 9 \mathrm { z } ^ { 8 } + \frac { 10 } { \mathrm { z } } \right) \mathbf { k }\)
• B) \({\nabla} f = \frac { 3 } { \mathrm { x } + \mathrm { y } } \mathrm { i } + \frac { 3 } { \mathrm { x } + \mathrm { y } } \mathrm { j } + \left( 9 \mathrm { z } ^ { 8 } + \frac { 10 } { \mathrm { z } } \right) \mathrm { k }\)
• C)\({\nabla} f = \frac { 3 } { x + y } i + \frac { 3 } { x + y } j + \left( 9 z ^ { 8 } - \frac { 10 } { z } \right) k\)
• D) \({\nabla} f = \frac { 3 } { \mathrm { x } } \mathrm { i } + \frac { 3 } { \mathrm { y } } \mathrm { j } + \left( 9 \mathrm { z } ^ { 8 } - \frac { 10 } { \mathrm { z } } \right) \mathbf { k }\)

Describe the eligibility and characteristics of Medicare and its significance for elderly Americans.

Explain the structure and benefits of employment-based health insurance in the United States.

Identify the characteristics that define common resources, private goods, public goods, and artificially scarce goods.

Differentiate between excludable and nonexcludable goods.

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