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

Apply the high-low method to estimate costs.

Examine the impact of pricing, advertising, and cost changes on profits through scenario analysis.

Utilize the concept of operating leverage to analyze changes in sales and profit.

Develop a contribution margin income statement and analyze financial outcomes.

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