Find the Formula and Limit as Requested $f ( x ) = 6 x ^ { 2 } + 2$

question 76

Multiple Choice

Find the formula and limit as requested.
-For the function $f ( x ) = 6 x ^ { 2 } + 2$ , find a formula for the upper sum obtained by dividing the interval $[ 0,3 ]$ into $n$ equal subintervals. Then take the limit as $n \rightarrow \infty$ to calculate the area under the curve over $[ 0,3 ]$ .

• A) \(6 + \frac { 324 n ^ { 3 } + 486 n ^ { 2 } + 162 n } { 6 n ^ { 3 } } ;\) Area \(= 60\)
• B) \(6 + \frac { 324 n ^ { 3 } + 486 n ^ { 2 } + 162 n } { 6 n ^ { 3 } } ;\) Area \(= 54\)
• C) \(6 + \frac { 324 n ^ { 3 } + 486 n ^ { 2 } + 162 n } { 6 n ^ { 4 } } ;\) Area \(= 6\)
• D) \(6 + \frac { 324 n ^ { 3 } + 486 n ^ { 2 } + 162 n } { 6 n ^ { 4 } } ;\) Area \(= 60\)

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Definitions:

Equilibrium Quantity

The quantity of goods or services bought and sold at the equilibrium price, where demand equals supply.

Negative Externality

occurs when the production or consumption of a good or service imposes costs on third parties not directly involved in the transaction.

Government Intervention

Actions taken by a government to influence or regulate the economy or specific industries, often to correct market failures or promote social welfare.

Equilibrium Quantity

The quantity of goods or services supplied that is equal to the quantity demanded at the market equilibrium price.