Calculus Examples

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Step 1
Find the equilibrium point.
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Find the equilibrium quantity.
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Find the equilibrium point by setting the supply function equal to the demand function.
Solve .
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Move all terms containing to the left side of the equation.
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Subtract from both sides of the equation.
Subtract from .
Subtract from both sides of the equation.
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Divide by .
Find the equilibrium price.
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Find the equilibrium price by substituting the equilibrium quantity for in .
Simplify .
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Multiply by .
Add and .
Write the equilibrium point.
Step 2
Set up the producer surplus where is the equilibrium quantity and is the equilibrium price.
Step 3
Evaluate the integral and simplify.
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Multiply by .
Since is constant with respect to , move out of the integral.
Multiply by .
By the Power Rule, the integral of with respect to is .
Simplify the answer.
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Combine and .
Substitute and simplify.
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Evaluate at and at .
Simplify.
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Raise to the power of .
Raising to any positive power yields .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Add and .
Combine and .
Multiply by .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Subtract from .
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