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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Subtract from both sides of the equation.
Factor the left side of the equation.
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Factor out of .
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Reorder the expression.
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Move .
Reorder and .
Factor out of .
Factor out of .
Rewrite as .
Factor out of .
Factor out of .
Factor.
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Factor using the AC method.
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Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Write the factored form using these integers.
Remove unnecessary parentheses.
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set equal to and solve for .
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Set equal to .
Add to both sides of the equation.
Set equal to and solve for .
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Set equal to .
Subtract from both sides of the equation.
The final solution is all the values that make true.
Ignore the negative solution.
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 .
Subtract from .
Write the equilibrium point.
Step 2
Set up the consumer surplus where is the equilibrium quantity and is the equilibrium price.
Step 3
Evaluate the integral and simplify.
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Multiply by .
Split the single integral into multiple integrals.
Apply the constant rule.
Since is constant with respect to , move out of the integral.
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 .
Evaluate at and at .
Simplify.
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Multiply by .
Multiply by .
Add and .
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 .
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Divide by .
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