Calculus Examples

$d (q) = - 0.8 q + 150$ , $s (q) = 5.2 q$

Step 1

Find the equilibrium point.

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Step 1.1

Find the equilibrium quantity.

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Step 1.1.1

Find the equilibrium point by setting the supply function equal to the demand function.

$- 0.8 q + 150 = 5.2 q$

Step 1.1.2

Solve $- 0.8 q + 150 = 5.2 q$ .

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Step 1.1.2.1

Move all terms containing $q$ to the left side of the equation.

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Step 1.1.2.1.1

Subtract $5.2 q$ from both sides of the equation.

$- 0.8 q + 150 - 5.2 q = 0$

Step 1.1.2.1.2

Subtract $5.2 q$ from $- 0.8 q$ .

$- 6 q + 150 = 0$

Step 1.1.2.2

Subtract $150$ from both sides of the equation.

$- 6 q = - 150$

Step 1.1.2.3

Divide each term in $- 6 q = - 150$ by $- 6$ and simplify.

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Step 1.1.2.3.1

Divide each term in $- 6 q = - 150$ by $- 6$ .

$\frac{- 6 q}{- 6} = \frac{- 150}{- 6}$

Step 1.1.2.3.2

Simplify the left side.

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Step 1.1.2.3.2.1

Cancel the common factor of $- 6$ .

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Step 1.1.2.3.2.1.1

Cancel the common factor.

$\frac{- 6 q}{- 6} = \frac{- 150}{- 6}$

Step 1.1.2.3.2.1.2

Divide $q$ by $1$ .

$q = \frac{- 150}{- 6}$

Step 1.1.2.3.3

Simplify the right side.

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Step 1.1.2.3.3.1

Divide $- 150$ by $- 6$ .

$q = 25$

Step 1.2

Find the equilibrium price.

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Step 1.2.1

Find the equilibrium price by substituting the equilibrium quantity $25$ for $q$ in $d (q) = - 0.8 q + 150$ .

$d \cdot 25 = - 0.8 \cdot 25 + 150$

Step 1.2.2

Simplify $d \cdot 25 = - 0.8 \cdot 25 + 150$ .

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Step 1.2.2.1

Multiply $- 0.8$ by $25$ .

$d \cdot 25 = - 20 + 150$

Step 1.2.2.2

Add $- 20$ and $150$ .

$d \cdot 25 = 130$

Step 1.3

Write the equilibrium point.

$(25, 130)$

Step 2

Set up the producer surplus $q_{eq} p_{eq} - \int_{0}^{q_{eq}} s (q) d q$ where $q_{eq}$ is the equilibrium quantity and $p_{eq}$ is the equilibrium price.

$25 \cdot 130 - \int_{0}^{25} 5.2 q d q$

Step 3

Evaluate the integral and simplify.

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Step 3.1

Multiply $25$ by $130$ .

$3250 - \int_{0}^{25} 5.2 q d q$

Step 3.2

Since $5.2$ is constant with respect to $q$ , move $5.2$ out of the integral.

$3250 - (5.2 \int_{0}^{25} q d q)$

Step 3.3

Multiply $5.2$ by $- 1$ .

$3250 - 5.2 \int_{0}^{25} q d q$

Step 3.4

By the Power Rule, the integral of $q$ with respect to $q$ is $\frac{1}{2} q^{2}$ .

$3250 - 5.2 (\frac{1}{2} q^{2}]_{0}^{25})$

Step 3.5

Simplify the answer.

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Step 3.5.1

Combine $\frac{1}{2}$ and $q^{2}$ .

$3250 - 5.2 (\frac{q^{2}}{2}]_{0}^{25})$

Step 3.5.2

Substitute and simplify.

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Step 3.5.2.1

Evaluate $\frac{q^{2}}{2}$ at $25$ and at $0$ .

$3250 - 5.2 ((\frac{25^{2}}{2}) - \frac{0^{2}}{2})$

Step 3.5.2.2

Simplify.

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Step 3.5.2.2.1

Raise $25$ to the power of $2$ .

$3250 - 5.2 (\frac{625}{2} - \frac{0^{2}}{2})$

Step 3.5.2.2.2

Raising $0$ to any positive power yields $0$ .

$3250 - 5.2 (\frac{625}{2} - \frac{0}{2})$

Step 3.5.2.2.3

Cancel the common factor of $0$ and $2$ .

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Step 3.5.2.2.3.1

Factor $2$ out of $0$ .

$3250 - 5.2 (\frac{625}{2} - \frac{2 (0)}{2})$

Step 3.5.2.2.3.2

Cancel the common factors.

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Step 3.5.2.2.3.2.1

Factor $2$ out of $2$ .

$3250 - 5.2 (\frac{625}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 3.5.2.2.3.2.2

Cancel the common factor.

$3250 - 5.2 (\frac{625}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 3.5.2.2.3.2.3

Rewrite the expression.

$3250 - 5.2 (\frac{625}{2} - \frac{0}{1})$

Step 3.5.2.2.3.2.4

Divide $0$ by $1$ .

$3250 - 5.2 (\frac{625}{2} - 0)$

Step 3.5.2.2.4

Multiply $- 1$ by $0$ .

$3250 - 5.2 (\frac{625}{2} + 0)$

Step 3.5.2.2.5

Add $\frac{625}{2}$ and $0$ .

$3250 - 5.2 (\frac{625}{2})$

Step 3.5.2.2.6

Combine $- 5.2$ and $\frac{625}{2}$ .

$3250 + \frac{- 5.2 \cdot 625}{2}$

Step 3.5.2.2.7

Multiply $- 5.2$ by $625$ .

$3250 + \frac{- 3250}{2}$

Step 3.5.2.2.8

Cancel the common factor of $- 3250$ and $2$ .

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Step 3.5.2.2.8.1

Factor $2$ out of $- 3250$ .

$3250 + \frac{2 \cdot - 1625}{2}$

Step 3.5.2.2.8.2

Cancel the common factors.

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Step 3.5.2.2.8.2.1

Factor $2$ out of $2$ .

$3250 + \frac{2 \cdot - 1625}{2 (1)}$

Step 3.5.2.2.8.2.2

Cancel the common factor.

$3250 + \frac{2 \cdot - 1625}{2 \cdot 1}$

Step 3.5.2.2.8.2.3

Rewrite the expression.

$3250 + \frac{- 1625}{1}$

Step 3.5.2.2.8.2.4

Divide $- 1625$ by $1$ .

$3250 - 1625$

Step 3.5.2.2.9

Subtract $1625$ from $3250$ .

$1625$

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