Calculus Examples

$d (q) = 300 - 5 q$ , $s (q) = q^{2}$

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.

$300 - 5 q = q^{2}$

Step 1.1.2

Solve $300 - 5 q = q^{2}$ .

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Step 1.1.2.1

Subtract $q^{2}$ from both sides of the equation.

$300 - 5 q - q^{2} = 0$

Step 1.1.2.2

Factor the left side of the equation.

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Step 1.1.2.2.1

Factor $- 1$ out of $300 - 5 q - q^{2}$ .

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Step 1.1.2.2.1.1

Reorder the expression.

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Step 1.1.2.2.1.1.1

Move $300$ .

$- 5 q - q^{2} + 300 = 0$

Step 1.1.2.2.1.1.2

Reorder $- 5 q$ and $- q^{2}$ .

$- q^{2} - 5 q + 300 = 0$

Step 1.1.2.2.1.2

Factor $- 1$ out of $- q^{2}$ .

$- (q^{2}) - 5 q + 300 = 0$

Step 1.1.2.2.1.3

Factor $- 1$ out of $- 5 q$ .

$- (q^{2}) - (5 q) + 300 = 0$

Step 1.1.2.2.1.4

Rewrite $300$ as $- 1 (- 300)$ .

$- (q^{2}) - (5 q) - 1 \cdot - 300 = 0$

Step 1.1.2.2.1.5

Factor $- 1$ out of $- (q^{2}) - (5 q)$ .

$- (q^{2} + 5 q) - 1 \cdot - 300 = 0$

Step 1.1.2.2.1.6

Factor $- 1$ out of $- (q^{2} + 5 q) - 1 (- 300)$ .

$- (q^{2} + 5 q - 300) = 0$

Step 1.1.2.2.2

Factor.

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Step 1.1.2.2.2.1

Factor $q^{2} + 5 q - 300$ using the AC method.

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Step 1.1.2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 300$ and whose sum is $5$ .

$- 15, 20$

Step 1.1.2.2.2.1.2

Write the factored form using these integers.

$- ((q - 15) (q + 20)) = 0$

Step 1.1.2.2.2.2

Remove unnecessary parentheses.

$- (q - 15) (q + 20) = 0$

Step 1.1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$q - 15 = 0$

$q + 20 = 0$

Step 1.1.2.4

Set $q - 15$ equal to $0$ and solve for $q$ .

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Step 1.1.2.4.1

Set $q - 15$ equal to $0$ .

$q - 15 = 0$

Step 1.1.2.4.2

Add $15$ to both sides of the equation.

$q = 15$

Step 1.1.2.5

Set $q + 20$ equal to $0$ and solve for $q$ .

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Step 1.1.2.5.1

Set $q + 20$ equal to $0$ .

$q + 20 = 0$

Step 1.1.2.5.2

Subtract $20$ from both sides of the equation.

$q = - 20$

Step 1.1.2.6

The final solution is all the values that make $- (q - 15) (q + 20) = 0$ true.

$q = 15, - 20$

Step 1.1.3

Ignore the negative solution.

$q = 15$

Step 1.2

Find the equilibrium price.

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Step 1.2.1

Find the equilibrium price by substituting the equilibrium quantity $15$ for $q$ in $d (q) = 300 - 5 q$ .

$d \cdot 15 = 300 - 5 \cdot 15$

Step 1.2.2

Simplify $d \cdot 15 = 300 - 5 \cdot 15$ .

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Step 1.2.2.1

Multiply $- 5$ by $15$ .

$d \cdot 15 = 300 - 75$

Step 1.2.2.2

Subtract $75$ from $300$ .

$d \cdot 15 = 225$

Step 1.3

Write the equilibrium point.

$(15, 225)$

Step 2

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

$\int_{0}^{15} 300 - 5 q d q - 15 \cdot 225$

Step 3

Evaluate the integral and simplify.

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Step 3.1

Multiply $- 15$ by $225$ .

$\int_{0}^{15} 300 - 5 q d q - 3375$

Step 3.2

Split the single integral into multiple integrals.

$\int_{0}^{15} 300 d q + \int_{0}^{15} - 5 q d q - 3375$

Step 3.3

Apply the constant rule.

$300 q]_{0}^{15} + \int_{0}^{15} - 5 q d q - 3375$

Step 3.4

Since $- 5$ is constant with respect to $q$ , move $- 5$ out of the integral.

$300 q]_{0}^{15} - 5 \int_{0}^{15} q d q - 3375$

Step 3.5

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

$300 q]_{0}^{15} - 5 (\frac{1}{2} q^{2}]_{0}^{15}) - 3375$

Step 3.6

Simplify the answer.

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Step 3.6.1

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

$300 q]_{0}^{15} - 5 (\frac{q^{2}}{2}]_{0}^{15}) - 3375$

Step 3.6.2

Substitute and simplify.

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Step 3.6.2.1

Evaluate $300 q$ at $15$ and at $0$ .

$(300 \cdot 15) - 300 \cdot 0 - 5 (\frac{q^{2}}{2}]_{0}^{15}) - 3375$

Step 3.6.2.2

Evaluate $\frac{q^{2}}{2}$ at $15$ and at $0$ .

$300 \cdot 15 - 300 \cdot 0 - 5 (\frac{15^{2}}{2} - \frac{0^{2}}{2}) - 3375$

Step 3.6.2.3

Simplify.

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Step 3.6.2.3.1

Multiply $300$ by $15$ .

$4500 - 300 \cdot 0 - 5 (\frac{15^{2}}{2} - \frac{0^{2}}{2}) - 3375$

Step 3.6.2.3.2

Multiply $- 300$ by $0$ .

$4500 + 0 - 5 (\frac{15^{2}}{2} - \frac{0^{2}}{2}) - 3375$

Step 3.6.2.3.3

Add $4500$ and $0$ .

$4500 - 5 (\frac{15^{2}}{2} - \frac{0^{2}}{2}) - 3375$

Step 3.6.2.3.4

Raise $15$ to the power of $2$ .

$4500 - 5 (\frac{225}{2} - \frac{0^{2}}{2}) - 3375$

Step 3.6.2.3.5

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

$4500 - 5 (\frac{225}{2} - \frac{0}{2}) - 3375$

Step 3.6.2.3.6

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

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Step 3.6.2.3.6.1

Factor $2$ out of $0$ .

$4500 - 5 (\frac{225}{2} - \frac{2 (0)}{2}) - 3375$

Step 3.6.2.3.6.2

Cancel the common factors.

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Step 3.6.2.3.6.2.1

Factor $2$ out of $2$ .

$4500 - 5 (\frac{225}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - 3375$

Step 3.6.2.3.6.2.2

Cancel the common factor.

$4500 - 5 (\frac{225}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - 3375$

Step 3.6.2.3.6.2.3

Rewrite the expression.

$4500 - 5 (\frac{225}{2} - \frac{0}{1}) - 3375$

Step 3.6.2.3.6.2.4

Divide $0$ by $1$ .

$4500 - 5 (\frac{225}{2} - 0) - 3375$

Step 3.6.2.3.7

Multiply $- 1$ by $0$ .

$4500 - 5 (\frac{225}{2} + 0) - 3375$

Step 3.6.2.3.8

Add $\frac{225}{2}$ and $0$ .

$4500 - 5 (\frac{225}{2}) - 3375$

Step 3.6.2.3.9

Combine $- 5$ and $\frac{225}{2}$ .

$4500 + \frac{- 5 \cdot 225}{2} - 3375$

Step 3.6.2.3.10

Multiply $- 5$ by $225$ .

$4500 + \frac{- 1125}{2} - 3375$

Step 3.6.2.3.11

Move the negative in front of the fraction.

$4500 - \frac{1125}{2} - 3375$

Step 3.6.2.3.12

To write $4500$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4500 \cdot \frac{2}{2} - \frac{1125}{2} - 3375$

Step 3.6.2.3.13

Combine $4500$ and $\frac{2}{2}$ .

$\frac{4500 \cdot 2}{2} - \frac{1125}{2} - 3375$

Step 3.6.2.3.14

Combine the numerators over the common denominator.

$\frac{4500 \cdot 2 - 1125}{2} - 3375$

Step 3.6.2.3.15

Simplify the numerator.

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Step 3.6.2.3.15.1

Multiply $4500$ by $2$ .

$\frac{9000 - 1125}{2} - 3375$

Step 3.6.2.3.15.2

Subtract $1125$ from $9000$ .

$\frac{7875}{2} - 3375$

Step 3.6.2.3.16

To write $- 3375$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7875}{2} - 3375 \cdot \frac{2}{2}$

Step 3.6.2.3.17

Combine $- 3375$ and $\frac{2}{2}$ .

$\frac{7875}{2} + \frac{- 3375 \cdot 2}{2}$

Step 3.6.2.3.18

Combine the numerators over the common denominator.

$\frac{7875 - 3375 \cdot 2}{2}$

Step 3.6.2.3.19

Simplify the numerator.

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Step 3.6.2.3.19.1

Multiply $- 3375$ by $2$ .

$\frac{7875 - 6750}{2}$

Step 3.6.2.3.19.2

Subtract $6750$ from $7875$ .

$\frac{1125}{2}$

Step 3.7

Divide $1125$ by $2$ .

$562.5$

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