Calculus Examples

$D (p) = 200 - p^{2}$ , $p = 10$

Step 1

Write $D (p) = 200 - p^{2}$ as an equation.

$q = 200 - p^{2}$

Step 2

To find elasticity of demand, use the formula $E = | \frac{p}{q} \frac{d q}{d p} |$ .

Step 3

Substitute $10$ for $p$ in $q = 200 - p^{2}$ and simplify to find $q$ .

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Step 3.1

Substitute $10$ for $p$ .

$q = 200 - 10^{2}$

Step 3.2

Simplify each term.

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Step 3.2.1

Raise $10$ to the power of $2$ .

$q = 200 - 1 \cdot 100$

Step 3.2.2

Multiply $- 1$ by $100$ .

$q = 200 - 100$

Step 3.3

Subtract $100$ from $200$ .

$q = 100$

Step 4

Find $\frac{d q}{d p}$ by differentiating the demand function.

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Step 4.1

Differentiate the demand function.

$\frac{d q}{d p} = \frac{d}{d p} [200 - p^{2}]$

Step 4.2

Differentiate.

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Step 4.2.1

By the Sum Rule, the derivative of $200 - p^{2}$ with respect to $p$ is $\frac{d}{d p} [200] + \frac{d}{d p} [- p^{2}]$ .

$\frac{d q}{d p} = \frac{d}{d p} [200] + \frac{d}{d p} [- p^{2}]$

Step 4.2.2

Since $200$ is constant with respect to $p$ , the derivative of $200$ with respect to $p$ is $0$ .

$\frac{d q}{d p} = 0 + \frac{d}{d p} [- p^{2}]$

Step 4.3

Evaluate $\frac{d}{d p} [- p^{2}]$ .

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Step 4.3.1

Since $- 1$ is constant with respect to $p$ , the derivative of $- p^{2}$ with respect to $p$ is $- \frac{d}{d p} [p^{2}]$ .

$\frac{d q}{d p} = 0 - \frac{d}{d p} [p^{2}]$

Step 4.3.2

Differentiate using the Power Rule which states that $\frac{d}{d p} [p^{n}]$ is $n p^{n - 1}$ where $n = 2$ .

$\frac{d q}{d p} = 0 - (2 p)$

Step 4.3.3

Multiply $2$ by $- 1$ .

$\frac{d q}{d p} = 0 - 2 p$

Step 4.4

Subtract $2 p$ from $0$ .

$\frac{d q}{d p} = - 2 p$

Step 5

Substitute into the formula for elasticity $E = | \frac{p}{q} \frac{d q}{d p} |$ and simplify.

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Step 5.1

Substitute $- 2 p$ for $\frac{d q}{d p}$ .

$E = | \frac{p}{q} (- 2 p) |$

Step 5.2

Substitute the values of $p$ and $q$ .

$E = | \frac{10}{100} (- 2 \cdot 10) |$

Step 5.3

Cancel the common factor of $10$ and $100$ .

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Step 5.3.1

Factor $10$ out of $10$ .

$E = | \frac{10 (1)}{100} (- 2 \cdot 10) |$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $10$ out of $100$ .

$E = | \frac{10 \cdot 1}{10 \cdot 10} (- 2 \cdot 10) |$

Step 5.3.2.2

Cancel the common factor.

$E = | \frac{10 \cdot 1}{10 \cdot 10} (- 2 \cdot 10) |$

Step 5.3.2.3

Rewrite the expression.

$E = | \frac{1}{10} (- 2 \cdot 10) |$

Step 5.4

Multiply $- 2$ by $10$ .

$E = | \frac{1}{10} \cdot - 20 |$

Step 5.5

Cancel the common factor of $10$ .

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Step 5.5.1

Factor $10$ out of $- 20$ .

$E = | \frac{1}{10} \cdot (10 (- 2)) |$

Step 5.5.2

Cancel the common factor.

$E = | \frac{1}{10} \cdot (10 \cdot - 2) |$

Step 5.5.3

Rewrite the expression.

$E = | - 2 |$

Step 5.6

The absolute value is the distance between a number and zero. The distance between $- 2$ and $0$ is $2$ .

$E = 2$

Step 6

Since $E > 1$ , the demand is elastic.

$E = 2$

$Elastic$

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