Calculus Examples

$D (p) = 1500 - 12 p$ , $p = 100$

Step 1

Write $D (p) = 1500 - 12 p$ as an equation.

$q = 1500 - 12 p$

Step 2

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

Step 3

Substitute $100$ for $p$ in $q = 1500 - 12 p$ and simplify to find $q$ .

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Step 3.1

Substitute $100$ for $p$ .

$q = 1500 - 12 \cdot 100$

Step 3.2

Multiply $- 12$ by $100$ .

$q = 1500 - 1200$

Step 3.3

Subtract $1200$ from $1500$ .

$q = 300$

Step 4

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

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Step 4.1

Differentiate the demand function.

$\frac{d q}{d p} = \frac{d}{d p} [1500 - 12 p]$

Step 4.2

Differentiate.

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Step 4.2.1

By the Sum Rule, the derivative of $1500 - 12 p$ with respect to $p$ is $\frac{d}{d p} [1500] + \frac{d}{d p} [- 12 p]$ .

$\frac{d q}{d p} = \frac{d}{d p} [1500] + \frac{d}{d p} [- 12 p]$

Step 4.2.2

Since $1500$ is constant with respect to $p$ , the derivative of $1500$ with respect to $p$ is $0$ .

$\frac{d q}{d p} = 0 + \frac{d}{d p} [- 12 p]$

Step 4.3

Evaluate $\frac{d}{d p} [- 12 p]$ .

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Step 4.3.1

Since $- 12$ is constant with respect to $p$ , the derivative of $- 12 p$ with respect to $p$ is $- 12 \frac{d}{d p} [p]$ .

$\frac{d q}{d p} = 0 - 12 \frac{d}{d p} [p]$

Step 4.3.2

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

$\frac{d q}{d p} = 0 - 12 \cdot 1$

Step 4.3.3

Multiply $- 12$ by $1$ .

$\frac{d q}{d p} = 0 - 12$

Step 4.4

Subtract $12$ from $0$ .

$\frac{d q}{d p} = - 12$

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 $- 12$ for $\frac{d q}{d p}$ .

$E = | \frac{p}{q} \cdot - 12 |$

Step 5.2

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

$E = | \frac{100}{300} \cdot - 12 |$

Step 5.3

Cancel the common factor of $12$ .

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Step 5.3.1

Factor $12$ out of $300$ .

$E = | \frac{100}{12 (25)} \cdot - 12 |$

Step 5.3.2

Factor $12$ out of $- 12$ .

$E = | \frac{100}{12 \cdot 25} \cdot (12 \cdot - 1) |$

Step 5.3.3

Cancel the common factor.

$E = | \frac{100}{12 \cdot 25} \cdot (12 \cdot - 1) |$

Step 5.3.4

Rewrite the expression.

$E = | \frac{100}{25} \cdot - 1 |$

Step 5.4

Combine $\frac{100}{25}$ and $- 1$ .

$E = | \frac{100 \cdot - 1}{25} |$

Step 5.5

Multiply $100$ by $- 1$ .

$E = | \frac{- 100}{25} |$

Step 5.6

Divide $- 100$ by $25$ .

$E = | - 4 |$

Step 5.7

The absolute value is the distance between a number and zero. The distance between $- 4$ and $0$ is $4$ .

$E = 4$

Step 6

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

$E = 4$

$Elastic$

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