Calculus Examples

$(- \frac{2}{3}, \frac{3}{2})$ , $(\frac{7}{3}, 2)$

Step 1

Use the distance formula to determine the distance between the two points.

$Distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Step 2

Substitute the actual values of the points into the distance formula.

$\sqrt{{(\frac{7}{3} - (- \frac{2}{3}))}^{2} + {(2 - \frac{3}{2})}^{2}}$

Step 3

Simplify.

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Step 3.1

Multiply $- (- \frac{2}{3})$ .

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Step 3.1.1

Multiply $- 1$ by $- 1$ .

$\sqrt{{(\frac{7}{3} + 1 (\frac{2}{3}))}^{2} + {(2 - \frac{3}{2})}^{2}}$

Step 3.1.2

Multiply $\frac{2}{3}$ by $1$ .

$\sqrt{{(\frac{7}{3} + \frac{2}{3})}^{2} + {(2 - \frac{3}{2})}^{2}}$

Step 3.2

Combine the numerators over the common denominator.

$\sqrt{{(\frac{7 + 2}{3})}^{2} + {(2 - \frac{3}{2})}^{2}}$

Step 3.3

Simplify the expression.

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Step 3.3.1

Add $7$ and $2$ .

$\sqrt{{(\frac{9}{3})}^{2} + {(2 - \frac{3}{2})}^{2}}$

Step 3.3.2

Divide $9$ by $3$ .

$\sqrt{3^{2} + {(2 - \frac{3}{2})}^{2}}$

Step 3.3.3

Raise $3$ to the power of $2$ .

$\sqrt{9 + {(2 - \frac{3}{2})}^{2}}$

Step 3.4

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

$\sqrt{9 + {(2 \cdot \frac{2}{2} - \frac{3}{2})}^{2}}$

Step 3.5

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

$\sqrt{9 + {(\frac{2 \cdot 2}{2} - \frac{3}{2})}^{2}}$

Step 3.6

Combine the numerators over the common denominator.

$\sqrt{9 + {(\frac{2 \cdot 2 - 3}{2})}^{2}}$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $2$ by $2$ .

$\sqrt{9 + {(\frac{4 - 3}{2})}^{2}}$

Step 3.7.2

Subtract $3$ from $4$ .

$\sqrt{9 + {(\frac{1}{2})}^{2}}$

Step 3.8

Apply the product rule to $\frac{1}{2}$ .

$\sqrt{9 + \frac{1^{2}}{2^{2}}}$

Step 3.9

One to any power is one.

$\sqrt{9 + \frac{1}{2^{2}}}$

Step 3.10

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

$\sqrt{9 + \frac{1}{4}}$

Step 3.11

To write $9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{9 \cdot \frac{4}{4} + \frac{1}{4}}$

Step 3.12

Combine $9$ and $\frac{4}{4}$ .

$\sqrt{\frac{9 \cdot 4}{4} + \frac{1}{4}}$

Step 3.13

Combine the numerators over the common denominator.

$\sqrt{\frac{9 \cdot 4 + 1}{4}}$

Step 3.14

Simplify the numerator.

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Step 3.14.1

Multiply $9$ by $4$ .

$\sqrt{\frac{36 + 1}{4}}$

Step 3.14.2

Add $36$ and $1$ .

$\sqrt{\frac{37}{4}}$

Step 3.15

Rewrite $\sqrt{\frac{37}{4}}$ as $\frac{\sqrt{37}}{\sqrt{4}}$ .

$\frac{\sqrt{37}}{\sqrt{4}}$

Step 3.16

Simplify the denominator.

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Step 3.16.1

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{37}}{\sqrt{2^{2}}}$

Step 3.16.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{37}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{37}}{2}$

Decimal Form:

$3.04138126 \dots$

Step 5