Precalculus Examples

$g (t) = \frac{t}{\sqrt{9 + t^{2}}}$ , $[4, 9]$

Step 1

Write $g (t) = \frac{t}{\sqrt{9 + t^{2}}}$ as an equation.

$y = \frac{t}{\sqrt{9 + t^{2}}}$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $t$ values of the two points.

$\frac{f (9) - f (4)}{(9) - (4)}$

Step 2.2

Substitute the equation $y = \frac{t}{\sqrt{9 + t^{2}}}$ for $f (9)$ and $f (4)$ , replacing $t$ in the function with the corresponding $t$ value.

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

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}}$ .

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Step 3.1.1

Multiply $\frac{\frac{9}{\sqrt{9 + 9^{2}}} - \frac{4}{\sqrt{9 + 4^{2}}}}{9 - (4)}$ by $\frac{\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}}}{\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}}}$ .

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $\sqrt{9 + 9^{2}}$ .

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Step 3.3.1.1

Factor $\sqrt{9 + 9^{2}}$ out of $\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}}$ .

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

Step 3.3.1.2

Cancel the common factor.

Step 3.3.1.3

Rewrite the expression.

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

Step 3.3.2

Cancel the common factor of $\sqrt{9 + 4^{2}}$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{4}{\sqrt{9 + 4^{2}}}$ into the numerator.

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

Step 3.3.2.2

Factor $\sqrt{9 + 4^{2}}$ out of $\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}}$ .

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

Step 3.3.2.3

Cancel the common factor.

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

Step 3.3.2.4

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Raise $4$ to the power of $2$ .

$\frac{\sqrt{9 + 16} \cdot 9 + \sqrt{9 + 9^{2}} \cdot - 4}{\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.4.2

Add $9$ and $16$ .

$\frac{\sqrt{25} \cdot 9 + \sqrt{9 + 9^{2}} \cdot - 4}{\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.4.3

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2}} \cdot 9 + \sqrt{9 + 9^{2}} \cdot - 4}{\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.4.4

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

$\frac{5 \cdot 9 + \sqrt{9 + 9^{2}} \cdot - 4}{\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.4.5

Multiply $5$ by $9$ .

$\frac{45 + \sqrt{9 + 9^{2}} \cdot - 4}{\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.4.6

Raise $9$ to the power of $2$ .

$\frac{45 + \sqrt{9 + 81} \cdot - 4}{\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.4.7

Add $9$ and $81$ .

$\frac{45 + \sqrt{90} \cdot - 4}{\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.4.8

Rewrite $90$ as $3^{2} \cdot 10$ .

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Step 3.4.8.1

Factor $9$ out of $90$ .

$\frac{45 + \sqrt{9 (10)} \cdot - 4}{\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.4.8.2

Rewrite $9$ as $3^{2}$ .

$\frac{45 + \sqrt{3^{2} \cdot 10} \cdot - 4}{\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.4.9

Pull terms out from under the radical.

$\frac{45 + 3 \sqrt{10} \cdot - 4}{\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.4.10

Multiply $- 4$ by $3$ .

$\frac{45 - 12 \sqrt{10}}{\sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Combine using the product rule for radicals.

$\frac{45 - 12 \sqrt{10}}{\sqrt{(9 + 9^{2}) (9 + 4^{2})} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.5.2

Raise $9$ to the power of $2$ .

$\frac{45 - 12 \sqrt{10}}{\sqrt{(9 + 81) (9 + 4^{2})} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.5.3

Add $9$ and $81$ .

$\frac{45 - 12 \sqrt{10}}{\sqrt{90 (9 + 4^{2})} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.5.4

Raise $4$ to the power of $2$ .

$\frac{45 - 12 \sqrt{10}}{\sqrt{90 (9 + 16)} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.5.5

Add $9$ and $16$ .

$\frac{45 - 12 \sqrt{10}}{\sqrt{90 \cdot 25} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.5.6

Multiply $90$ by $25$ .

$\frac{45 - 12 \sqrt{10}}{\sqrt{2250} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.5.7

Rewrite $2250$ as $15^{2} \cdot 10$ .

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Step 3.5.7.1

Factor $225$ out of $2250$ .

$\frac{45 - 12 \sqrt{10}}{\sqrt{225 (10)} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.5.7.2

Rewrite $225$ as $15^{2}$ .

$\frac{45 - 12 \sqrt{10}}{\sqrt{15^{2} \cdot 10} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.5.8

Pull terms out from under the radical.

$\frac{45 - 12 \sqrt{10}}{15 \sqrt{10} \cdot 9 + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.5.9

Multiply $9$ by $15$ .

$\frac{45 - 12 \sqrt{10}}{135 \sqrt{10} + \sqrt{9 + 9^{2}} \sqrt{9 + 4^{2}} (- (4))}$

Step 3.5.10

Combine using the product rule for radicals.

$\frac{45 - 12 \sqrt{10}}{135 \sqrt{10} + \sqrt{(9 + 9^{2}) (9 + 4^{2})} (- (4))}$

Step 3.5.11

Raise $9$ to the power of $2$ .

$\frac{45 - 12 \sqrt{10}}{135 \sqrt{10} + \sqrt{(9 + 81) (9 + 4^{2})} (- (4))}$

Step 3.5.12

Add $9$ and $81$ .

$\frac{45 - 12 \sqrt{10}}{135 \sqrt{10} + \sqrt{90 (9 + 4^{2})} (- (4))}$

Step 3.5.13

Raise $4$ to the power of $2$ .

$\frac{45 - 12 \sqrt{10}}{135 \sqrt{10} + \sqrt{90 (9 + 16)} (- (4))}$

Step 3.5.14

Add $9$ and $16$ .

$\frac{45 - 12 \sqrt{10}}{135 \sqrt{10} + \sqrt{90 \cdot 25} (- (4))}$

Step 3.5.15

Multiply $90$ by $25$ .

$\frac{45 - 12 \sqrt{10}}{135 \sqrt{10} + \sqrt{2250} (- (4))}$

Step 3.5.16

Rewrite $2250$ as $15^{2} \cdot 10$ .

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Step 3.5.16.1

Factor $225$ out of $2250$ .

$\frac{45 - 12 \sqrt{10}}{135 \sqrt{10} + \sqrt{225 (10)} (- (4))}$

Step 3.5.16.2

Rewrite $225$ as $15^{2}$ .

$\frac{45 - 12 \sqrt{10}}{135 \sqrt{10} + \sqrt{15^{2} \cdot 10} (- (4))}$

Step 3.5.17

Pull terms out from under the radical.

$\frac{45 - 12 \sqrt{10}}{135 \sqrt{10} + 15 \sqrt{10} (- (4))}$

Step 3.5.18

Multiply $- 1$ by $4$ .

$\frac{45 - 12 \sqrt{10}}{135 \sqrt{10} + 15 \sqrt{10} \cdot - 4}$

Step 3.5.19

Multiply $- 4$ by $15$ .

$\frac{45 - 12 \sqrt{10}}{135 \sqrt{10} - 60 \sqrt{10}}$

Step 3.5.20

Subtract $60 \sqrt{10}$ from $135 \sqrt{10}$ .

$\frac{45 - 12 \sqrt{10}}{75 \sqrt{10}}$

Step 3.6

Cancel the common factor of $45 - 12 \sqrt{10}$ and $75$ .

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Step 3.6.1

Factor $3$ out of $45$ .

$\frac{3 (15) - 12 \sqrt{10}}{75 \sqrt{10}}$

Step 3.6.2

Factor $3$ out of $- 12 \sqrt{10}$ .

$\frac{3 (15) + 3 (- 4 \sqrt{10})}{75 \sqrt{10}}$

Step 3.6.3

Factor $3$ out of $3 (15) + 3 (- 4 \sqrt{10})$ .

$\frac{3 (15 - 4 \sqrt{10})}{75 \sqrt{10}}$

Step 3.6.4

Cancel the common factors.

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Step 3.6.4.1

Factor $3$ out of $75 \sqrt{10}$ .

$\frac{3 (15 - 4 \sqrt{10})}{3 (25 \sqrt{10})}$

Step 3.6.4.2

Cancel the common factor.

$\frac{3 (15 - 4 \sqrt{10})}{3 (25 \sqrt{10})}$

Step 3.6.4.3

Rewrite the expression.

$\frac{15 - 4 \sqrt{10}}{25 \sqrt{10}}$

Step 3.7

Multiply $\frac{15 - 4 \sqrt{10}}{25 \sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{15 - 4 \sqrt{10}}{25 \sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 3.8

Combine and simplify the denominator.

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Step 3.8.1

Multiply $\frac{15 - 4 \sqrt{10}}{25 \sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{(15 - 4 \sqrt{10}) \sqrt{10}}{25 \sqrt{10} \sqrt{10}}$

Step 3.8.2

Move $\sqrt{10}$ .

$\frac{(15 - 4 \sqrt{10}) \sqrt{10}}{25 (\sqrt{10} \sqrt{10})}$

Step 3.8.3

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{(15 - 4 \sqrt{10}) \sqrt{10}}{25 ({\sqrt{10}}^{1} \sqrt{10})}$

Step 3.8.4

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{(15 - 4 \sqrt{10}) \sqrt{10}}{25 ({\sqrt{10}}^{1} {\sqrt{10}}^{1})}$

Step 3.8.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(15 - 4 \sqrt{10}) \sqrt{10}}{25 {\sqrt{10}}^{1 + 1}}$

Step 3.8.6

Add $1$ and $1$ .

$\frac{(15 - 4 \sqrt{10}) \sqrt{10}}{25 {\sqrt{10}}^{2}}$

Step 3.8.7

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 3.8.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{(15 - 4 \sqrt{10}) \sqrt{10}}{25 {(10^{\frac{1}{2}})}^{2}}$

Step 3.8.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{(15 - 4 \sqrt{10}) \sqrt{10}}{25 \cdot 10^{\frac{1}{2} \cdot 2}}$

Step 3.8.7.3

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

$\frac{(15 - 4 \sqrt{10}) \sqrt{10}}{25 \cdot 10^{\frac{2}{2}}}$

Step 3.8.7.4

Cancel the common factor of $2$ .

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Step 3.8.7.4.1

Cancel the common factor.

$\frac{(15 - 4 \sqrt{10}) \sqrt{10}}{25 \cdot 10^{\frac{2}{2}}}$

Step 3.8.7.4.2

Rewrite the expression.

$\frac{(15 - 4 \sqrt{10}) \sqrt{10}}{25 \cdot 10^{1}}$

Step 3.8.7.5

Evaluate the exponent.

$\frac{(15 - 4 \sqrt{10}) \sqrt{10}}{25 \cdot 10}$

Step 3.9

Multiply $25$ by $10$ .

$\frac{(15 - 4 \sqrt{10}) \sqrt{10}}{250}$

Step 3.10

Apply the distributive property.

$\frac{15 \sqrt{10} - 4 \sqrt{10} \sqrt{10}}{250}$

Step 3.11

Multiply $- 4 \sqrt{10} \sqrt{10}$ .

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Step 3.11.1

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{15 \sqrt{10} - 4 ({\sqrt{10}}^{1} \sqrt{10})}{250}$

Step 3.11.2

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{15 \sqrt{10} - 4 ({\sqrt{10}}^{1} {\sqrt{10}}^{1})}{250}$

Step 3.11.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{15 \sqrt{10} - 4 {\sqrt{10}}^{1 + 1}}{250}$

Step 3.11.4

Add $1$ and $1$ .

$\frac{15 \sqrt{10} - 4 {\sqrt{10}}^{2}}{250}$

Step 3.12

Simplify each term.

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Step 3.12.1

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 3.12.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{15 \sqrt{10} - 4 {(10^{\frac{1}{2}})}^{2}}{250}$

Step 3.12.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{15 \sqrt{10} - 4 \cdot 10^{\frac{1}{2} \cdot 2}}{250}$

Step 3.12.1.3

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

$\frac{15 \sqrt{10} - 4 \cdot 10^{\frac{2}{2}}}{250}$

Step 3.12.1.4

Cancel the common factor of $2$ .

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Step 3.12.1.4.1

Cancel the common factor.

$\frac{15 \sqrt{10} - 4 \cdot 10^{\frac{2}{2}}}{250}$

Step 3.12.1.4.2

Rewrite the expression.

$\frac{15 \sqrt{10} - 4 \cdot 10^{1}}{250}$

Step 3.12.1.5

Evaluate the exponent.

$\frac{15 \sqrt{10} - 4 \cdot 10}{250}$

Step 3.12.2

Multiply $- 4$ by $10$ .

$\frac{15 \sqrt{10} - 40}{250}$

Step 3.13

Cancel the common factor of $15 \sqrt{10} - 40$ and $250$ .

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Step 3.13.1

Factor $5$ out of $15 \sqrt{10}$ .

$\frac{5 (3 \sqrt{10}) - 40}{250}$

Step 3.13.2

Factor $5$ out of $- 40$ .

$\frac{5 (3 \sqrt{10}) + 5 (- 8)}{250}$

Step 3.13.3

Factor $5$ out of $5 (3 \sqrt{10}) + 5 (- 8)$ .

$\frac{5 (3 \sqrt{10} - 8)}{250}$

Step 3.13.4

Cancel the common factors.

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Step 3.13.4.1

Factor $5$ out of $250$ .

$\frac{5 (3 \sqrt{10} - 8)}{5 (50)}$

Step 3.13.4.2

Cancel the common factor.

$\frac{5 (3 \sqrt{10} - 8)}{5 \cdot 50}$

Step 3.13.4.3

Rewrite the expression.

$\frac{3 \sqrt{10} - 8}{50}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{10} - 8}{50}$

Decimal Form:

$0.02973665 \dots$