Calculus Examples

$y = \sqrt{2 x}$ , $(2, 8)$

Step 1

Write $y = \sqrt{2 x}$ as a function.

$f (x) = \sqrt{2 x}$

Step 2

Find the derivative of $f (x) = \sqrt{2 x}$ .

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Step 2.1

Find the first derivative.

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Step 2.1.1

Simplify with factoring out.

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Step 2.1.1.1

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

$\frac{d}{d x} [{(2 x)}^{\frac{1}{2}}]$

Step 2.1.1.2

Factor $2$ out of $2 x$ .

$\frac{d}{d x} [{(2 (x))}^{\frac{1}{2}}]$

Step 2.1.1.3

Apply the product rule to $2 (x)$ .

$\frac{d}{d x} [2^{\frac{1}{2}} x^{\frac{1}{2}}]$

Step 2.1.2

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

$2^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 2.1.3

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

$2^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 2.1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 2.1.5

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

$2^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 2.1.6

Combine the numerators over the common denominator.

$2^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 2.1.7

Simplify the numerator.

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Step 2.1.7.1

Multiply $- 1$ by $2$ .

$2^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 2.1.7.2

Subtract $2$ from $1$ .

$2^{\frac{1}{2}} (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 2.1.8

Move the negative in front of the fraction.

$2^{\frac{1}{2}} (\frac{1}{2} x^{- \frac{1}{2}})$

Step 2.1.9

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$2^{\frac{1}{2}} \frac{x^{- \frac{1}{2}}}{2}$

Step 2.1.10

Combine $2^{\frac{1}{2}}$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{2^{\frac{1}{2}} x^{- \frac{1}{2}}}{2}$

Step 2.1.11

Simplify the expression.

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Step 2.1.11.1

Move $2^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{x^{- \frac{1}{2}}}{2 \cdot 2^{- \frac{1}{2}}}$

Step 2.1.11.2

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 \cdot 2^{- \frac{1}{2}} x^{\frac{1}{2}}}$

Step 2.1.12

Multiply $2$ by $2^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.1.12.1

Multiply $2$ by $2^{- \frac{1}{2}}$ .

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Step 2.1.12.1.1

Raise $2$ to the power of $1$ .

$\frac{1}{2^{1} \cdot 2^{- \frac{1}{2}} x^{\frac{1}{2}}}$

Step 2.1.12.1.2

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

$\frac{1}{2^{1 - \frac{1}{2}} x^{\frac{1}{2}}}$

Step 2.1.12.2

Write $1$ as a fraction with a common denominator.

$\frac{1}{2^{\frac{2}{2} - \frac{1}{2}} x^{\frac{1}{2}}}$

Step 2.1.12.3

Combine the numerators over the common denominator.

$\frac{1}{2^{\frac{2 - 1}{2}} x^{\frac{1}{2}}}$

Step 2.1.12.4

Subtract $1$ from $2$ .

$f' (x) = \frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$ .

$\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$

Step 3

To find the average value of a function, the function should be continuous on the closed interval $[a, b]$ . To find whether $f' (x)$ is continuous on $[2, 8]$ or not, find the domain of $f' (x) = \frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$ .

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{1}{\sqrt{2^{1}} x^{\frac{1}{2}}}$

Step 3.1.2

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{1}{\sqrt{2^{1}} \sqrt{x^{1}}}$

Step 3.1.3

Anything raised to $1$ is the base itself.

$\frac{1}{\sqrt{2} \sqrt{x^{1}}}$

Step 3.1.4

Anything raised to $1$ is the base itself.

$\frac{1}{\sqrt{2} \sqrt{x}}$

Step 3.2

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 3.3

Set the denominator in $\frac{1}{\sqrt{2} \sqrt{x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{2} \sqrt{x} = 0$

Step 3.4

Solve for $x$ .

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Step 3.4.1

To remove the radical on the left side of the equation, square both sides of the equation.

${(\sqrt{2} \sqrt{x})}^{2} = 0^{2}$

Step 3.4.2

Simplify each side of the equation.

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Step 3.4.2.1

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

${(\sqrt{2} x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Simplify ${(\sqrt{2} x^{\frac{1}{2}})}^{2}$ .

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Step 3.4.2.2.1.1

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

${\sqrt{2}}^{2} {(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.4.2.2.1.2

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

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Step 3.4.2.2.1.2.1

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

${(2^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.4.2.2.1.2.2

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

$2^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.4.2.2.1.2.3

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

$2^{\frac{2}{2}} {(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.4.2.2.1.2.4

Cancel the common factor of $2$ .

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Step 3.4.2.2.1.2.4.1

Cancel the common factor.

$2^{\frac{2}{2}} {(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.4.2.2.1.2.4.2

Rewrite the expression.

$2^{1} {(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.4.2.2.1.2.5

Evaluate the exponent.

$2 {(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.4.2.2.1.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 3.4.2.2.1.3.1

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

$2 x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.4.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.4.2.2.1.3.2.1

Cancel the common factor.

$2 x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.4.2.2.1.3.2.2

Rewrite the expression.

$2 x^{1} = 0^{2}$

Step 3.4.2.2.1.4

Simplify.

$2 x = 0^{2}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

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

$2 x = 0$

Step 3.4.3

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 3.4.3.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.4.3.2

Simplify the left side.

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Step 3.4.3.2.1

Cancel the common factor of $2$ .

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Step 3.4.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 3.4.3.3

Simplify the right side.

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Step 3.4.3.3.1

Divide $0$ by $2$ .

$x = 0$

Step 3.5

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Step 4

$f' (x)$ is continuous on $[2, 8]$ .

$f' (x)$ is continuous

Step 5

The average value of function $f'$ over the interval $[a, b]$ is defined as $A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

$A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

Step 6

Substitute the actual values into the formula for the average value of a function.

$A (x) = \frac{1}{8 - 2} (\int_{2}^{8} \frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}} d x)$

Step 7

Since $\frac{1}{2^{\frac{1}{2}}}$ is constant with respect to $x$ , move $\frac{1}{2^{\frac{1}{2}}}$ out of the integral.

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot \int_{2}^{8} \frac{1}{x^{\frac{1}{2}}} d x)$

Step 8

Apply basic rules of exponents.

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Step 8.1

Move $x^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot \int_{2}^{8} {(x^{\frac{1}{2}})}^{- 1} d x)$

Step 8.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 8.2.1

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

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot \int_{2}^{8} x^{\frac{1}{2} \cdot - 1} d x)$

Step 8.2.2

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

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot \int_{2}^{8} x^{\frac{- 1}{2}} d x)$

Step 8.2.3

Move the negative in front of the fraction.

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot \int_{2}^{8} x^{- \frac{1}{2}} d x)$

Step 9

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

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot 2 x^{\frac{1}{2}}]_{2}^{8})$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $2 x^{\frac{1}{2}}$ at $8$ and at $2$ .

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 2^{\frac{1}{2}}))$

Step 10.2

Simplify.

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Step 10.2.1

Rewrite $8$ as $2^{3}$ .

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2 \cdot {(2^{3})}^{\frac{1}{2}} - 2 \cdot 2^{\frac{1}{2}}))$

Step 10.2.2

Multiply the exponents in ${(2^{3})}^{\frac{1}{2}}$ .

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Step 10.2.2.1

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

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2 \cdot 2^{3 (\frac{1}{2})} - 2 \cdot 2^{\frac{1}{2}}))$

Step 10.2.2.2

Combine $3$ and $\frac{1}{2}$ .

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2 \cdot 2^{\frac{3}{2}} - 2 \cdot 2^{\frac{1}{2}}))$

Step 10.2.3

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

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2^{1 + \frac{3}{2}} - 2 \cdot 2^{\frac{1}{2}}))$

Step 10.2.4

Write $1$ as a fraction with a common denominator.

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2^{\frac{2}{2} + \frac{3}{2}} - 2 \cdot 2^{\frac{1}{2}}))$

Step 10.2.5

Combine the numerators over the common denominator.

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2^{\frac{2 + 3}{2}} - 2 \cdot 2^{\frac{1}{2}}))$

Step 10.2.6

Add $2$ and $3$ .

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2^{\frac{5}{2}} - 2 \cdot 2^{\frac{1}{2}}))$

Step 10.2.7

Factor out negative.

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2^{\frac{5}{2}} - (2 \cdot 2^{\frac{1}{2}})))$

Step 10.2.8

Raise $2$ to the power of $1$ .

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2^{\frac{5}{2}} - (2 \cdot 2^{\frac{1}{2}})))$

Step 10.2.9

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

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2^{\frac{5}{2}} - 2^{1 + \frac{1}{2}}))$

Step 10.2.10

Write $1$ as a fraction with a common denominator.

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2^{\frac{5}{2}} - 2^{\frac{2}{2} + \frac{1}{2}}))$

Step 10.2.11

Combine the numerators over the common denominator.

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2^{\frac{5}{2}} - 2^{\frac{2 + 1}{2}}))$

Step 10.2.12

Add $2$ and $1$ .

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2^{\frac{5}{2}} - 2^{\frac{3}{2}}))$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot 2^{\frac{5}{2}} + \frac{1}{2^{\frac{1}{2}}} \cdot (- 2^{\frac{3}{2}}))$

Step 11.2

Cancel the common factor of $2^{\frac{1}{2}}$ .

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Step 11.2.1

Factor $2^{\frac{1}{2}}$ out of $2^{\frac{5}{2}}$ .

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2^{\frac{1}{2}} \cdot 2^{\frac{4}{2}}) + \frac{1}{2^{\frac{1}{2}}} \cdot (- 2^{\frac{3}{2}}))$

Step 11.2.2

Cancel the common factor.

$A (x) = \frac{1}{8 - 2} (\frac{1}{2^{\frac{1}{2}}} \cdot (2^{\frac{1}{2}} \cdot 2^{\frac{4}{2}}) + \frac{1}{2^{\frac{1}{2}}} \cdot (- 2^{\frac{3}{2}}))$

Step 11.2.3

Rewrite the expression.

$A (x) = \frac{1}{8 - 2} (2^{\frac{4}{2}} + \frac{1}{2^{\frac{1}{2}}} \cdot (- 2^{\frac{3}{2}}))$

Step 11.3

Cancel the common factor of $2^{\frac{1}{2}}$ .

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Step 11.3.1

Factor $2^{\frac{1}{2}}$ out of $- 2^{\frac{3}{2}}$ .

$A (x) = \frac{1}{8 - 2} (2^{\frac{4}{2}} + \frac{1}{2^{\frac{1}{2}}} \cdot (2^{\frac{1}{2}} (- 2^{\frac{2}{2}})))$

Step 11.3.2

Cancel the common factor.

$A (x) = \frac{1}{8 - 2} (2^{\frac{4}{2}} + \frac{1}{2^{\frac{1}{2}}} \cdot (2^{\frac{1}{2}} (- 2^{\frac{2}{2}})))$

Step 11.3.3

Rewrite the expression.

$A (x) = \frac{1}{8 - 2} (2^{\frac{4}{2}} - 2^{\frac{2}{2}})$

Step 11.4

Simplify each term.

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Step 11.4.1

Divide $4$ by $2$ .

$A (x) = \frac{1}{8 - 2} (2^{2} - 2^{\frac{2}{2}})$

Step 11.4.2

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

$A (x) = \frac{1}{8 - 2} (4 - 2^{\frac{2}{2}})$

Step 11.4.3

Divide $2$ by $2$ .

$A (x) = \frac{1}{8 - 2} (4 - 2)$

Step 11.4.4

Evaluate the exponent.

$A (x) = \frac{1}{8 - 2} (4 - 1 \cdot 2)$

Step 11.4.5

Multiply $- 1$ by $2$ .

$A (x) = \frac{1}{8 - 2} (4 - 2)$

Step 11.5

Subtract $2$ from $4$ .

$A (x) = \frac{1}{8 - 2} (2)$

Step 12

Subtract $2$ from $8$ .

$A (x) = \frac{1}{6} \cdot 2$

Step 13

Cancel the common factor of $2$ .

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Step 13.1

Factor $2$ out of $6$ .

$A (x) = \frac{1}{2 (3)} \cdot 2$

Step 13.2

Cancel the common factor.

$A (x) = \frac{1}{2 \cdot 3} \cdot 2$

Step 13.3

Rewrite the expression.

$A (x) = \frac{1}{3}$

Step 14