Calculus Examples

$f (x) = 2 \sqrt{x}$ , $[4, 9]$

Step 1

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

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

Combine the numerators over the common denominator.

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

Step 1.1.7

Simplify the numerator.

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Step 1.1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.1.7.2

Subtract $2$ from $1$ .

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

Step 1.1.8

Move the negative in front of the fraction.

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

Step 1.1.9

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

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

Step 1.1.10

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

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

Step 1.1.11

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

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

Step 1.1.12

Cancel the common factor.

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

Step 1.1.13

Rewrite the expression.

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

Step 1.2

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

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

Step 2

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 $[4, 9]$ or not, find the domain of $f' (x) = \frac{1}{x^{\frac{1}{2}}}$ .

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Step 2.1

Convert expressions with fractional exponents to radicals.

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Step 2.1.1

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

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

Step 2.1.2

Anything raised to $1$ is the base itself.

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

Step 2.2

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

$x \geq 0$

Step 2.3

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

$\sqrt{x} = 0$

Step 2.4

Solve for $x$ .

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Step 2.4.1

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

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

Step 2.4.2

Simplify each side of the equation.

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Step 2.4.2.1

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

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

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

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

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Step 2.4.2.2.1.1

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

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Step 2.4.2.2.1.1.1

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

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

Step 2.4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.4.2.2.1.1.2.1

Cancel the common factor.

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

Step 2.4.2.2.1.1.2.2

Rewrite the expression.

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

Step 2.4.2.2.1.2

Simplify.

$x = 0^{2}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

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

$x = 0$

Step 2.5

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 3

$f' (x)$ is continuous on $[4, 9]$ .

$f' (x)$ is continuous

Step 4

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 5

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

$A (x) = \frac{1}{9 - 4} (\int_{4}^{9} \frac{1}{x^{\frac{1}{2}}} d x)$

Step 6

Apply basic rules of exponents.

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Step 6.1

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

$A (x) = \frac{1}{9 - 4} (\int_{4}^{9} {(x^{\frac{1}{2}})}^{- 1} d x)$

Step 6.2

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

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Step 6.2.1

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

$A (x) = \frac{1}{9 - 4} (\int_{4}^{9} x^{\frac{1}{2} \cdot - 1} d x)$

Step 6.2.2

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

$A (x) = \frac{1}{9 - 4} (\int_{4}^{9} x^{\frac{- 1}{2}} d x)$

Step 6.2.3

Move the negative in front of the fraction.

$A (x) = \frac{1}{9 - 4} (\int_{4}^{9} x^{- \frac{1}{2}} d x)$

Step 7

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

$A (x) = \frac{1}{9 - 4} (2 x^{\frac{1}{2}}]_{4}^{9})$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $2 x^{\frac{1}{2}}$ at $9$ and at $4$ .

$A (x) = \frac{1}{9 - 4} ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 8.2

Simplify.

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Step 8.2.1

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

$A (x) = \frac{1}{9 - 4} (2 \cdot {(3^{2})}^{\frac{1}{2}} - 2 \cdot 4^{\frac{1}{2}})$

Step 8.2.2

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

$A (x) = \frac{1}{9 - 4} (2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 4^{\frac{1}{2}})$

Step 8.2.3

Cancel the common factor of $2$ .

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Step 8.2.3.1

Cancel the common factor.

$A (x) = \frac{1}{9 - 4} (2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 4^{\frac{1}{2}})$

Step 8.2.3.2

Rewrite the expression.

$A (x) = \frac{1}{9 - 4} (2 \cdot 3 - 2 \cdot 4^{\frac{1}{2}})$

Step 8.2.4

Evaluate the exponent.

$A (x) = \frac{1}{9 - 4} (2 \cdot 3 - 2 \cdot 4^{\frac{1}{2}})$

Step 8.2.5

Multiply $2$ by $3$ .

$A (x) = \frac{1}{9 - 4} (6 - 2 \cdot 4^{\frac{1}{2}})$

Step 8.2.6

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

$A (x) = \frac{1}{9 - 4} (6 - 2 \cdot {(2^{2})}^{\frac{1}{2}})$

Step 8.2.7

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

$A (x) = \frac{1}{9 - 4} (6 - 2 \cdot 2^{2 (\frac{1}{2})})$

Step 8.2.8

Cancel the common factor of $2$ .

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Step 8.2.8.1

Cancel the common factor.

$A (x) = \frac{1}{9 - 4} (6 - 2 \cdot 2^{2 (\frac{1}{2})})$

Step 8.2.8.2

Rewrite the expression.

$A (x) = \frac{1}{9 - 4} (6 - 2 \cdot 2)$

Step 8.2.9

Evaluate the exponent.

$A (x) = \frac{1}{9 - 4} (6 - 2 \cdot 2)$

Step 8.2.10

Multiply $- 2$ by $2$ .

$A (x) = \frac{1}{9 - 4} (6 - 4)$

Step 8.2.11

Subtract $4$ from $6$ .

$A (x) = \frac{1}{9 - 4} (2)$

Step 9

Subtract $4$ from $9$ .

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

Step 10

Combine $\frac{1}{5}$ and $2$ .

$A (x) = \frac{2}{5}$

Step 11