Calculus Examples

\sqrt{x}

$\sqrt{x}$ ;

4 \leq x \leq 9

$4 \leq x \leq 9$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

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

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

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

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

Step 1.1.1.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = \frac{1}{2}

$n = \frac{1}{2}$ .

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

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

Step 1.1.1.3

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 1.1.1.4

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 1.1.1.5

Combine the numerators over the common denominator.

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

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

Step 1.1.1.6

Simplify the numerator.

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Step 1.1.1.6.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 1.1.1.6.2

Subtract

2

$2$ from

1

$1$ .

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

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

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

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

Step 1.1.1.7

Move the negative in front of the fraction.

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

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

Step 1.1.1.8

Simplify.

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Step 1.1.1.8.1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 1.1.1.8.2

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

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

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

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

Step 1.1.2

The first derivative of

$f (x)$ with respect to

$x$ is

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

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

Step 1.2

Set the first derivative equal to

$0$ then solve the equation

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

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Step 1.2.1

Set the first derivative equal to

$0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$1 = 0$

Step 1.2.3

Since

$1 \neq 0$ , there are no solutions.

No solution

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Convert expressions with fractional exponents to radicals.

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Step 1.3.1.1

Apply the rule

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

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

Step 1.3.1.2

Anything raised to

$1$ is the base itself.

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

Step 1.3.2

Set the denominator in

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

$0$ to find where the expression is undefined.

$2 \sqrt{x} = 0$

Step 1.3.3

Solve for

$x$ .

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Step 1.3.3.1

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

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

Step 1.3.3.2

Simplify each side of the equation.

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Step 1.3.3.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x}$ as

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

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

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Simplify

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

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Step 1.3.3.2.2.1.1

Apply the product rule to

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

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

Step 1.3.3.2.2.1.2

Raise

$2$ to the power of

$2$ .

$4 {(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.3.3.2.2.1.3

Multiply the exponents in

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

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Step 1.3.3.2.2.1.3.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$4 x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.3.3.2.2.1.3.2

Cancel the common factor of

$2$ .

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Step 1.3.3.2.2.1.3.2.1

Cancel the common factor.

$4 x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.3.3.2.2.1.3.2.2

Rewrite the expression.

$4 x^{1} = 0^{2}$

Step 1.3.3.2.2.1.4

Simplify.

$4 x = 0^{2}$

Step 1.3.3.2.3

Simplify the right side.

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Step 1.3.3.2.3.1

Raising

$0$ to any positive power yields

$0$ .

$4 x = 0$

Step 1.3.3.3

Divide each term in

$4 x = 0$ by

$4$ and simplify.

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Step 1.3.3.3.1

Divide each term in

$4 x = 0$ by

$4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 1.3.3.3.2

Simplify the left side.

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Step 1.3.3.3.2.1

Cancel the common factor of

$4$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 1.3.3.3.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{0}{4}$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide

$0$ by

$4$ .

$x = 0$

Step 1.3.4

Set the radicand in

$\sqrt{x}$ less than

$0$ to find where the expression is undefined.

$x < 0$

Step 1.3.5

The equation is undefined where the denominator equals

$0$ , the argument of a square root is less than

$0$ , or the argument of a logarithm is less than or equal to

$0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 1.4

Evaluate

$\sqrt{x}$ at each

$x$ value where the derivative is

$0$ or undefined.

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Step 1.4.1

Evaluate at

$x = 0$ .

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Step 1.4.1.1

Substitute

$0$ for

$x$ .

$\sqrt{0}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Remove parentheses.

$\sqrt{0}$

Step 1.4.1.2.2

Rewrite

$0$ as

$0^{2}$ .

$\sqrt{0^{2}}$

Step 1.4.1.2.3

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

$0$

Step 1.4.2

List all of the points.

$(0, 0)$

Step 2

Exclude the points that are not on the interval.

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at

$x = 4$ .

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Step 3.1.1

Substitute

$4$ for

$x$ .

$\sqrt{4}$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Remove parentheses.

$\sqrt{4}$

Step 3.1.2.2

Rewrite

$4$ as

$2^{2}$ .

$\sqrt{2^{2}}$

Step 3.1.2.3

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

$2$

Step 3.2

Evaluate at

$x = 9$ .

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Step 3.2.1

Substitute

$9$ for

$x$ .

$\sqrt{9}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Remove parentheses.

$\sqrt{9}$

Step 3.2.2.2

Rewrite

$9$ as

$3^{2}$ .

$\sqrt{3^{2}}$

Step 3.2.2.3

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

$3$

Step 3.3

List all of the points.

$(4, 2), (9, 3)$

Step 4

Compare the

$f (x)$ values found for each value of

$x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest

$f (x)$ value and the minimum will occur at the lowest

$f (x)$ value.

Absolute Maximum:

$(9, 3)$

Absolute Minimum:

$(4, 2)$

Step 5