Calculus Examples

$c (x) = \sqrt{x}$ , given $\frac{1}{121} \leq x \leq 17$

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

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

Step 1.1.1.2

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

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

Step 1.1.1.3

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

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

Step 1.1.1.4

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

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

Step 1.1.1.5

Combine the numerators over the common denominator.

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

Step 1.1.1.6

Simplify the numerator.

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Step 1.1.1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.1.1.6.2

Subtract $2$ from $1$ .

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

Step 1.1.1.7

Move the negative in front of the fraction.

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

Step 1.1.1.8

Simplify.

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Step 1.1.1.8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.1.1.8.2

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

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

Step 1.1.2

The first derivative of $c (x)$ 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 $x = \frac{1}{121}$ .

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Step 3.1.1

Substitute $\frac{1}{121}$ for $x$ .

$\sqrt{\frac{1}{121}}$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Remove parentheses.

$\sqrt{\frac{1}{121}}$

Step 3.1.2.2

Rewrite $\sqrt{\frac{1}{121}}$ as $\frac{\sqrt{1}}{\sqrt{121}}$ .

$\frac{\sqrt{1}}{\sqrt{121}}$

Step 3.1.2.3

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{121}}$

Step 3.1.2.4

Simplify the denominator.

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Step 3.1.2.4.1

Rewrite $121$ as $11^{2}$ .

$\frac{1}{\sqrt{11^{2}}}$

Step 3.1.2.4.2

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

$\frac{1}{11}$

Step 3.2

Evaluate at $x = 17$ .

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Step 3.2.1

Substitute $17$ for $x$ .

$\sqrt{17}$

Step 3.2.2

Remove parentheses.

$\sqrt{17}$

Step 3.3

List all of the points.

$(\frac{1}{121}, \frac{1}{11}), (17, \sqrt{17})$

Step 4

Compare the $c (x)$ 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 $c (x)$ value and the minimum will occur at the lowest $c (x)$ value.

Absolute Maximum: $(17, \sqrt{17})$

Absolute Minimum: $(\frac{1}{121}, \frac{1}{11})$

Step 5