Calculus Examples

$f (x) = (\frac{4}{5}) x^{\frac{5}{4}}$ , $[0, 4]$

Step 1

Check if $f (x) = (\frac{4}{5}) x^{\frac{5}{4}}$ is continuous.

Tap for more steps...

Step 1.1

To find whether the function is continuous on $[0, 4]$ or not, find the domain of $f (x) = (\frac{4}{5}) x^{\frac{5}{4}}$ .

Tap for more steps...

Step 1.1.1

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

$\frac{4}{5} \sqrt[4]{x^{5}}$

Step 1.1.2

Set the radicand in $\sqrt[4]{x^{5}}$ greater than or equal to $0$ to find where the expression is defined.

$x^{5} \geq 0$

Step 1.1.3

Solve for $x$ .

Tap for more steps...

Step 1.1.3.1

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt[5]{x^{5}} \geq \sqrt[5]{0}$

Step 1.1.3.2

Simplify the equation.

Tap for more steps...

Step 1.1.3.2.1

Simplify the left side.

Tap for more steps...

Step 1.1.3.2.1.1

Pull terms out from under the radical.

$x \geq \sqrt[5]{0}$

Step 1.1.3.2.2

Simplify the right side.

Tap for more steps...

Step 1.1.3.2.2.1

Simplify $\sqrt[5]{0}$ .

Tap for more steps...

Step 1.1.3.2.2.1.1

Rewrite $0$ as $0^{5}$ .

$x \geq \sqrt[5]{0^{5}}$

Step 1.1.3.2.2.1.2

Pull terms out from under the radical.

$x \geq 0$

Step 1.1.4

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 1.2

$f (x)$ is continuous on $[0, 4]$ .

The function is continuous.

Step 2

Check if $f (x) = (\frac{4}{5}) x^{\frac{5}{4}}$ is differentiable.

Tap for more steps...

Step 2.1

Find the derivative.

Tap for more steps...

Step 2.1.1

Find the first derivative.

Tap for more steps...

Step 2.1.1.1

Combine $\frac{4}{5}$ and $x^{\frac{5}{4}}$ .

$\frac{d}{d x} [\frac{4 x^{\frac{5}{4}}}{5}]$

Step 2.1.1.2

Since $\frac{4}{5}$ is constant with respect to $x$ , the derivative of $\frac{4 x^{\frac{5}{4}}}{5}$ with respect to $x$ is $\frac{4}{5} \frac{d}{d x} [x^{\frac{5}{4}}]$ .

$\frac{4}{5} \frac{d}{d x} [x^{\frac{5}{4}}]$

Step 2.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{4}$ .

$\frac{4}{5} (\frac{5}{4} x^{\frac{5}{4} - 1})$

Step 2.1.1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{4}{5} (\frac{5}{4} x^{\frac{5}{4} - 1 \cdot \frac{4}{4}})$

Step 2.1.1.5

Combine $- 1$ and $\frac{4}{4}$ .

$\frac{4}{5} (\frac{5}{4} x^{\frac{5}{4} + \frac{- 1 \cdot 4}{4}})$

Step 2.1.1.6

Combine the numerators over the common denominator.

$\frac{4}{5} (\frac{5}{4} x^{\frac{5 - 1 \cdot 4}{4}})$

Step 2.1.1.7

Simplify the numerator.

Tap for more steps...

Step 2.1.1.7.1

Multiply $- 1$ by $4$ .

$\frac{4}{5} (\frac{5}{4} x^{\frac{5 - 4}{4}})$

Step 2.1.1.7.2

Subtract $4$ from $5$ .

$\frac{4}{5} (\frac{5}{4} x^{\frac{1}{4}})$

Step 2.1.1.8

Combine $\frac{5}{4}$ and $x^{\frac{1}{4}}$ .

$\frac{4}{5} \cdot \frac{5 x^{\frac{1}{4}}}{4}$

Step 2.1.1.9

Multiply $\frac{4}{5}$ by $\frac{5 x^{\frac{1}{4}}}{4}$ .

$\frac{4 (5 x^{\frac{1}{4}})}{5 \cdot 4}$

Step 2.1.1.10

Multiply.

Tap for more steps...

Step 2.1.1.10.1

Multiply $5$ by $4$ .

$\frac{20 x^{\frac{1}{4}}}{5 \cdot 4}$

Step 2.1.1.10.2

Multiply $5$ by $4$ .

$\frac{20 x^{\frac{1}{4}}}{20}$

Step 2.1.1.11

Cancel the common factor.

$\frac{20 x^{\frac{1}{4}}}{20}$

Step 2.1.1.12

Divide $x^{\frac{1}{4}}$ by $1$ .

$f' (x) = x^{\frac{1}{4}}$

Step 2.1.2

The first derivative of $f (x)$ with respect to $x$ is $x^{\frac{1}{4}}$ .

$x^{\frac{1}{4}}$

Step 2.2

Find if the derivative is continuous on $[0, 4]$ .

Tap for more steps...

Step 2.2.1

To find whether the function is continuous on $[0, 4]$ or not, find the domain of $f' (x) = x^{\frac{1}{4}}$ .

Tap for more steps...

Step 2.2.1.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 2.2.1.1.1

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

$\sqrt[4]{x^{1}}$

Step 2.2.1.1.2

Anything raised to $1$ is the base itself.

$\sqrt[4]{x}$

Step 2.2.1.2

Set the radicand in $\sqrt[4]{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 2.2.1.3

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 2.2.2

$f' (x)$ is continuous on $[0, 4]$ .

The function is continuous.

Step 2.3

The function is differentiable on $[0, 4]$ because the derivative is continuous on $[0, 4]$ .

The function is differentiable.

Step 3

For arc length to be guaranteed, the function and its derivative must both be continuous on the closed interval $[0, 4]$ .

The function and its derivative are continuous on the closed interval $[0, 4]$ .

Step 4

Find the derivative of $f (x) = (\frac{4}{5}) x^{\frac{5}{4}}$ .

Tap for more steps...

Step 4.1

Combine $\frac{4}{5}$ and $x^{\frac{5}{4}}$ .

$\frac{d}{d x} [\frac{4 x^{\frac{5}{4}}}{5}]$

Step 4.2

Since $\frac{4}{5}$ is constant with respect to $x$ , the derivative of $\frac{4 x^{\frac{5}{4}}}{5}$ with respect to $x$ is $\frac{4}{5} \frac{d}{d x} [x^{\frac{5}{4}}]$ .

$\frac{4}{5} \frac{d}{d x} [x^{\frac{5}{4}}]$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{4}$ .

$\frac{4}{5} (\frac{5}{4} x^{\frac{5}{4} - 1})$

Step 4.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{4}{5} (\frac{5}{4} x^{\frac{5}{4} - 1 \cdot \frac{4}{4}})$

Step 4.5

Combine $- 1$ and $\frac{4}{4}$ .

$\frac{4}{5} (\frac{5}{4} x^{\frac{5}{4} + \frac{- 1 \cdot 4}{4}})$

Step 4.6

Combine the numerators over the common denominator.

$\frac{4}{5} (\frac{5}{4} x^{\frac{5 - 1 \cdot 4}{4}})$

Step 4.7

Simplify the numerator.

Tap for more steps...

Step 4.7.1

Multiply $- 1$ by $4$ .

$\frac{4}{5} (\frac{5}{4} x^{\frac{5 - 4}{4}})$

Step 4.7.2

Subtract $4$ from $5$ .

$\frac{4}{5} (\frac{5}{4} x^{\frac{1}{4}})$

Step 4.8

Combine $\frac{5}{4}$ and $x^{\frac{1}{4}}$ .

$\frac{4}{5} \cdot \frac{5 x^{\frac{1}{4}}}{4}$

Step 4.9

Multiply $\frac{4}{5}$ by $\frac{5 x^{\frac{1}{4}}}{4}$ .

$\frac{4 (5 x^{\frac{1}{4}})}{5 \cdot 4}$

Step 4.10

Multiply.

Tap for more steps...

Step 4.10.1

Multiply $5$ by $4$ .

$\frac{20 x^{\frac{1}{4}}}{5 \cdot 4}$

Step 4.10.2

Multiply $5$ by $4$ .

$\frac{20 x^{\frac{1}{4}}}{20}$

Step 4.11

Cancel the common factor.

$\frac{20 x^{\frac{1}{4}}}{20}$

Step 4.12

Divide $x^{\frac{1}{4}}$ by $1$ .

$x^{\frac{1}{4}}$

Step 5

To find the arc length of a function, use the formula $L = \int_{a}^{b} \sqrt{1 + {(f' (x))}^{2}} d x$ .

$\int_{0}^{4} \sqrt{1 + {(x^{\frac{1}{4}})}^{2}} d x$

Step 6