Calculus Examples

$f (x) = \frac{2}{3} x^{\frac{3}{2}} + 1$ , $[0, 4]$

Step 1

Check if $f (x) = \frac{2}{3} x^{\frac{3}{2}} + 1$ is continuous.

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Step 1.1

To find whether the function is continuous on $[0, 4]$ or not, find the domain of $f (x) = \frac{2}{3} x^{\frac{3}{2}} + 1$ .

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Step 1.1.1

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

$\frac{2}{3} \sqrt{x^{3}} + 1$

Step 1.1.2

Set the radicand in $\sqrt{x^{3}}$ greater than or equal to $0$ to find where the expression is defined.

$x^{3} \geq 0$

Step 1.1.3

Solve for $x$ .

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Step 1.1.3.1

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

$\sqrt[3]{x^{3}} \geq \sqrt[3]{0}$

Step 1.1.3.2

Simplify the equation.

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Step 1.1.3.2.1

Simplify the left side.

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Step 1.1.3.2.1.1

Pull terms out from under the radical.

$x \geq \sqrt[3]{0}$

Step 1.1.3.2.2

Simplify the right side.

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Step 1.1.3.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 1.1.3.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x \geq \sqrt[3]{0^{3}}$

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 $f (x) = \frac{2}{3} x^{\frac{3}{2}} + 1$ is differentiable.

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Step 2.1

Find the derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

By the Sum Rule, the derivative of $\frac{2}{3} x^{\frac{3}{2}} + 1$ with respect to $x$ is $\frac{d}{d x} [\frac{2}{3} x^{\frac{3}{2}}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [\frac{2}{3} x^{\frac{3}{2}}] + \frac{d}{d x} [1]$

Step 2.1.1.2

Evaluate $\frac{d}{d x} [\frac{2}{3} x^{\frac{3}{2}}]$ .

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Step 2.1.1.2.1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2}{3} x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$\frac{2}{3} \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [1]$

Step 2.1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{2}{3} (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} [1]$

Step 2.1.1.2.3

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

$\frac{2}{3} (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [1]$

Step 2.1.1.2.4

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

$\frac{2}{3} (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [1]$

Step 2.1.1.2.5

Combine the numerators over the common denominator.

$\frac{2}{3} (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} [1]$

Step 2.1.1.2.6

Simplify the numerator.

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Step 2.1.1.2.6.1

Multiply $- 1$ by $2$ .

$\frac{2}{3} (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} [1]$

Step 2.1.1.2.6.2

Subtract $2$ from $3$ .

$\frac{2}{3} (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} [1]$

Step 2.1.1.2.7

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

$\frac{2}{3} \cdot \frac{3 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [1]$

Step 2.1.1.2.8

Multiply $\frac{2}{3}$ by $\frac{3 x^{\frac{1}{2}}}{2}$ .

$\frac{2 (3 x^{\frac{1}{2}})}{3 \cdot 2} + \frac{d}{d x} [1]$

Step 2.1.1.2.9

Multiply $3$ by $2$ .

$\frac{6 x^{\frac{1}{2}}}{3 \cdot 2} + \frac{d}{d x} [1]$

Step 2.1.1.2.10

Multiply $3$ by $2$ .

$\frac{6 x^{\frac{1}{2}}}{6} + \frac{d}{d x} [1]$

Step 2.1.1.2.11

Cancel the common factor.

$\frac{6 x^{\frac{1}{2}}}{6} + \frac{d}{d x} [1]$

Step 2.1.1.2.12

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

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

Step 2.1.1.3

Differentiate using the Constant Rule.

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Step 2.1.1.3.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 2.1.1.3.2

Add $x^{\frac{1}{2}}$ and $0$ .

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

To find whether the function is continuous on $[0, 4]$ or not, find the domain of $f' (x) = x^{\frac{1}{2}}$ .

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Step 2.2.1.1

Convert expressions with fractional exponents to radicals.

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Step 2.2.1.1.1

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

$\sqrt{x^{1}}$

Step 2.2.1.1.2

Anything raised to $1$ is the base itself.

$\sqrt{x}$

Step 2.2.1.2

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

$x \geq 0$

Step 2.2.1.3

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 $f (x) = \frac{2}{3} x^{\frac{3}{2}} + 1$ .

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Step 4.1

By the Sum Rule, the derivative of $\frac{2}{3} x^{\frac{3}{2}} + 1$ with respect to $x$ is $\frac{d}{d x} [\frac{2}{3} x^{\frac{3}{2}}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [\frac{2}{3} x^{\frac{3}{2}}] + \frac{d}{d x} [1]$

Step 4.2

Evaluate $\frac{d}{d x} [\frac{2}{3} x^{\frac{3}{2}}]$ .

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Step 4.2.1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2}{3} x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$\frac{2}{3} \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [1]$

Step 4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{2}{3} (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} [1]$

Step 4.2.3

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

$\frac{2}{3} (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [1]$

Step 4.2.4

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

$\frac{2}{3} (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [1]$

Step 4.2.5

Combine the numerators over the common denominator.

$\frac{2}{3} (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} [1]$

Step 4.2.6

Simplify the numerator.

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Step 4.2.6.1

Multiply $- 1$ by $2$ .

$\frac{2}{3} (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} [1]$

Step 4.2.6.2

Subtract $2$ from $3$ .

$\frac{2}{3} (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} [1]$

Step 4.2.7

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

$\frac{2}{3} \cdot \frac{3 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [1]$

Step 4.2.8

Multiply $\frac{2}{3}$ by $\frac{3 x^{\frac{1}{2}}}{2}$ .

$\frac{2 (3 x^{\frac{1}{2}})}{3 \cdot 2} + \frac{d}{d x} [1]$

Step 4.2.9

Multiply $3$ by $2$ .

$\frac{6 x^{\frac{1}{2}}}{3 \cdot 2} + \frac{d}{d x} [1]$

Step 4.2.10

Multiply $3$ by $2$ .

$\frac{6 x^{\frac{1}{2}}}{6} + \frac{d}{d x} [1]$

Step 4.2.11

Cancel the common factor.

$\frac{6 x^{\frac{1}{2}}}{6} + \frac{d}{d x} [1]$

Step 4.2.12

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

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

Step 4.3

Differentiate using the Constant Rule.

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Step 4.3.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 4.3.2

Add $x^{\frac{1}{2}}$ and $0$ .

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

Step 5

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

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

Step 6

Evaluate the integral.

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Step 6.1

Let $u = 1 + x$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1.1

Let $u = 1 + x$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1.1

Differentiate $1 + x$ .

$\frac{d}{d x} [1 + x]$

Step 6.1.1.2

By the Sum Rule, the derivative of $1 + x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [x]$

Step 6.1.1.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x]$

Step 6.1.1.4

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

$0 + 1$

Step 6.1.1.5

Add $0$ and $1$ .

$1$

Step 6.1.2

Substitute the lower limit in for $x$ in $u = 1 + x$ .

$u_{lower} = 1 + 0$

Step 6.1.3

Add $1$ and $0$ .

$u_{lower} = 1$

Step 6.1.4

Substitute the upper limit in for $x$ in $u = 1 + x$ .

$u_{upper} = 1 + 4$

Step 6.1.5

Add $1$ and $4$ .

$u_{upper} = 5$

Step 6.1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 5$

Step 6.1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{5} \sqrt{u} d u$

Step 6.2

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

$\int_{1}^{5} u^{\frac{1}{2}} d u$

Step 6.3

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

$\frac{2}{3} u^{\frac{3}{2}}]_{1}^{5}$

Step 6.4

Substitute and simplify.

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Step 6.4.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $5$ and at $1$ .

$(\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 6.4.2

Simplify.

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Step 6.4.2.1

Combine $\frac{2}{3}$ and $5^{\frac{3}{2}}$ .

$\frac{2 \cdot 5^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 6.4.2.2

One to any power is one.

$\frac{2 \cdot 5^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 1$

Step 6.4.2.3

Multiply $- 1$ by $1$ .

$\frac{2 \cdot 5^{\frac{3}{2}}}{3} - \frac{2}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \cdot 5^{\frac{3}{2}}}{3} - \frac{2}{3}$

Decimal Form:

$6.78689325 \dots$

Step 8