Calculus Examples

$y = \frac{x^{3}}{3} + \frac{1}{4 x}$ , $1 \leq x \leq 3$

Step 1

Check if $f (x) = \frac{x^{3}}{3} + \frac{1}{4 x}$ is continuous.

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Step 1.1

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

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Step 1.1.1

Set the denominator in $\frac{1}{4 x}$ equal to $0$ to find where the expression is undefined.

$4 x = 0$

Step 1.1.2

Divide each term in $4 x = 0$ by $4$ and simplify.

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Step 1.1.2.1

Divide each term in $4 x = 0$ by $4$ .

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $4$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Divide $0$ by $4$ .

$x = 0$

Step 1.1.3

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 1.2

$f (x)$ is continuous on $[1, 3]$ .

The function is continuous.

Step 2

Check if $f (x) = \frac{x^{3}}{3} + \frac{1}{4 x}$ 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 $\frac{x^{3}}{3} + \frac{1}{4 x}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{3}}{3}] + \frac{d}{d x} [\frac{1}{4 x}]$ .

$\frac{d}{d x} [\frac{x^{3}}{3}] + \frac{d}{d x} [\frac{1}{4 x}]$

Step 2.1.1.2

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

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Step 2.1.1.2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x^{3}}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{3}]$ .

$\frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{1}{4 x}]$

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 $n = 3$ .

$\frac{1}{3} (3 x^{2}) + \frac{d}{d x} [\frac{1}{4 x}]$

Step 2.1.1.2.3

Combine $3$ and $\frac{1}{3}$ .

$\frac{3}{3} x^{2} + \frac{d}{d x} [\frac{1}{4 x}]$

Step 2.1.1.2.4

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

$\frac{3 x^{2}}{3} + \frac{d}{d x} [\frac{1}{4 x}]$

Step 2.1.1.2.5

Cancel the common factor of $3$ .

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Step 2.1.1.2.5.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{d}{d x} [\frac{1}{4 x}]$

Step 2.1.1.2.5.2

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

$x^{2} + \frac{d}{d x} [\frac{1}{4 x}]$

Step 2.1.1.3

Evaluate $\frac{d}{d x} [\frac{1}{4 x}]$ .

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Step 2.1.1.3.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4 x}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [\frac{1}{x}]$ .

$x^{2} + \frac{1}{4} \frac{d}{d x} [\frac{1}{x}]$

Step 2.1.1.3.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$x^{2} + \frac{1}{4} \frac{d}{d x} [x^{- 1}]$

Step 2.1.1.3.3

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

$x^{2} + \frac{1}{4} (- x^{- 2})$

Step 2.1.1.3.4

Combine $\frac{1}{4}$ and $x^{- 2}$ .

$x^{2} - \frac{x^{- 2}}{4}$

Step 2.1.1.3.5

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

$f' (x) = x^{2} - \frac{1}{4 x^{2}}$

Step 2.1.2

The first derivative of $f (x)$ with respect to $x$ is $x^{2} - \frac{1}{4 x^{2}}$ .

$x^{2} - \frac{1}{4 x^{2}}$

Step 2.2

Find if the derivative is continuous on $[1, 3]$ .

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Step 2.2.1

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

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Step 2.2.1.1

Set the denominator in $\frac{1}{4 x^{2}}$ equal to $0$ to find where the expression is undefined.

$4 x^{2} = 0$

Step 2.2.1.2

Solve for $x$ .

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Step 2.2.1.2.1

Divide each term in $4 x^{2} = 0$ by $4$ and simplify.

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Step 2.2.1.2.1.1

Divide each term in $4 x^{2} = 0$ by $4$ .

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

Step 2.2.1.2.1.2

Simplify the left side.

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Step 2.2.1.2.1.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.2.1.2.1.1

Cancel the common factor.

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

Step 2.2.1.2.1.2.1.2

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

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

Step 2.2.1.2.1.3

Simplify the right side.

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Step 2.2.1.2.1.3.1

Divide $0$ by $4$ .

$x^{2} = 0$

Step 2.2.1.2.2

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

$x = \pm \sqrt{0}$

Step 2.2.1.2.3

Simplify $\pm \sqrt{0}$ .

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Step 2.2.1.2.3.1

Rewrite $0$ as $0^{2}$ .

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

Step 2.2.1.2.3.2

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

$x = \pm 0$

Step 2.2.1.2.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.2.1.3

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 2.2.2

$f' (x)$ is continuous on $[1, 3]$ .

The function is continuous.

Step 2.3

The function is differentiable on $[1, 3]$ because the derivative is continuous on $[1, 3]$ .

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 $[1, 3]$ .

The function and its derivative are continuous on the closed interval $[1, 3]$ .

Step 4

Find the derivative of $f (x) = \frac{x^{3}}{3} + \frac{1}{4 x}$ .

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Step 4.1

By the Sum Rule, the derivative of $\frac{x^{3}}{3} + \frac{1}{4 x}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{3}}{3}] + \frac{d}{d x} [\frac{1}{4 x}]$ .

$\frac{d}{d x} [\frac{x^{3}}{3}] + \frac{d}{d x} [\frac{1}{4 x}]$

Step 4.2

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

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Step 4.2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x^{3}}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{3}]$ .

$\frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{1}{4 x}]$

Step 4.2.2

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

$\frac{1}{3} (3 x^{2}) + \frac{d}{d x} [\frac{1}{4 x}]$

Step 4.2.3

Combine $3$ and $\frac{1}{3}$ .

$\frac{3}{3} x^{2} + \frac{d}{d x} [\frac{1}{4 x}]$

Step 4.2.4

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

$\frac{3 x^{2}}{3} + \frac{d}{d x} [\frac{1}{4 x}]$

Step 4.2.5

Cancel the common factor of $3$ .

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Step 4.2.5.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{d}{d x} [\frac{1}{4 x}]$

Step 4.2.5.2

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

$x^{2} + \frac{d}{d x} [\frac{1}{4 x}]$

Step 4.3

Evaluate $\frac{d}{d x} [\frac{1}{4 x}]$ .

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Step 4.3.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4 x}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [\frac{1}{x}]$ .

$x^{2} + \frac{1}{4} \frac{d}{d x} [\frac{1}{x}]$

Step 4.3.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$x^{2} + \frac{1}{4} \frac{d}{d x} [x^{- 1}]$

Step 4.3.3

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

$x^{2} + \frac{1}{4} (- x^{- 2})$

Step 4.3.4

Combine $\frac{1}{4}$ and $x^{- 2}$ .

$x^{2} - \frac{x^{- 2}}{4}$

Step 4.3.5

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

$x^{2} - \frac{1}{4 x^{2}}$

Step 5

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

$\int_{1}^{3} \sqrt{1 + {(x^{2} - \frac{1}{4 x^{2}})}^{2}} d x$

Step 6

Evaluate the integral.

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Step 6.1

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{1}^{3} \frac{4 x^{4} + 1}{x^{2}} d x$

Step 6.2

Apply basic rules of exponents.

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Step 6.2.1

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

$\frac{1}{4} \int_{1}^{3} (4 x^{4} + 1) {(x^{2})}^{- 1} d x$

Step 6.2.2

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

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Step 6.2.2.1

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

$\frac{1}{4} \int_{1}^{3} (4 x^{4} + 1) x^{2 \cdot - 1} d x$

Step 6.2.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{4} \int_{1}^{3} (4 x^{4} + 1) x^{- 2} d x$

Step 6.3

Multiply $(4 x^{4} + 1) x^{- 2}$ .

$\frac{1}{4} \int_{1}^{3} 4 x^{4} x^{- 2} + 1 x^{- 2} d x$

Step 6.4

Simplify.

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Step 6.4.1

Multiply $x^{4}$ by $x^{- 2}$ by adding the exponents.

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Step 6.4.1.1

Move $x^{- 2}$ .

$\frac{1}{4} \int_{1}^{3} 4 (x^{- 2} x^{4}) + 1 x^{- 2} d x$

Step 6.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{4} \int_{1}^{3} 4 x^{- 2 + 4} + 1 x^{- 2} d x$

Step 6.4.1.3

Add $- 2$ and $4$ .

$\frac{1}{4} \int_{1}^{3} 4 x^{2} + 1 x^{- 2} d x$

Step 6.4.2

Multiply $x^{- 2}$ by $1$ .

$\frac{1}{4} \int_{1}^{3} 4 x^{2} + x^{- 2} d x$

Step 6.5

Split the single integral into multiple integrals.

$\frac{1}{4} (\int_{1}^{3} 4 x^{2} d x + \int_{1}^{3} x^{- 2} d x)$

Step 6.6

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\frac{1}{4} (4 \int_{1}^{3} x^{2} d x + \int_{1}^{3} x^{- 2} d x)$

Step 6.7

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

$\frac{1}{4} (4 (\frac{1}{3} x^{3}]_{1}^{3}) + \int_{1}^{3} x^{- 2} d x)$

Step 6.8

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

$\frac{1}{4} (4 (\frac{x^{3}}{3}]_{1}^{3}) + \int_{1}^{3} x^{- 2} d x)$

Step 6.9

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

$\frac{1}{4} (4 (\frac{x^{3}}{3}]_{1}^{3}) + - x^{- 1}]_{1}^{3})$

Step 6.10

Substitute and simplify.

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Step 6.10.1

Evaluate $\frac{x^{3}}{3}$ at $3$ and at $1$ .

$\frac{1}{4} (4 ((\frac{3^{3}}{3}) - \frac{1^{3}}{3}) + - x^{- 1}]_{1}^{3})$

Step 6.10.2

Evaluate $- x^{- 1}$ at $3$ and at $1$ .

$\frac{1}{4} (4 (\frac{3^{3}}{3} - \frac{1^{3}}{3}) + - 3^{- 1} + 1^{- 1})$

Step 6.10.3

Simplify.

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Step 6.10.3.1

Raise $3$ to the power of $3$ .

$\frac{1}{4} (4 (\frac{27}{3} - \frac{1^{3}}{3}) - 3^{- 1} + 1^{- 1})$

Step 6.10.3.2

Cancel the common factor of $27$ and $3$ .

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Step 6.10.3.2.1

Factor $3$ out of $27$ .

$\frac{1}{4} (4 (\frac{3 \cdot 9}{3} - \frac{1^{3}}{3}) - 3^{- 1} + 1^{- 1})$

Step 6.10.3.2.2

Cancel the common factors.

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Step 6.10.3.2.2.1

Factor $3$ out of $3$ .

$\frac{1}{4} (4 (\frac{3 \cdot 9}{3 (1)} - \frac{1^{3}}{3}) - 3^{- 1} + 1^{- 1})$

Step 6.10.3.2.2.2

Cancel the common factor.

$\frac{1}{4} (4 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{1^{3}}{3}) - 3^{- 1} + 1^{- 1})$

Step 6.10.3.2.2.3

Rewrite the expression.

$\frac{1}{4} (4 (\frac{9}{1} - \frac{1^{3}}{3}) - 3^{- 1} + 1^{- 1})$

Step 6.10.3.2.2.4

Divide $9$ by $1$ .

$\frac{1}{4} (4 (9 - \frac{1^{3}}{3}) - 3^{- 1} + 1^{- 1})$

Step 6.10.3.3

One to any power is one.

$\frac{1}{4} (4 (9 - \frac{1}{3}) - 3^{- 1} + 1^{- 1})$

Step 6.10.3.4

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{4} (4 (9 \cdot \frac{3}{3} - \frac{1}{3}) - 3^{- 1} + 1^{- 1})$

Step 6.10.3.5

Combine $9$ and $\frac{3}{3}$ .

$\frac{1}{4} (4 (\frac{9 \cdot 3}{3} - \frac{1}{3}) - 3^{- 1} + 1^{- 1})$

Step 6.10.3.6

Combine the numerators over the common denominator.

$\frac{1}{4} (4 \frac{9 \cdot 3 - 1}{3} - 3^{- 1} + 1^{- 1})$

Step 6.10.3.7

Simplify the numerator.

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Step 6.10.3.7.1

Multiply $9$ by $3$ .

$\frac{1}{4} (4 \frac{27 - 1}{3} - 3^{- 1} + 1^{- 1})$

Step 6.10.3.7.2

Subtract $1$ from $27$ .

$\frac{1}{4} (4 (\frac{26}{3}) - 3^{- 1} + 1^{- 1})$

Step 6.10.3.8

Combine $4$ and $\frac{26}{3}$ .

$\frac{1}{4} (\frac{4 \cdot 26}{3} - 3^{- 1} + 1^{- 1})$

Step 6.10.3.9

Multiply $4$ by $26$ .

$\frac{1}{4} (\frac{104}{3} - 3^{- 1} + 1^{- 1})$

Step 6.10.3.10

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

$\frac{1}{4} (\frac{104}{3} - \frac{1}{3} + 1^{- 1})$

Step 6.10.3.11

One to any power is one.

$\frac{1}{4} (\frac{104}{3} - \frac{1}{3} + 1)$

Step 6.10.3.12

Write $1$ as a fraction with a common denominator.

$\frac{1}{4} (\frac{104}{3} - \frac{1}{3} + \frac{3}{3})$

Step 6.10.3.13

Combine the numerators over the common denominator.

$\frac{1}{4} (\frac{104}{3} + \frac{- 1 + 3}{3})$

Step 6.10.3.14

Add $- 1$ and $3$ .

$\frac{1}{4} (\frac{104}{3} + \frac{2}{3})$

Step 6.10.3.15

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{104 + 2}{3}$

Step 6.10.3.16

Add $104$ and $2$ .

$\frac{1}{4} \cdot \frac{106}{3}$

Step 6.10.3.17

Multiply $\frac{1}{4}$ by $\frac{106}{3}$ .

$\frac{106}{4 \cdot 3}$

Step 6.10.3.18

Multiply $4$ by $3$ .

$\frac{106}{12}$

Step 6.10.3.19

Cancel the common factor of $106$ and $12$ .

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Step 6.10.3.19.1

Factor $2$ out of $106$ .

$\frac{2 (53)}{12}$

Step 6.10.3.19.2

Cancel the common factors.

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Step 6.10.3.19.2.1

Factor $2$ out of $12$ .

$\frac{2 \cdot 53}{2 \cdot 6}$

Step 6.10.3.19.2.2

Cancel the common factor.

$\frac{2 \cdot 53}{2 \cdot 6}$

Step 6.10.3.19.2.3

Rewrite the expression.

$\frac{53}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{53}{6}$

Decimal Form:

$8.8\overline{3}$

Mixed Number Form:

$8 \frac{5}{6}$

Step 8