Calculus Examples

$f (x) = x^{3} - x - 1$ ; between $1$ and $2$

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

Differentiate.

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Step 1.1.1.1.1

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

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

Step 1.1.1.1.2

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

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

Step 1.1.1.2

Evaluate $\frac{d}{d x} [- x]$ .

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

Multiply $- 1$ by $1$ .

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

Step 1.1.1.3

Differentiate using the Constant Rule.

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Step 1.1.1.3.1

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

$3 x^{2} - 1 + 0$

Step 1.1.1.3.2

Add $3 x^{2} - 1$ and $0$ .

$f' (x) = 3 x^{2} - 1$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} - 1$ .

$3 x^{2} - 1$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} - 1 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$3 x^{2} - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$3 x^{2} = 1$

Step 1.2.3

Divide each term in $3 x^{2} = 1$ by $3$ and simplify.

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Step 1.2.3.1

Divide each term in $3 x^{2} = 1$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{1}{3}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $3$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{1}{3}$

Step 1.2.3.2.1.2

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

$x^{2} = \frac{1}{3}$

Step 1.2.4

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

$x = \pm \sqrt{\frac{1}{3}}$

Step 1.2.5

Simplify $\pm \sqrt{\frac{1}{3}}$ .

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Step 1.2.5.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{3}}$

Step 1.2.5.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{3}}$

Step 1.2.5.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 1.2.5.4

Combine and simplify the denominator.

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Step 1.2.5.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 1.2.5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 1.2.5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 1.2.5.4.4

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

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 1.2.5.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 1.2.5.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.2.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 1.2.5.4.6.2

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

$x = \pm \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 1.2.5.4.6.3

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

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.2.5.4.6.4

Cancel the common factor of $2$ .

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Step 1.2.5.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.2.5.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{3}}{3^{1}}$

Step 1.2.5.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{3}}{3}$

Step 1.2.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{3}}{3}$

Step 1.2.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{3}}{3}$

Step 1.2.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 1.4

Evaluate $x^{3} - x - 1$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \frac{\sqrt{3}}{3}$ .

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Step 1.4.1.1

Substitute $\frac{\sqrt{3}}{3}$ for $x$ .

${(\frac{\sqrt{3}}{3})}^{3} - (\frac{\sqrt{3}}{3}) - 1$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

Apply the product rule to $\frac{\sqrt{3}}{3}$ .

$\frac{{\sqrt{3}}^{3}}{3^{3}} - (\frac{\sqrt{3}}{3}) - 1$

Step 1.4.1.2.1.2

Simplify the numerator.

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Step 1.4.1.2.1.2.1

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$\frac{\sqrt{3^{3}}}{3^{3}} - (\frac{\sqrt{3}}{3}) - 1$

Step 1.4.1.2.1.2.2

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

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

Step 1.4.1.2.1.2.3

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.4.1.2.1.2.3.1

Factor $9$ out of $27$ .

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

Step 1.4.1.2.1.2.3.2

Rewrite $9$ as $3^{2}$ .

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

Step 1.4.1.2.1.2.4

Pull terms out from under the radical.

$\frac{3 \sqrt{3}}{3^{3}} - (\frac{\sqrt{3}}{3}) - 1$

Step 1.4.1.2.1.3

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

$\frac{3 \sqrt{3}}{27} - (\frac{\sqrt{3}}{3}) - 1$

Step 1.4.1.2.1.4

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

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Step 1.4.1.2.1.4.1

Factor $3$ out of $3 \sqrt{3}$ .

$\frac{3 (\sqrt{3})}{27} - (\frac{\sqrt{3}}{3}) - 1$

Step 1.4.1.2.1.4.2

Cancel the common factors.

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Step 1.4.1.2.1.4.2.1

Factor $3$ out of $27$ .

$\frac{3 \sqrt{3}}{3 \cdot 9} - (\frac{\sqrt{3}}{3}) - 1$

Step 1.4.1.2.1.4.2.2

Cancel the common factor.

$\frac{3 \sqrt{3}}{3 \cdot 9} - (\frac{\sqrt{3}}{3}) - 1$

Step 1.4.1.2.1.4.2.3

Rewrite the expression.

$\frac{\sqrt{3}}{9} - (\frac{\sqrt{3}}{3}) - 1$

$\frac{\sqrt{3}}{9} - \frac{\sqrt{3}}{3} - 1$

Step 1.4.1.2.2

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

$\frac{\sqrt{3}}{9} - \frac{\sqrt{3}}{3} \cdot \frac{3}{3} - 1$

Step 1.4.1.2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1.2.3.1

Multiply $\frac{\sqrt{3}}{3}$ by $\frac{3}{3}$ .

$\frac{\sqrt{3}}{9} - \frac{\sqrt{3} \cdot 3}{3 \cdot 3} - 1$

Step 1.4.1.2.3.2

Multiply $3$ by $3$ .

$\frac{\sqrt{3}}{9} - \frac{\sqrt{3} \cdot 3}{9} - 1$

Step 1.4.1.2.4

Combine the numerators over the common denominator.

$\frac{\sqrt{3} - \sqrt{3} \cdot 3}{9} - 1$

Step 1.4.1.2.5

Simplify each term.

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Step 1.4.1.2.5.1

Simplify the numerator.

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Step 1.4.1.2.5.1.1

Multiply $3$ by $- 1$ .

$\frac{\sqrt{3} - 3 \sqrt{3}}{9} - 1$

Step 1.4.1.2.5.1.2

Subtract $3 \sqrt{3}$ from $\sqrt{3}$ .

$\frac{- 2 \sqrt{3}}{9} - 1$

Step 1.4.1.2.5.2

Move the negative in front of the fraction.

$- \frac{2 \sqrt{3}}{9} - 1$

Step 1.4.2

Evaluate at $x = - \frac{\sqrt{3}}{3}$ .

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Step 1.4.2.1

Substitute $- \frac{\sqrt{3}}{3}$ for $x$ .

${(- \frac{\sqrt{3}}{3})}^{3} - (- \frac{\sqrt{3}}{3}) - 1$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify each term.

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Step 1.4.2.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.4.2.2.1.1.1

Apply the product rule to $- \frac{\sqrt{3}}{3}$ .

${(- 1)}^{3} {(\frac{\sqrt{3}}{3})}^{3} - (- \frac{\sqrt{3}}{3}) - 1$

Step 1.4.2.2.1.1.2

Apply the product rule to $\frac{\sqrt{3}}{3}$ .

${(- 1)}^{3} \frac{{\sqrt{3}}^{3}}{3^{3}} - (- \frac{\sqrt{3}}{3}) - 1$

Step 1.4.2.2.1.2

Raise $- 1$ to the power of $3$ .

$- \frac{{\sqrt{3}}^{3}}{3^{3}} - (- \frac{\sqrt{3}}{3}) - 1$

Step 1.4.2.2.1.3

Simplify the numerator.

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Step 1.4.2.2.1.3.1

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$- \frac{\sqrt{3^{3}}}{3^{3}} - (- \frac{\sqrt{3}}{3}) - 1$

Step 1.4.2.2.1.3.2

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

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

Step 1.4.2.2.1.3.3

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.4.2.2.1.3.3.1

Factor $9$ out of $27$ .

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

Step 1.4.2.2.1.3.3.2

Rewrite $9$ as $3^{2}$ .

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

Step 1.4.2.2.1.3.4

Pull terms out from under the radical.

$- \frac{3 \sqrt{3}}{3^{3}} - (- \frac{\sqrt{3}}{3}) - 1$

Step 1.4.2.2.1.4

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

$- \frac{3 \sqrt{3}}{27} - (- \frac{\sqrt{3}}{3}) - 1$

Step 1.4.2.2.1.5

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

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Step 1.4.2.2.1.5.1

Factor $3$ out of $3 \sqrt{3}$ .

$- \frac{3 (\sqrt{3})}{27} - (- \frac{\sqrt{3}}{3}) - 1$

Step 1.4.2.2.1.5.2

Cancel the common factors.

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Step 1.4.2.2.1.5.2.1

Factor $3$ out of $27$ .

$- \frac{3 \sqrt{3}}{3 \cdot 9} - (- \frac{\sqrt{3}}{3}) - 1$

Step 1.4.2.2.1.5.2.2

Cancel the common factor.

$- \frac{3 \sqrt{3}}{3 \cdot 9} - (- \frac{\sqrt{3}}{3}) - 1$

Step 1.4.2.2.1.5.2.3

Rewrite the expression.

$- \frac{\sqrt{3}}{9} - (- \frac{\sqrt{3}}{3}) - 1$

Step 1.4.2.2.1.6

Multiply $- (- \frac{\sqrt{3}}{3})$ .

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Step 1.4.2.2.1.6.1

Multiply $- 1$ by $- 1$ .

$- \frac{\sqrt{3}}{9} + 1 \frac{\sqrt{3}}{3} - 1$

Step 1.4.2.2.1.6.2

Multiply $\frac{\sqrt{3}}{3}$ by $1$ .

$- \frac{\sqrt{3}}{9} + \frac{\sqrt{3}}{3} - 1$

Step 1.4.2.2.2

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

$- \frac{\sqrt{3}}{9} + \frac{\sqrt{3}}{3} \cdot \frac{3}{3} - 1$

Step 1.4.2.2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.2.2.3.1

Multiply $\frac{\sqrt{3}}{3}$ by $\frac{3}{3}$ .

$- \frac{\sqrt{3}}{9} + \frac{\sqrt{3} \cdot 3}{3 \cdot 3} - 1$

Step 1.4.2.2.3.2

Multiply $3$ by $3$ .

$- \frac{\sqrt{3}}{9} + \frac{\sqrt{3} \cdot 3}{9} - 1$

Step 1.4.2.2.4

Simplify the expression.

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Step 1.4.2.2.4.1

Combine the numerators over the common denominator.

$\frac{- \sqrt{3} + \sqrt{3} \cdot 3}{9} - 1$

Step 1.4.2.2.4.2

Reorder the factors of $\sqrt{3} \cdot 3$ .

$\frac{- \sqrt{3} + 3 \sqrt{3}}{9} - 1$

Step 1.4.2.2.5

Add $- \sqrt{3}$ and $3 \sqrt{3}$ .

$\frac{2 \sqrt{3}}{9} - 1$

Step 1.4.3

List all of the points.

$(\frac{\sqrt{3}}{3}, - \frac{2 \sqrt{3}}{9} - 1), (- \frac{\sqrt{3}}{3}, \frac{2 \sqrt{3}}{9} - 1)$

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 $x = 1$ .

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Step 3.1.1

Substitute $1$ for $x$ .

${(1)}^{3} - (1) - 1$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

One to any power is one.

$1 - (1) - 1$

Step 3.1.2.1.2

Multiply $- 1$ by $1$ .

$1 - 1 - 1$

Step 3.1.2.2

Simplify by subtracting numbers.

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Step 3.1.2.2.1

Subtract $1$ from $1$ .

$0 - 1$

Step 3.1.2.2.2

Subtract $1$ from $0$ .

$- 1$

Step 3.2

Evaluate at $x = 2$ .

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Step 3.2.1

Substitute $2$ for $x$ .

${(2)}^{3} - (2) - 1$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Raise $2$ to the power of $3$ .

$8 - (2) - 1$

Step 3.2.2.1.2

Multiply $- 1$ by $2$ .

$8 - 2 - 1$

Step 3.2.2.2

Simplify by subtracting numbers.

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Step 3.2.2.2.1

Subtract $2$ from $8$ .

$6 - 1$

Step 3.2.2.2.2

Subtract $1$ from $6$ .

$5$

Step 3.3

List all of the points.

$(1, - 1), (2, 5)$

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: $(2, 5)$

Absolute Minimum: $(1, - 1)$

Step 5