Calculus Examples

$g (x) = \frac{8 x^{2}}{x - 2}$ , $[- 2, 1]$

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

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8 x^{2}}{x - 2}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{x^{2}}{x - 2}]$ .

$8 \frac{d}{d x} [\frac{x^{2}}{x - 2}]$

Step 1.1.1.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2}$ and $g (x) = x - 2$ .

$8 \frac{(x - 2) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.1.1.3

Differentiate.

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Step 1.1.1.3.1

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

$8 \frac{(x - 2) (2 x) - x^{2} \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.1.1.3.2

Move $2$ to the left of $x - 2$ .

$8 \frac{2 \cdot (x - 2) x - x^{2} \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.1.1.3.3

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

$8 \frac{2 (x - 2) x - x^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 2])}{{(x - 2)}^{2}}$

Step 1.1.1.3.4

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

$8 \frac{2 (x - 2) x - x^{2} (1 + \frac{d}{d x} [- 2])}{{(x - 2)}^{2}}$

Step 1.1.1.3.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$8 \frac{2 (x - 2) x - x^{2} (1 + 0)}{{(x - 2)}^{2}}$

Step 1.1.1.3.6

Combine fractions.

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Step 1.1.1.3.6.1

Add $1$ and $0$ .

$8 \frac{2 (x - 2) x - x^{2} \cdot 1}{{(x - 2)}^{2}}$

Step 1.1.1.3.6.2

Multiply $- 1$ by $1$ .

$8 \frac{2 (x - 2) x - x^{2}}{{(x - 2)}^{2}}$

Step 1.1.1.3.6.3

Combine $8$ and $\frac{2 (x - 2) x - x^{2}}{{(x - 2)}^{2}}$ .

$\frac{8 (2 (x - 2) x - x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4

Simplify.

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Step 1.1.1.4.1

Apply the distributive property.

$\frac{8 ((2 x + 2 \cdot - 2) x - x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.2

Apply the distributive property.

$\frac{8 (2 x \cdot x + 2 \cdot - 2 x - x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.3

Apply the distributive property.

$\frac{8 (2 x \cdot x) + 8 (2 \cdot - 2 x) + 8 (- x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.4

Simplify the numerator.

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Step 1.1.1.4.4.1

Simplify each term.

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Step 1.1.1.4.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.1.4.4.1.1.1

Move $x$ .

$\frac{8 (2 (x \cdot x)) + 8 (2 \cdot - 2 x) + 8 (- x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.4.1.1.2

Multiply $x$ by $x$ .

$\frac{8 (2 x^{2}) + 8 (2 \cdot - 2 x) + 8 (- x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.4.1.2

Multiply $2$ by $8$ .

$\frac{16 x^{2} + 8 (2 \cdot - 2 x) + 8 (- x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.4.1.3

Multiply $2$ by $- 2$ .

$\frac{16 x^{2} + 8 (- 4 x) + 8 (- x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.4.1.4

Multiply $- 4$ by $8$ .

$\frac{16 x^{2} - 32 x + 8 (- x^{2})}{{(x - 2)}^{2}}$

Step 1.1.1.4.4.1.5

Multiply $- 1$ by $8$ .

$\frac{16 x^{2} - 32 x - 8 x^{2}}{{(x - 2)}^{2}}$

Step 1.1.1.4.4.2

Subtract $8 x^{2}$ from $16 x^{2}$ .

$\frac{8 x^{2} - 32 x}{{(x - 2)}^{2}}$

Step 1.1.1.4.5

Factor $8 x$ out of $8 x^{2} - 32 x$ .

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Step 1.1.1.4.5.1

Factor $8 x$ out of $8 x^{2}$ .

$\frac{8 x (x) - 32 x}{{(x - 2)}^{2}}$

Step 1.1.1.4.5.2

Factor $8 x$ out of $- 32 x$ .

$\frac{8 x (x) + 8 x (- 4)}{{(x - 2)}^{2}}$

Step 1.1.1.4.5.3

Factor $8 x$ out of $8 x (x) + 8 x (- 4)$ .

$f' (x) = \frac{8 x (x - 4)}{{(x - 2)}^{2}}$

Step 1.1.2

The first derivative of $g (x)$ with respect to $x$ is $\frac{8 x (x - 4)}{{(x - 2)}^{2}}$ .

$\frac{8 x (x - 4)}{{(x - 2)}^{2}}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{8 x (x - 4)}{{(x - 2)}^{2}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{8 x (x - 4)}{{(x - 2)}^{2}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$8 x (x - 4) = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 4 = 0$

Step 1.2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.3.3

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 1.2.3.3.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 1.2.3.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 1.2.3.4

The final solution is all the values that make $8 x (x - 4) = 0$ true.

$x = 0, 4$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Set the denominator in $\frac{8 x (x - 4)}{{(x - 2)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x - 2)}^{2} = 0$

Step 1.3.2

Solve for $x$ .

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Step 1.3.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.3.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.4

Evaluate $\frac{8 x^{2}}{x - 2}$ 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 $x$ .

$\frac{8 {(0)}^{2}}{(0) - 2}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Cancel the common factor of $8$ and $(0) - 2$ .

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Step 1.4.1.2.1.1

Factor $2$ out of $8 {(0)}^{2}$ .

$\frac{2 (4 {(0)}^{2})}{(0) - 2}$

Step 1.4.1.2.1.2

Cancel the common factors.

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Step 1.4.1.2.1.2.1

Factor $2$ out of $0$ .

$\frac{2 (4 {(0)}^{2})}{2 (0) - 2}$

Step 1.4.1.2.1.2.2

Factor $2$ out of $- 2$ .

$\frac{2 (4 {(0)}^{2})}{2 \cdot 0 + 2 \cdot - 1}$

Step 1.4.1.2.1.2.3

Factor $2$ out of $2 \cdot 0 + 2 \cdot - 1$ .

$\frac{2 (4 {(0)}^{2})}{2 \cdot (0 - 1)}$

Step 1.4.1.2.1.2.4

Cancel the common factor.

$\frac{2 (4 {(0)}^{2})}{2 \cdot (0 - 1)}$

Step 1.4.1.2.1.2.5

Rewrite the expression.

$\frac{4 {(0)}^{2}}{0 - 1}$

Step 1.4.1.2.2

Simplify the expression.

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Step 1.4.1.2.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{4 \cdot 0}{0 - 1}$

Step 1.4.1.2.2.2

Subtract $1$ from $0$ .

$\frac{4 \cdot 0}{- 1}$

Step 1.4.1.2.2.3

Multiply $4$ by $0$ .

$\frac{0}{- 1}$

Step 1.4.1.2.2.4

Divide $0$ by $- 1$ .

$0$

Step 1.4.2

Evaluate at $x = 4$ .

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Step 1.4.2.1

Substitute $4$ for $x$ .

$\frac{8 {(4)}^{2}}{(4) - 2}$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Cancel the common factor of $8$ and $(4) - 2$ .

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Step 1.4.2.2.1.1

Factor $2$ out of $8 {(4)}^{2}$ .

$\frac{2 (4 {(4)}^{2})}{(4) - 2}$

Step 1.4.2.2.1.2

Cancel the common factors.

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Step 1.4.2.2.1.2.1

Factor $2$ out of $4$ .

$\frac{2 (4 {(4)}^{2})}{2 (2) - 2}$

Step 1.4.2.2.1.2.2

Factor $2$ out of $- 2$ .

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

Step 1.4.2.2.1.2.3

Factor $2$ out of $2 \cdot 2 + 2 \cdot - 1$ .

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

Step 1.4.2.2.1.2.4

Cancel the common factor.

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

Step 1.4.2.2.1.2.5

Rewrite the expression.

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

Step 1.4.2.2.2

Multiply $4$ by ${(4)}^{2}$ by adding the exponents.

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Step 1.4.2.2.2.1

Multiply $4$ by ${(4)}^{2}$ .

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Step 1.4.2.2.2.1.1

Raise $4$ to the power of $1$ .

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

Step 1.4.2.2.2.1.2

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

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

Step 1.4.2.2.2.2

Add $1$ and $2$ .

$\frac{4^{3}}{2 - 1}$

Step 1.4.2.2.3

Raise $4$ to the power of $3$ .

$\frac{64}{2 - 1}$

Step 1.4.2.2.4

Subtract $1$ from $2$ .

$\frac{64}{1}$

Step 1.4.2.2.5

Divide $64$ by $1$ .

$64$

Step 1.4.3

Evaluate at $x = 2$ .

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Step 1.4.3.1

Substitute $2$ for $x$ .

$\frac{8 {(2)}^{2}}{(2) - 2}$

Step 1.4.3.2

Simplify.

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Step 1.4.3.2.1

Subtract $2$ from $2$ .

$\frac{8 {(2)}^{2}}{0}$

Step 1.4.3.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 1.4.4

List all of the points.

$(0, 0), (4, 64)$

Step 2

Exclude the points that are not on the interval.

$(0, 0)$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = - 2$ .

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Step 3.1.1

Substitute $- 2$ for $x$ .

$\frac{8 {(- 2)}^{2}}{(- 2) - 2}$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Cancel the common factor of $8$ and $(- 2) - 2$ .

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Step 3.1.2.1.1

Factor $2$ out of $8 {(- 2)}^{2}$ .

$\frac{2 (4 {(- 2)}^{2})}{(- 2) - 2}$

Step 3.1.2.1.2

Cancel the common factors.

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Step 3.1.2.1.2.1

Factor $2$ out of $- 2$ .

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

Step 3.1.2.1.2.2

Factor $2$ out of $- 2$ .

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

Step 3.1.2.1.2.3

Factor $2$ out of $2 \cdot - 1 + 2 \cdot - 1$ .

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

Step 3.1.2.1.2.4

Cancel the common factor.

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

Step 3.1.2.1.2.5

Rewrite the expression.

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

Step 3.1.2.2

Simplify the expression.

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Step 3.1.2.2.1

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

$\frac{4 \cdot 4}{- 1 - 1}$

Step 3.1.2.2.2

Subtract $1$ from $- 1$ .

$\frac{4 \cdot 4}{- 2}$

Step 3.1.2.2.3

Multiply $4$ by $4$ .

$\frac{16}{- 2}$

Step 3.1.2.2.4

Divide $16$ by $- 2$ .

$- 8$

Step 3.2

Evaluate at $x = 1$ .

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Step 3.2.1

Substitute $1$ for $x$ .

$\frac{8 {(1)}^{2}}{(1) - 2}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

One to any power is one.

$\frac{8 \cdot 1}{1 - 2}$

Step 3.2.2.2

Subtract $2$ from $1$ .

$\frac{8 \cdot 1}{- 1}$

Step 3.2.2.3

Multiply $8$ by $1$ .

$\frac{8}{- 1}$

Step 3.2.2.4

Divide $8$ by $- 1$ .

$- 8$

Step 3.3

List all of the points.

$(- 2, - 8), (1, - 8)$

Step 4

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

Absolute Maximum: $(0, 0)$

Absolute Minimum: $(- 2, - 8), (1, - 8)$

Step 5