Calculus Examples

$f (x) = \frac{x^{2} - 36}{x^{2} + 36}$ , $[- 36, 36]$

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 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 $f (x) = x^{2} - 36$ and $g (x) = x^{2} + 36$ .

$\frac{(x^{2} + 36) \frac{d}{d x} [x^{2} - 36] - (x^{2} - 36) \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

By the Sum Rule, the derivative of $x^{2} - 36$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 36]$ .

$\frac{(x^{2} + 36) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 36]) - (x^{2} - 36) \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

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 $n = 2$ .

$\frac{(x^{2} + 36) (2 x + \frac{d}{d x} [- 36]) - (x^{2} - 36) \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.2.3

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36$ with respect to $x$ is $0$ .

$\frac{(x^{2} + 36) (2 x + 0) - (x^{2} - 36) \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.2.4

Simplify the expression.

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Step 1.1.1.2.4.1

Add $2 x$ and $0$ .

$\frac{(x^{2} + 36) (2 x) - (x^{2} - 36) \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.2.4.2

Move $2$ to the left of $x^{2} + 36$ .

$\frac{2 \cdot (x^{2} + 36) x - (x^{2} - 36) \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.2.5

By the Sum Rule, the derivative of $x^{2} + 36$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [36]$ .

$\frac{2 (x^{2} + 36) x - (x^{2} - 36) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [36])}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.2.6

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

$\frac{2 (x^{2} + 36) x - (x^{2} - 36) (2 x + \frac{d}{d x} [36])}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.2.7

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$\frac{2 (x^{2} + 36) x - (x^{2} - 36) (2 x + 0)}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.2.8

Simplify the expression.

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Step 1.1.1.2.8.1

Add $2 x$ and $0$ .

$\frac{2 (x^{2} + 36) x - (x^{2} - 36) (2 x)}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.2.8.2

Multiply $2$ by $- 1$ .

$\frac{2 (x^{2} + 36) x - 2 (x^{2} - 36) x}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.3

Simplify.

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Step 1.1.1.3.1

Apply the distributive property.

$\frac{(2 x^{2} + 2 \cdot 36) x - 2 (x^{2} - 36) x}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.3.2

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot 36 x - 2 (x^{2} - 36) x}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.3.3

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot 36 x + (- 2 x^{2} - 2 \cdot - 36) x}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.3.4

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot 36 x - 2 x^{2} x - 2 \cdot - 36 x}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.3.5

Simplify the numerator.

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Step 1.1.1.3.5.1

Combine the opposite terms in $2 x^{2} x + 2 \cdot 36 x - 2 x^{2} x - 2 \cdot - 36 x$ .

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Step 1.1.1.3.5.1.1

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

$\frac{2 \cdot 36 x + 0 - 2 \cdot - 36 x}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.3.5.1.2

Add $2 \cdot 36 x$ and $0$ .

$\frac{2 \cdot 36 x - 2 \cdot - 36 x}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.3.5.2

Simplify each term.

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Step 1.1.1.3.5.2.1

Multiply $2$ by $36$ .

$\frac{72 x - 2 \cdot - 36 x}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.3.5.2.2

Multiply $- 2$ by $- 36$ .

$\frac{72 x + 72 x}{{(x^{2} + 36)}^{2}}$

Step 1.1.1.3.5.3

Add $72 x$ and $72 x$ .

$f' (x) = \frac{144 x}{{(x^{2} + 36)}^{2}}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{144 x}{{(x^{2} + 36)}^{2}}$ .

$\frac{144 x}{{(x^{2} + 36)}^{2}}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{144 x}{{(x^{2} + 36)}^{2}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{144 x}{{(x^{2} + 36)}^{2}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$144 x = 0$

Step 1.2.3

Divide each term in $144 x = 0$ by $144$ and simplify.

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Step 1.2.3.1

Divide each term in $144 x = 0$ by $144$ .

$\frac{144 x}{144} = \frac{0}{144}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $144$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{144 x}{144} = \frac{0}{144}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{144}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $144$ .

$x = 0$

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 $\frac{x^{2} - 36}{x^{2} + 36}$ 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 $x$ .

$\frac{{(0)}^{2} - 36}{{(0)}^{2} + 36}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify the numerator.

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Step 1.4.1.2.1.1

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

$\frac{0 - 36}{0^{2} + 36}$

Step 1.4.1.2.1.2

Subtract $36$ from $0$ .

$\frac{- 36}{0^{2} + 36}$

Step 1.4.1.2.2

Simplify the denominator.

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Step 1.4.1.2.2.1

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

$\frac{- 36}{0 + 36}$

Step 1.4.1.2.2.2

Add $0$ and $36$ .

$\frac{- 36}{36}$

Step 1.4.1.2.3

Divide $- 36$ by $36$ .

$- 1$

Step 1.4.2

List all of the points.

$(0, - 1)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 36$ .

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Step 2.1.1

Substitute $- 36$ for $x$ .

$\frac{{(- 36)}^{2} - 36}{{(- 36)}^{2} + 36}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Reduce the expression by cancelling the common factors.

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Step 2.1.2.1.1

Cancel the common factor of ${(- 36)}^{2} - 36$ and ${(- 36)}^{2} + 36$ .

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Step 2.1.2.1.1.1

Factor $- 1$ out of ${(- 36)}^{2}$ .

$\frac{{(- 36)}^{2} - 36}{- 1 (- {(- 36)}^{2}) + 36}$

Step 2.1.2.1.1.2

Rewrite $36$ as $- 1 (- 36)$ .

$\frac{{(- 36)}^{2} - 36}{- 1 (- {(- 36)}^{2}) - 1 (- 36)}$

Step 2.1.2.1.1.3

Factor $- 1$ out of $- 1 (- {(- 36)}^{2}) - 1 (- 36)$ .

$\frac{{(- 36)}^{2} - 36}{- 1 (- {(- 36)}^{2} - 36)}$

Step 2.1.2.1.1.4

Factor $- 36$ out of ${(- 36)}^{2}$ .

$\frac{- 36 \cdot - 36 - 36}{- 1 (- {(- 36)}^{2} - 36)}$

Step 2.1.2.1.1.5

Factor $- 36$ out of $- 36$ .

$\frac{- 36 \cdot - 36 - 36 (1)}{- 1 (- {(- 36)}^{2} - 36)}$

Step 2.1.2.1.1.6

Factor $- 36$ out of $- 36 \cdot - 36 - 36 (1)$ .

$\frac{- 36 \cdot (- 36 + 1)}{- 1 (- {(- 36)}^{2} - 36)}$

Step 2.1.2.1.1.7

Cancel the common factors.

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Step 2.1.2.1.1.7.1

Factor $- 36$ out of $- 1 (- {(- 36)}^{2} - 36)$ .

$\frac{- 36 \cdot (- 36 + 1)}{- 36 (- 1 (- - 36 + 1))}$

Step 2.1.2.1.1.7.2

Cancel the common factor.

$\frac{- 36 \cdot (- 36 + 1)}{- 36 (- 1 (- - 36 + 1))}$

Step 2.1.2.1.1.7.3

Rewrite the expression.

$\frac{- 36 + 1}{- 1 (- - 36 + 1)}$

Step 2.1.2.1.2

Add $- 36$ and $1$ .

$\frac{- 35}{- 1 (- - 36 + 1)}$

Step 2.1.2.2

Simplify the denominator.

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Step 2.1.2.2.1

Multiply $- 1$ by $- 36$ .

$\frac{- 35}{- 1 (36 + 1)}$

Step 2.1.2.2.2

Add $36$ and $1$ .

$\frac{- 35}{- 1 \cdot 37}$

Step 2.1.2.3

Reduce the expression by cancelling the common factors.

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Step 2.1.2.3.1

Multiply $- 1$ by $37$ .

$\frac{- 35}{- 37}$

Step 2.1.2.3.2

Dividing two negative values results in a positive value.

$\frac{35}{37}$

Step 2.2

Evaluate at $x = 36$ .

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Step 2.2.1

Substitute $36$ for $x$ .

$\frac{{(36)}^{2} - 36}{{(36)}^{2} + 36}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Cancel the common factor of ${(36)}^{2} - 36$ and ${(36)}^{2} + 36$ .

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Step 2.2.2.1.1

Factor $- 1$ out of ${(36)}^{2}$ .

$\frac{- 1 (- {(36)}^{2}) - 36}{{(36)}^{2} + 36}$

Step 2.2.2.1.2

Rewrite $- 36$ as $- 1 (36)$ .

$\frac{- 1 (- {(36)}^{2}) - 1 (36)}{{(36)}^{2} + 36}$

Step 2.2.2.1.3

Factor $- 1$ out of $- 1 (- {(36)}^{2}) - 1 (36)$ .

$\frac{- 1 (- {(36)}^{2} + 36)}{{(36)}^{2} + 36}$

Step 2.2.2.1.4

Factor $36$ out of $- 1 (- 36^{2} + 36)$ .

$\frac{36 (- 1 (- 1 \cdot 36 + 1))}{36^{2} + 36}$

Step 2.2.2.1.5

Cancel the common factors.

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Step 2.2.2.1.5.1

Factor $36$ out of $36^{2}$ .

$\frac{36 (- 1 (- 1 \cdot 36 + 1))}{36 \cdot 36 + 36}$

Step 2.2.2.1.5.2

Factor $36$ out of $36$ .

$\frac{36 (- 1 (- 1 \cdot 36 + 1))}{36 \cdot 36 + 36 (1)}$

Step 2.2.2.1.5.3

Factor $36$ out of $36 \cdot 36 + 36 (1)$ .

$\frac{36 (- 1 (- 1 \cdot 36 + 1))}{36 \cdot (36 + 1)}$

Step 2.2.2.1.5.4

Cancel the common factor.

$\frac{36 (- 1 (- 1 \cdot 36 + 1))}{36 \cdot (36 + 1)}$

Step 2.2.2.1.5.5

Rewrite the expression.

$\frac{- 1 (- 1 \cdot 36 + 1)}{36 + 1}$

Step 2.2.2.2

Simplify the numerator.

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Step 2.2.2.2.1

Multiply $- 1$ by $36$ .

$\frac{- 1 (- 36 + 1)}{36 + 1}$

Step 2.2.2.2.2

Add $- 36$ and $1$ .

$\frac{- 1 \cdot - 35}{36 + 1}$

Step 2.2.2.3

Simplify the expression.

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Step 2.2.2.3.1

Add $36$ and $1$ .

$\frac{- 1 \cdot - 35}{37}$

Step 2.2.2.3.2

Multiply $- 1$ by $- 35$ .

$\frac{35}{37}$

Step 2.3

List all of the points.

$(- 36, \frac{35}{37}), (36, \frac{35}{37})$

Step 3

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: $(- 36, \frac{35}{37}), (36, \frac{35}{37})$

Absolute Minimum: $(0, - 1)$

Step 4