Calculus Examples

$f (t) = \frac{t^{2}}{t - 8}$ ; $- 2 \leq t \leq - \frac{1}{4}$

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 $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = t^{2}$ and $g (t) = t - 8$ .

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

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

Move $2$ to the left of $t - 8$ .

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

Step 1.1.1.2.3

By the Sum Rule, the derivative of $t - 8$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [- 8]$ .

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

Step 1.1.1.2.4

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

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

Step 1.1.1.2.5

Since $- 8$ is constant with respect to $t$ , the derivative of $- 8$ with respect to $t$ is $0$ .

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

Step 1.1.1.2.6

Simplify the expression.

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Step 1.1.1.2.6.1

Add $1$ and $0$ .

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

Step 1.1.1.2.6.2

Multiply $- 1$ by $1$ .

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

Step 1.1.1.3

Simplify.

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Step 1.1.1.3.1

Apply the distributive property.

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

Step 1.1.1.3.2

Apply the distributive property.

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

Step 1.1.1.3.3

Simplify the numerator.

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Step 1.1.1.3.3.1

Simplify each term.

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Step 1.1.1.3.3.1.1

Multiply $t$ by $t$ by adding the exponents.

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Step 1.1.1.3.3.1.1.1

Move $t$ .

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

Step 1.1.1.3.3.1.1.2

Multiply $t$ by $t$ .

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

Step 1.1.1.3.3.1.2

Multiply $2$ by $- 8$ .

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

Step 1.1.1.3.3.2

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

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

Step 1.1.1.3.4

Factor $t$ out of $t^{2} - 16 t$ .

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Step 1.1.1.3.4.1

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

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

Step 1.1.1.3.4.2

Factor $t$ out of $- 16 t$ .

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

Step 1.1.1.3.4.3

Factor $t$ out of $t \cdot t + t \cdot - 16$ .

$f' (t) = \frac{t (t - 16)}{{(t - 8)}^{2}}$

Step 1.1.2

The first derivative of $f (t)$ with respect to $t$ is $\frac{t (t - 16)}{{(t - 8)}^{2}}$ .

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

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{t (t - 16)}{{(t - 8)}^{2}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{t (t - 16)}{{(t - 8)}^{2}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$t (t - 16) = 0$

Step 1.2.3

Solve the equation for $t$ .

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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 $0$ .

$t = 0$

$t - 16 = 0$

Step 1.2.3.2

Set $t$ equal to $0$ .

$t = 0$

Step 1.2.3.3

Set $t - 16$ equal to $0$ and solve for $t$ .

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Step 1.2.3.3.1

Set $t - 16$ equal to $0$ .

$t - 16 = 0$

Step 1.2.3.3.2

Add $16$ to both sides of the equation.

$t = 16$

Step 1.2.3.4

The final solution is all the values that make $t (t - 16) = 0$ true.

$t = 0, 16$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Set the denominator in $\frac{t (t - 16)}{{(t - 8)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(t - 8)}^{2} = 0$

Step 1.3.2

Solve for $t$ .

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Step 1.3.2.1

Set the $t - 8$ equal to $0$ .

$t - 8 = 0$

Step 1.3.2.2

Add $8$ to both sides of the equation.

$t = 8$

Step 1.4

Evaluate $\frac{t^{2}}{t - 8}$ at each $t$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $t = 0$ .

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Step 1.4.1.1

Substitute $0$ for $t$ .

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

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

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

$\frac{0}{0 - 8}$

Step 1.4.1.2.2

Subtract $8$ from $0$ .

$\frac{0}{- 8}$

Step 1.4.1.2.3

Divide $0$ by $- 8$ .

$0$

Step 1.4.2

Evaluate at $t = 16$ .

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Step 1.4.2.1

Substitute $16$ for $t$ .

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

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Raise $16$ to the power of $2$ .

$\frac{256}{16 - 8}$

Step 1.4.2.2.2

Subtract $8$ from $16$ .

$\frac{256}{8}$

Step 1.4.2.2.3

Divide $256$ by $8$ .

$32$

Step 1.4.3

Evaluate at $t = 8$ .

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Step 1.4.3.1

Substitute $8$ for $t$ .

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

Step 1.4.3.2

Simplify.

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Step 1.4.3.2.1

Subtract $8$ from $8$ .

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

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.

$(0, 0), (16, 32)$

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 $t = - 2$ .

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Step 3.1.1

Substitute $- 2$ for $t$ .

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

Step 3.1.2

Simplify.

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Step 3.1.2.1

Simplify the expression.

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Step 3.1.2.1.1

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

$\frac{4}{- 2 - 8}$

Step 3.1.2.1.2

Subtract $8$ from $- 2$ .

$\frac{4}{- 10}$

Step 3.1.2.2

Cancel the common factor of $4$ and $- 10$ .

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Step 3.1.2.2.1

Factor $2$ out of $4$ .

$\frac{2 (2)}{- 10}$

Step 3.1.2.2.2

Cancel the common factors.

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Step 3.1.2.2.2.1

Factor $2$ out of $- 10$ .

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

Step 3.1.2.2.2.2

Cancel the common factor.

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

Step 3.1.2.2.2.3

Rewrite the expression.

$\frac{2}{- 5}$

Step 3.1.2.3

Move the negative in front of the fraction.

$- \frac{2}{5}$

Step 3.2

Evaluate at $t = - \frac{1}{4}$ .

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Step 3.2.1

Substitute $- \frac{1}{4}$ for $t$ .

$\frac{{(- \frac{1}{4})}^{2}}{(- \frac{1}{4}) - 8}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Simplify the numerator.

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Step 3.2.2.1.1

Apply the product rule to $- \frac{1}{4}$ .

$\frac{{(- 1)}^{2} {(\frac{1}{4})}^{2}}{- \frac{1}{4} - 8}$

Step 3.2.2.1.2

Raise $- 1$ to the power of $2$ .

$\frac{1 {(\frac{1}{4})}^{2}}{- \frac{1}{4} - 8}$

Step 3.2.2.1.3

Apply the product rule to $\frac{1}{4}$ .

$\frac{1 \frac{1^{2}}{4^{2}}}{- \frac{1}{4} - 8}$

Step 3.2.2.1.4

One to any power is one.

$\frac{1 \frac{1}{4^{2}}}{- \frac{1}{4} - 8}$

Step 3.2.2.1.5

Raise $4$ to the power of $2$ .

$\frac{1 (\frac{1}{16})}{- \frac{1}{4} - 8}$

Step 3.2.2.1.6

Multiply $\frac{1}{16}$ by $1$ .

$\frac{\frac{1}{16}}{- \frac{1}{4} - 8}$

Step 3.2.2.2

Simplify the denominator.

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Step 3.2.2.2.1

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{1}{16}}{- \frac{1}{4} - 8 \cdot \frac{4}{4}}$

Step 3.2.2.2.2

Combine $- 8$ and $\frac{4}{4}$ .

$\frac{\frac{1}{16}}{- \frac{1}{4} + \frac{- 8 \cdot 4}{4}}$

Step 3.2.2.2.3

Combine the numerators over the common denominator.

$\frac{\frac{1}{16}}{\frac{- 1 - 8 \cdot 4}{4}}$

Step 3.2.2.2.4

Simplify the numerator.

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Step 3.2.2.2.4.1

Multiply $- 8$ by $4$ .

$\frac{\frac{1}{16}}{\frac{- 1 - 32}{4}}$

Step 3.2.2.2.4.2

Subtract $32$ from $- 1$ .

$\frac{\frac{1}{16}}{\frac{- 33}{4}}$

Step 3.2.2.2.5

Move the negative in front of the fraction.

$\frac{\frac{1}{16}}{- \frac{33}{4}}$

Step 3.2.2.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{16} (- \frac{4}{33})$

Step 3.2.2.4

Cancel the common factor of $4$ .

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Step 3.2.2.4.1

Move the leading negative in $- \frac{4}{33}$ into the numerator.

$\frac{1}{16} \cdot \frac{- 4}{33}$

Step 3.2.2.4.2

Factor $4$ out of $16$ .

$\frac{1}{4 (4)} \cdot \frac{- 4}{33}$

Step 3.2.2.4.3

Factor $4$ out of $- 4$ .

$\frac{1}{4 \cdot 4} \cdot \frac{4 \cdot - 1}{33}$

Step 3.2.2.4.4

Cancel the common factor.

$\frac{1}{4 \cdot 4} \cdot \frac{4 \cdot - 1}{33}$

Step 3.2.2.4.5

Rewrite the expression.

$\frac{1}{4} \cdot \frac{- 1}{33}$

Step 3.2.2.5

Multiply $\frac{1}{4}$ by $\frac{- 1}{33}$ .

$\frac{- 1}{4 \cdot 33}$

Step 3.2.2.6

Simplify the expression.

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Step 3.2.2.6.1

Multiply $4$ by $33$ .

$\frac{- 1}{132}$

Step 3.2.2.6.2

Move the negative in front of the fraction.

$- \frac{1}{132}$

Step 3.3

List all of the points.

$(- 2, - \frac{2}{5}), (- \frac{1}{4}, - \frac{1}{132})$

Step 4

Compare the $f (t)$ values found for each value of $t$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (t)$ value and the minimum will occur at the lowest $f (t)$ value.

Absolute Maximum: $(- \frac{1}{4}, - \frac{1}{132})$

Absolute Minimum: $(- 2, - \frac{2}{5})$

Step 5