Calculus Examples

$g (t) = \frac{t}{t - 8}$ on $10$ , $12$

Step 1

Find the critical points.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Find the first derivative.

Tap for more steps...

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 $f (t) = t$ and $g (t) = t - 8$ .

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

Step 1.1.1.2

Differentiate.

Tap for more steps...

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 $n = 1$ .

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

Step 1.1.1.2.2

Multiply $t - 8$ by $1$ .

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

Step 1.1.1.2.3

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

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

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 $n = 1$ .

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

Step 1.1.1.2.5

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

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

Step 1.1.1.2.6

Simplify by adding terms.

Tap for more steps...

Step 1.1.1.2.6.1

Add $1$ and $0$ .

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

Step 1.1.1.2.6.2

Multiply $- 1$ by $1$ .

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

Step 1.1.1.2.6.3

Subtract $t$ from $t$ .

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

Step 1.1.1.2.6.4

Simplify the expression.

Tap for more steps...

Step 1.1.1.2.6.4.1

Subtract $8$ from $0$ .

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

Step 1.1.1.2.6.4.2

Move the negative in front of the fraction.

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

Step 1.1.2

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

Set the first derivative equal to $0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$8 = 0$

Step 1.2.3

Since $8 \neq 0$ , there are no solutions.

No solution

Step 1.3

Find the values where the derivative is undefined.

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

Solve for $t$ .

Tap for more steps...

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

Tap for more steps...

Step 1.4.1

Evaluate at $t = 8$ .

Tap for more steps...

Step 1.4.1.1

Substitute $8$ for $t$ .

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

Step 1.4.1.2

Simplify.

Tap for more steps...

Step 1.4.1.2.1

Subtract $8$ from $8$ .

$\frac{8}{0}$

Step 1.4.1.2.2

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

Undefined

Step 1.5

There are no values of $t$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

Step 2

Evaluate at the included endpoints.

Tap for more steps...

Step 2.1

Evaluate at $t = 10$ .

Tap for more steps...

Step 2.1.1

Substitute $10$ for $t$ .

$\frac{10}{(10) - 8}$

Step 2.1.2

Simplify.

Tap for more steps...

Step 2.1.2.1

Cancel the common factor of $10$ and $(10) - 8$ .

Tap for more steps...

Step 2.1.2.1.1

Factor $2$ out of $10$ .

$\frac{2 (5)}{(10) - 8}$

Step 2.1.2.1.2

Cancel the common factors.

Tap for more steps...

Step 2.1.2.1.2.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 5}{2 \cdot 5 - 8}$

Step 2.1.2.1.2.2

Factor $2$ out of $- 8$ .

$\frac{2 (5)}{2 \cdot 5 + 2 \cdot - 4}$

Step 2.1.2.1.2.3

Factor $2$ out of $2 \cdot 5 + 2 \cdot - 4$ .

$\frac{2 (5)}{2 \cdot (5 - 4)}$

Step 2.1.2.1.2.4

Cancel the common factor.

$\frac{2 \cdot 5}{2 \cdot (5 - 4)}$

Step 2.1.2.1.2.5

Rewrite the expression.

$\frac{5}{5 - 4}$

Step 2.1.2.2

Simplify the expression.

Tap for more steps...

Step 2.1.2.2.1

Subtract $4$ from $5$ .

$\frac{5}{1}$

Step 2.1.2.2.2

Divide $5$ by $1$ .

$5$

Step 2.2

Evaluate at $t = 12$ .

Tap for more steps...

Step 2.2.1

Substitute $12$ for $t$ .

$\frac{12}{(12) - 8}$

Step 2.2.2

Simplify.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $12$ and $(12) - 8$ .

Tap for more steps...

Step 2.2.2.1.1

Factor $4$ out of $12$ .

$\frac{4 (3)}{(12) - 8}$

Step 2.2.2.1.2

Cancel the common factors.

Tap for more steps...

Step 2.2.2.1.2.1

Factor $4$ out of $12$ .

$\frac{4 \cdot 3}{4 \cdot 3 - 8}$

Step 2.2.2.1.2.2

Factor $4$ out of $- 8$ .

$\frac{4 (3)}{4 \cdot 3 + 4 \cdot - 2}$

Step 2.2.2.1.2.3

Factor $4$ out of $4 \cdot 3 + 4 \cdot - 2$ .

$\frac{4 (3)}{4 \cdot (3 - 2)}$

Step 2.2.2.1.2.4

Cancel the common factor.

$\frac{4 \cdot 3}{4 \cdot (3 - 2)}$

Step 2.2.2.1.2.5

Rewrite the expression.

$\frac{3}{3 - 2}$

Step 2.2.2.2

Simplify the expression.

Tap for more steps...

Step 2.2.2.2.1

Subtract $2$ from $3$ .

$\frac{3}{1}$

Step 2.2.2.2.2

Divide $3$ by $1$ .

$3$

Step 2.3

List all of the points.

$(10, 5), (12, 3)$

Step 3

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

Absolute Maximum: $(10, 5)$

Absolute Minimum: $(12, 3)$

Step 4