Calculus Examples

$C (t) = 3 t e^{- \frac{1}{30} t}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1.1

Combine $t$ and $\frac{1}{30}$ .

$f' (t) = \frac{d}{d t} (3 t e^{- \frac{t}{30}})$

Step 1.1.1.2

Since $3$ is constant with respect to $t$ , the derivative of $3 t e^{- \frac{t}{30}}$ with respect to $t$ is $3 \frac{d}{d t} [t e^{- \frac{t}{30}}]$ .

$f' (t) = 3 \frac{d}{d t} (t e^{- \frac{t}{30}})$

Step 1.1.2

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = t$ and $g (t) = e^{- \frac{t}{30}}$ .

$f' (t) = 3 (t \frac{d}{d t} (e^{- \frac{t}{30}}) + e^{- \frac{t}{30}} \frac{d}{d t} (t))$

Step 1.1.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = e^{t}$ and $g (t) = - \frac{t}{30}$ .

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Step 1.1.3.1

To apply the Chain Rule, set $u$ as $- \frac{t}{30}$ .

$f' (t) = 3 (t (\frac{d}{d u} (e^{u}) \frac{d}{d t} (- \frac{t}{30})) + e^{- \frac{t}{30}} \frac{d}{d t} (t))$

Step 1.1.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$f' (t) = 3 (t (e^{u} \frac{d}{d t} (- \frac{t}{30})) + e^{- \frac{t}{30}} \frac{d}{d t} (t))$

Step 1.1.3.3

Replace all occurrences of $u$ with $- \frac{t}{30}$ .

$f' (t) = 3 (t (e^{- \frac{t}{30}} \frac{d}{d t} (- \frac{t}{30})) + e^{- \frac{t}{30}} \frac{d}{d t} (t))$

Step 1.1.4

Differentiate.

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Step 1.1.4.1

Since $- \frac{1}{30}$ is constant with respect to $t$ , the derivative of $- \frac{t}{30}$ with respect to $t$ is $- \frac{1}{30} \frac{d}{d t} [t]$ .

$f' (t) = 3 (t (e^{- \frac{t}{30}} (- \frac{1}{30} \cdot \frac{d}{d t} t)) + e^{- \frac{t}{30}} \frac{d}{d t} (t))$

Step 1.1.4.2

Combine fractions.

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Step 1.1.4.2.1

Combine $e^{- \frac{t}{30}}$ and $\frac{1}{30}$ .

$f' (t) = 3 (t (- \frac{e^{- \frac{t}{30}}}{30} \cdot \frac{d}{d t} t) + e^{- \frac{t}{30}} \frac{d}{d t} (t))$

Step 1.1.4.2.2

Combine $t$ and $\frac{e^{- \frac{t}{30}}}{30}$ .

$f' (t) = 3 (- \frac{t e^{- \frac{t}{30}}}{30} \cdot \frac{d}{d t} t + e^{- \frac{t}{30}} \frac{d}{d t} (t))$

Step 1.1.4.3

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

$f' (t) = 3 (- \frac{t e^{- \frac{t}{30}}}{30} \cdot 1 + e^{- \frac{t}{30}} \frac{d}{d t} (t))$

Step 1.1.4.4

Multiply $- 1$ by $1$ .

$f' (t) = 3 (- \frac{t e^{- \frac{t}{30}}}{30} + e^{- \frac{t}{30}} \frac{d}{d t} (t))$

Step 1.1.4.5

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

$f' (t) = 3 (- \frac{t e^{- \frac{t}{30}}}{30} + e^{- \frac{t}{30}} \cdot 1)$

Step 1.1.4.6

Multiply $e^{- \frac{t}{30}}$ by $1$ .

$f' (t) = 3 (- \frac{t e^{- \frac{t}{30}}}{30} + e^{- \frac{t}{30}})$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Apply the distributive property.

$f' (t) = 3 (- \frac{t e^{- \frac{t}{30}}}{30}) + 3 e^{- \frac{t}{30}}$

Step 1.1.5.2

Combine terms.

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Step 1.1.5.2.1

Multiply $- 1$ by $3$ .

$f' (t) = - 3 \frac{t e^{- \frac{t}{30}}}{30} + 3 e^{- \frac{t}{30}}$

Step 1.1.5.2.2

Combine $- 3$ and $\frac{t e^{- \frac{t}{30}}}{30}$ .

$f' (t) = \frac{- 3 (t e^{- \frac{t}{30}})}{30} + 3 e^{- \frac{t}{30}}$

Step 1.1.5.2.3

Cancel the common factor of $- 3$ and $30$ .

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Step 1.1.5.2.3.1

Factor $3$ out of $- 3 t e^{- \frac{t}{30}}$ .

$f' (t) = \frac{3 (- t e^{- \frac{t}{30}})}{30} + 3 e^{- \frac{t}{30}}$

Step 1.1.5.2.3.2

Cancel the common factors.

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Step 1.1.5.2.3.2.1

Factor $3$ out of $30$ .

$f' (t) = \frac{3 (- t e^{- \frac{t}{30}})}{3 (10)} + 3 e^{- \frac{t}{30}}$

Step 1.1.5.2.3.2.2

Cancel the common factor.

$f' (t) = \frac{3 (- t e^{- \frac{t}{30}})}{3 \cdot 10} + 3 e^{- \frac{t}{30}}$

Step 1.1.5.2.3.2.3

Rewrite the expression.

$f' (t) = \frac{- t e^{- \frac{t}{30}}}{10} + 3 e^{- \frac{t}{30}}$

Step 1.1.5.2.4

Move the negative in front of the fraction.

$f' (t) = - \frac{t e^{- \frac{t}{30}}}{10} + 3 e^{- \frac{t}{30}}$

Step 1.2

The first derivative of $C (t)$ with respect to $t$ is $- \frac{t e^{- \frac{t}{30}}}{10} + 3 e^{- \frac{t}{30}}$ .

$- \frac{t e^{- \frac{t}{30}}}{10} + 3 e^{- \frac{t}{30}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{t e^{- \frac{t}{30}}}{10} + 3 e^{- \frac{t}{30}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{t e^{- \frac{t}{30}}}{10} + 3 e^{- \frac{t}{30}} = 0$

Step 2.2

Factor $e^{- \frac{t}{30}}$ out of $- \frac{t e^{- \frac{t}{30}}}{10} + 3 e^{- \frac{t}{30}}$ .

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Step 2.2.1

Factor $e^{- \frac{t}{30}}$ out of $- \frac{t e^{- \frac{t}{30}}}{10}$ .

$e^{- \frac{t}{30}} (- \frac{t}{10}) + 3 e^{- \frac{t}{30}} = 0$

Step 2.2.2

Factor $e^{- \frac{t}{30}}$ out of $3 e^{- \frac{t}{30}}$ .

$e^{- \frac{t}{30}} (- \frac{t}{10}) + e^{- \frac{t}{30}} \cdot 3 = 0$

Step 2.2.3

Factor $e^{- \frac{t}{30}}$ out of $e^{- \frac{t}{30}} (- \frac{t}{10}) + e^{- \frac{t}{30}} \cdot 3$ .

$e^{- \frac{t}{30}} (- \frac{t}{10} + 3) = 0$

Step 2.3

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

$e^{- \frac{t}{30}} = 0$

$- \frac{t}{10} + 3 = 0$

Step 2.4

Set $e^{- \frac{t}{30}}$ equal to $0$ and solve for $t$ .

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Step 2.4.1

Set $e^{- \frac{t}{30}}$ equal to $0$ .

$e^{- \frac{t}{30}} = 0$

Step 2.4.2

Solve $e^{- \frac{t}{30}} = 0$ for $t$ .

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Step 2.4.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- \frac{t}{30}}) = \ln (0)$

Step 2.4.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.4.2.3

There is no solution for $e^{- \frac{t}{30}} = 0$

No solution

Step 2.5

Set $- \frac{t}{10} + 3$ equal to $0$ and solve for $t$ .

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Step 2.5.1

Set $- \frac{t}{10} + 3$ equal to $0$ .

$- \frac{t}{10} + 3 = 0$

Step 2.5.2

Solve $- \frac{t}{10} + 3 = 0$ for $t$ .

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Step 2.5.2.1

Subtract $3$ from both sides of the equation.

$- \frac{t}{10} = - 3$

Step 2.5.2.2

Multiply both sides of the equation by $- 10$ .

$- 10 (- \frac{t}{10}) = - 10 \cdot - 3$

Step 2.5.2.3

Simplify both sides of the equation.

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Step 2.5.2.3.1

Simplify the left side.

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Step 2.5.2.3.1.1

Simplify $- 10 (- \frac{t}{10})$ .

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Step 2.5.2.3.1.1.1

Cancel the common factor of $10$ .

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Step 2.5.2.3.1.1.1.1

Move the leading negative in $- \frac{t}{10}$ into the numerator.

$- 10 \frac{- t}{10} = - 10 \cdot - 3$

Step 2.5.2.3.1.1.1.2

Factor $10$ out of $- 10$ .

$10 (- 1) \frac{- t}{10} = - 10 \cdot - 3$

Step 2.5.2.3.1.1.1.3

Cancel the common factor.

$10 \cdot - 1 \frac{- t}{10} = - 10 \cdot - 3$

Step 2.5.2.3.1.1.1.4

Rewrite the expression.

$- - t = - 10 \cdot - 3$

Step 2.5.2.3.1.1.2

Multiply.

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Step 2.5.2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 t = - 10 \cdot - 3$

Step 2.5.2.3.1.1.2.2

Multiply $t$ by $1$ .

$t = - 10 \cdot - 3$

Step 2.5.2.3.2

Simplify the right side.

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Step 2.5.2.3.2.1

Multiply $- 10$ by $- 3$ .

$t = 30$

Step 2.6

The final solution is all the values that make $e^{- \frac{t}{30}} (- \frac{t}{10} + 3) = 0$ true.

$t = 30$

Step 3

Find the values where the derivative is undefined.

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Step 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 4

Evaluate $3 t e^{- \frac{1}{30} t}$ at each $t$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $t = 30$ .

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Step 4.1.1

Substitute $30$ for $t$ .

$3 (30) \cdot e^{- \frac{1}{30} \cdot 30}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $3$ by $30$ .

$90 \cdot e^{- \frac{1}{30} \cdot 30}$

Step 4.1.2.2

Cancel the common factor of $30$ .

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Step 4.1.2.2.1

Move the leading negative in $- \frac{1}{30}$ into the numerator.

$90 \cdot e^{\frac{- 1}{30} \cdot 30}$

Step 4.1.2.2.2

Cancel the common factor.

$90 \cdot e^{\frac{- 1}{30} \cdot 30}$

Step 4.1.2.2.3

Rewrite the expression.

$90 \cdot e^{- 1}$

Step 4.1.2.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$90 \cdot \frac{1}{e}$

Step 4.1.2.4

Combine $90$ and $\frac{1}{e}$ .

$\frac{90}{e}$

Step 4.2

List all of the points.

$(30, \frac{90}{e})$

Step 5