Calculus Examples

f (t) = 70 {(\frac{2}{5})}^{t}

$f (t) = 70 {(\frac{2}{5})}^{t}$

Step 1

Since

0.4

$0.4$ is less than

1

$1$ , there is an exponential decay.

a {(1 - r)}^{t}

$a {(1 - r)}^{t}$

Step 2

Setup the rate equation.

1 - r = 0.4

$1 - r = 0.4$

Step 3

Solve

1 - r = 0.4

$1 - r = 0.4$ for

r

$r$ to determine the decay factor.

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Step 3.1

Move all terms not containing

r

$r$ to the right side of the equation.

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Step 3.1.1

Subtract

1

$1$ from both sides of the equation.

- r = 0.4 - 1

$- r = 0.4 - 1$

Step 3.1.2

Subtract

1

$1$ from

0.4

$0.4$ .

- r = - 0.6

$- r = - 0.6$

- r = - 0.6

$- r = - 0.6$

Step 3.2

Divide each term in

- r = - 0.6

$- r = - 0.6$ by

- 1

$- 1$ and simplify.

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Step 3.2.1

Divide each term in

- r = - 0.6

$- r = - 0.6$ by

- 1

$- 1$ .

\frac{- r}{- 1} = \frac{- 0.6}{- 1}

$\frac{- r}{- 1} = \frac{- 0.6}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

\frac{r}{1} = \frac{- 0.6}{- 1}

$\frac{r}{1} = \frac{- 0.6}{- 1}$

Step 3.2.2.2

Divide

r

$r$ by

1

$1$ .

r = \frac{- 0.6}{- 1}

$r = \frac{- 0.6}{- 1}$

r = \frac{- 0.6}{- 1}

$r = \frac{- 0.6}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide

- 0.6

$- 0.6$ by

- 1

$- 1$ .

r = 0.6

$r = 0.6$

r = 0.6

$r = 0.6$

r = 0.6

$r = 0.6$

r = 0.6

$r = 0.6$

Step 4

Convert to a percentage.

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Step 4.1

Multiply

0.6

$0.6$ by

100

$100$ to convert to a percentage.

0.6 \cdot 100

$0.6 \cdot 100$

Step 4.2

Simplify

0.6 \cdot 100

$0.6 \cdot 100$ .

60 %

$60 %$

60 %

$60 %$

⎡ ⎢ ⎣ \begin{matrix} x^{2} & \frac{1}{2} \sqrt{π} & \int x d x \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$