Calculus Examples

$\frac{d P}{d t} = 500 - 0.12 P$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{500 - 0.12 P}$ .

$\frac{1}{500 - 0.12 P} \frac{d P}{d t} = \frac{1}{500 - 0.12 P} (500 - 0.12 P)$

Step 1.2

Cancel the common factor of $500 - 0.12 P$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{500 - 0.12 P} \frac{d P}{d t} = \frac{1}{500 - 0.12 P} (500 - 0.12 P)$

Step 1.2.2

Rewrite the expression.

$\frac{1}{500 - 0.12 P} \frac{d P}{d t} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{500 - 0.12 P} d P = 1 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{500 - 0.12 P} d P = \int d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = 500 - 0.12 P$ . Then $d u = - 0.12 d P$ , so $\frac{1}{- 0.12} d u = d P$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = 500 - 0.12 P$ . Find $\frac{d u}{d P}$ .

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Step 2.2.1.1.1

Differentiate $500 - 0.12 P$ .

$\frac{d}{d P} [500 - 0.12 P]$

Step 2.2.1.1.2

Differentiate.

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Step 2.2.1.1.2.1

By the Sum Rule, the derivative of $500 - 0.12 P$ with respect to $P$ is $\frac{d}{d P} [500] + \frac{d}{d P} [- 0.12 P]$ .

$\frac{d}{d P} [500] + \frac{d}{d P} [- 0.12 P]$

Step 2.2.1.1.2.2

Since $500$ is constant with respect to $P$ , the derivative of $500$ with respect to $P$ is $0$ .

$0 + \frac{d}{d P} [- 0.12 P]$

Step 2.2.1.1.3

Evaluate $\frac{d}{d P} [- 0.12 P]$ .

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Step 2.2.1.1.3.1

Since $- 0.12$ is constant with respect to $P$ , the derivative of $- 0.12 P$ with respect to $P$ is $- 0.12 \frac{d}{d P} [P]$ .

$0 - 0.12 \frac{d}{d P} [P]$

Step 2.2.1.1.3.2

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

$0 - 0.12 \cdot 1$

Step 2.2.1.1.3.3

Multiply $- 0.12$ by $1$ .

$0 - 0.12$

Step 2.2.1.1.4

Subtract $0.12$ from $0$ .

$- 0.12$

Step 2.2.1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{u} \cdot \frac{1}{- 0.12} d u = \int d t$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Move the negative in front of the fraction.

$\int \frac{1}{u} (- \frac{1}{0.12}) d u = \int d t$

Step 2.2.2.2

Multiply $\frac{1}{u}$ by $\frac{1}{0.12}$ .

$\int - \frac{1}{u \cdot 0.12} d u = \int d t$

Step 2.2.2.3

Move $0.12$ to the left of $u$ .

$\int - \frac{1}{0.12 u} d u = \int d t$

Step 2.2.3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int \frac{1}{0.12 u} d u = \int d t$

Step 2.2.4

Since $\frac{1}{0.12}$ is constant with respect to $u$ , move $\frac{1}{0.12}$ out of the integral.

$- (\frac{1}{0.12} \int \frac{1}{u} d u) = \int d t$

Step 2.2.5

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$- \frac{1}{0.12} (\ln (| u |) + C_{1}) = \int d t$

Step 2.2.6

Simplify.

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Step 2.2.6.1

Rewrite $- \frac{1}{0.12} (\ln (| u |) + C_{1})$ as $- 1 \cdot 8.\overline{3} \ln (| u |) + C_{1}$ .

$- 1 \cdot 8.\overline{3} \ln (| u |) + C_{1} = \int d t$

Step 2.2.6.2

Multiply $- 1$ by $8.\overline{3}$ .

$- 8.\overline{3} \ln (| u |) + C_{1} = \int d t$

Step 2.2.7

Replace all occurrences of $u$ with $500 - 0.12 P$ .

$- 8.\overline{3} \ln (| 500 - 0.12 P |) + C_{1} = \int d t$

Step 2.3

Apply the constant rule.

$- 8.\overline{3} \ln (| 500 - 0.12 P |) + C_{1} = t + C_{2}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$- 8.\overline{3} \ln (| 500 - 0.12 P |) = t + C$

Step 3

Solve for $P$ .

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Step 3.1

Divide each term in $- 8.\overline{3} \ln (| 500 - 0.12 P |) = t + C$ by $- 8.\overline{3}$ and simplify.

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Step 3.1.1

Divide each term in $- 8.\overline{3} \ln (| 500 - 0.12 P |) = t + C$ by $- 8.\overline{3}$ .

$\frac{- 8.\overline{3} \ln (| 500 - 0.12 P |)}{- 8.\overline{3}} = \frac{t}{- 8.\overline{3}} + \frac{C}{- 8.\overline{3}}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $- 8.\overline{3}$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{- 8.\overline{3} \ln (| 500 - 0.12 P |)}{- 8.\overline{3}} = \frac{t}{- 8.\overline{3}} + \frac{C}{- 8.\overline{3}}$

Step 3.1.2.1.2

Divide $\ln (| 500 - 0.12 P |)$ by $1$ .

$\ln (| 500 - 0.12 P |) = \frac{t}{- 8.\overline{3}} + \frac{C}{- 8.\overline{3}}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Move the negative in front of the fraction.

$\ln (| 500 - 0.12 P |) = - \frac{t}{8.\overline{3}} + \frac{C}{- 8.\overline{3}}$

Step 3.1.3.1.2

Multiply by $1$ .

$\ln (| 500 - 0.12 P |) = - \frac{1 t}{8.\overline{3}} + \frac{C}{- 8.\overline{3}}$

Step 3.1.3.1.3

Factor $8.\overline{3}$ out of $8.\overline{3}$ .

$\ln (| 500 - 0.12 P |) = - \frac{1 t}{8.\overline{3} (1)} + \frac{C}{- 8.\overline{3}}$

Step 3.1.3.1.4

Separate fractions.

$\ln (| 500 - 0.12 P |) = - (\frac{1}{8.\overline{3}} \cdot \frac{t}{1}) + \frac{C}{- 8.\overline{3}}$

Step 3.1.3.1.5

Divide $1$ by $8.\overline{3}$ .

$\ln (| 500 - 0.12 P |) = - (0.12 \frac{t}{1}) + \frac{C}{- 8.\overline{3}}$

Step 3.1.3.1.6

Divide $t$ by $1$ .

$\ln (| 500 - 0.12 P |) = - (0.12 t) + \frac{C}{- 8.\overline{3}}$

Step 3.1.3.1.7

Multiply $0.12$ by $- 1$ .

$\ln (| 500 - 0.12 P |) = - 0.12 t + \frac{C}{- 8.\overline{3}}$

Step 3.1.3.1.8

Move the negative in front of the fraction.

$\ln (| 500 - 0.12 P |) = - 0.12 t - \frac{C}{8.\overline{3}}$

Step 3.1.3.1.9

Multiply by $1$ .

$\ln (| 500 - 0.12 P |) = - 0.12 t - \frac{1 C}{8.\overline{3}}$

Step 3.1.3.1.10

Factor $8.\overline{3}$ out of $8.\overline{3}$ .

$\ln (| 500 - 0.12 P |) = - 0.12 t - \frac{1 C}{8.\overline{3} (1)}$

Step 3.1.3.1.11

Separate fractions.

$\ln (| 500 - 0.12 P |) = - 0.12 t - (\frac{1}{8.\overline{3}} \cdot \frac{C}{1})$

Step 3.1.3.1.12

Divide $1$ by $8.\overline{3}$ .

$\ln (| 500 - 0.12 P |) = - 0.12 t - (0.12 \frac{C}{1})$

Step 3.1.3.1.13

Divide $C$ by $1$ .

$\ln (| 500 - 0.12 P |) = - 0.12 t - (0.12 C)$

Step 3.1.3.1.14

Multiply $0.12$ by $- 1$ .

$\ln (| 500 - 0.12 P |) = - 0.12 t - 0.12 C$

Step 3.2

To solve for $P$ , rewrite the equation using properties of logarithms.

$e^{\ln (| 500 - 0.12 P |)} = e^{- 0.12 t - 0.12 C}$

Step 3.3

Rewrite $\ln (| 500 - 0.12 P |) = - 0.12 t - 0.12 C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- 0.12 t - 0.12 C} = | 500 - 0.12 P |$

Step 3.4

Solve for $P$ .

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Step 3.4.1

Rewrite the equation as $| 500 - 0.12 P | = e^{- 0.12 t - 0.12 C}$ .

$| 500 - 0.12 P | = e^{- 0.12 t - 0.12 C}$

Step 3.4.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$500 - 0.12 P = \pm e^{- 0.12 t - 0.12 C}$

Step 3.4.3

Subtract $500$ from both sides of the equation.

$- 0.12 P = \pm e^{- 0.12 t - 0.12 C} - 500$

Step 3.4.4

Divide each term in $- 0.12 P = \pm e^{- 0.12 t - 0.12 C} - 500$ by $- 0.12$ and simplify.

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Step 3.4.4.1

Divide each term in $- 0.12 P = \pm e^{- 0.12 t - 0.12 C} - 500$ by $- 0.12$ .

$\frac{- 0.12 P}{- 0.12} = \frac{\pm e^{- 0.12 t - 0.12 C}}{- 0.12} + \frac{- 500}{- 0.12}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $- 0.12$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{- 0.12 P}{- 0.12} = \frac{\pm e^{- 0.12 t - 0.12 C}}{- 0.12} + \frac{- 500}{- 0.12}$

Step 3.4.4.2.1.2

Divide $P$ by $1$ .

$P = \frac{\pm e^{- 0.12 t - 0.12 C}}{- 0.12} + \frac{- 500}{- 0.12}$

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Simplify each term.

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Step 3.4.4.3.1.1

Simplify $\frac{\pm e^{- 0.12 t - 0.12 C}}{- 0.12}$ .

$P = \frac{\pm e^{- 0.12 t - 0.12 C}}{0.12} + \frac{- 500}{- 0.12}$

Step 3.4.4.3.1.2

Multiply by $1$ .

$P = \frac{1 (\pm e^{- 0.12 t - 0.12 C})}{0.12} + \frac{- 500}{- 0.12}$

Step 3.4.4.3.1.3

Factor $0.12$ out of $0.12$ .

$P = \frac{1 (\pm e^{- 0.12 t - 0.12 C})}{0.12 (1)} + \frac{- 500}{- 0.12}$

Step 3.4.4.3.1.4

Separate fractions.

$P = \frac{1}{0.12} \cdot \frac{\pm e^{- 0.12 t - 0.12 C}}{1} + \frac{- 500}{- 0.12}$

Step 3.4.4.3.1.5

Divide $1$ by $0.12$ .

$P = 8.\overline{3} \frac{\pm e^{- 0.12 t - 0.12 C}}{1} + \frac{- 500}{- 0.12}$

Step 3.4.4.3.1.6

Divide $\pm e^{- 0.12 t - 0.12 C}$ by $1$ .

$P = 8.\overline{3} (\pm e^{- 0.12 t - 0.12 C}) + \frac{- 500}{- 0.12}$

Step 3.4.4.3.1.7

Divide $- 500$ by $- 0.12$ .

$P = 8.\overline{3} (\pm e^{- 0.12 t - 0.12 C}) + 4166.\overline{6}$

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$P = 8.\overline{3} (\pm e^{- 0.12 t + C}) + 4166.\overline{6}$

Step 4.2

Rewrite $e^{- 0.12 t + C}$ as $e^{- 0.12 t} e^{C}$ .

$P = 8.\overline{3} (\pm e^{- 0.12 t} e^{C}) + 4166.\overline{6}$

Step 4.3

Reorder $e^{- 0.12 t}$ and $e^{C}$ .

$P = 8.\overline{3} (\pm e^{C} e^{- 0.12 t}) + 4166.\overline{6}$

Step 4.4

Combine constants with the plus or minus.

$P = 8.\overline{3} (C e^{- 0.12 t}) + 4166.\overline{6}$