Calculus Examples

$\frac{d A}{d t} = 0.05 A + 3200$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{0.05 A + 3200}$ .

$\frac{1}{0.05 A + 3200} \frac{d A}{d t} = \frac{1}{0.05 A + 3200} (0.05 A + 3200)$

Step 1.2

Cancel the common factor of $0.05 A + 3200$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{0.05 A + 3200} \frac{d A}{d t} = \frac{1}{0.05 A + 3200} (0.05 A + 3200)$

Step 1.2.2

Rewrite the expression.

$\frac{1}{0.05 A + 3200} \frac{d A}{d t} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{0.05 A + 3200} d A = 1 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{0.05 A + 3200} d A = \int d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = 0.05 A + 3200$ . Then $d u = 0.05 d A$ , so $\frac{1}{0.05} d u = d A$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = 0.05 A + 3200$ . Find $\frac{d u}{d A}$ .

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Step 2.2.1.1.1

Rewrite.

$\frac{0.05}{1}$

Step 2.2.1.1.2

Divide $0.05$ by $1$ .

$0.05$

Step 2.2.1.2

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

$\int \frac{20}{u} d u = \int d t$

Step 2.2.2

Since $20$ is constant with respect to $u$ , move $20$ out of the integral.

$20 \int \frac{1}{u} d u = \int d t$

Step 2.2.3

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

$20 (\ln (| u |) + C_{1}) = \int d t$

Step 2.2.4

Simplify.

$20 \ln (| u |) + C_{1} = \int d t$

Step 2.2.5

Replace all occurrences of $u$ with $0.05 A + 3200$ .

$20 \ln (| 0.05 A + 3200 |) + C_{1} = \int d t$

Step 2.3

Apply the constant rule.

$20 \ln (| 0.05 A + 3200 |) + C_{1} = t + C_{2}$

Step 2.4

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

$20 \ln (| 0.05 A + 3200 |) = t + C$

Step 3

Solve for $A$ .

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Step 3.1

Divide each term in $20 \ln (| 0.05 A + 3200 |) = t + C$ by $20$ and simplify.

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Step 3.1.1

Divide each term in $20 \ln (| 0.05 A + 3200 |) = t + C$ by $20$ .

$\frac{20 \ln (| 0.05 A + 3200 |)}{20} = \frac{t}{20} + \frac{C}{20}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $20$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{20 \ln (| 0.05 A + 3200 |)}{20} = \frac{t}{20} + \frac{C}{20}$

Step 3.1.2.1.2

Divide $\ln (| 0.05 A + 3200 |)$ by $1$ .

$\ln (| 0.05 A + 3200 |) = \frac{t}{20} + \frac{C}{20}$

Step 3.2

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

$e^{\ln (| 0.05 A + 3200 |)} = e^{\frac{t}{20} + \frac{C}{20}}$

Step 3.3

Rewrite $\ln (| 0.05 A + 3200 |) = \frac{t}{20} + \frac{C}{20}$ 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 $b^{y} = x$ .

$e^{\frac{t}{20} + \frac{C}{20}} = | 0.05 A + 3200 |$

Step 3.4

Solve for $A$ .

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Step 3.4.1

Rewrite the equation as $| 0.05 A + 3200 | = e^{\frac{t}{20} + \frac{C}{20}}$ .

$| 0.05 A + 3200 | = e^{\frac{t}{20} + \frac{C}{20}}$

Step 3.4.2

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

$0.05 A + 3200 = \pm e^{\frac{t}{20} + \frac{C}{20}}$

Step 3.4.3

Subtract $3200$ from both sides of the equation.

$0.05 A = \pm e^{\frac{t}{20} + \frac{C}{20}} - 3200$

Step 3.4.4

Divide each term in $0.05 A = \pm e^{\frac{t}{20} + \frac{C}{20}} - 3200$ by $0.05$ and simplify.

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Step 3.4.4.1

Divide each term in $0.05 A = \pm e^{\frac{t}{20} + \frac{C}{20}} - 3200$ by $0.05$ .

$\frac{0.05 A}{0.05} = \frac{\pm e^{\frac{t}{20} + \frac{C}{20}}}{0.05} + \frac{- 3200}{0.05}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $0.05$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{0.05 A}{0.05} = \frac{\pm e^{\frac{t}{20} + \frac{C}{20}}}{0.05} + \frac{- 3200}{0.05}$

Step 3.4.4.2.1.2

Divide $A$ by $1$ .

$A = \frac{\pm e^{\frac{t}{20} + \frac{C}{20}}}{0.05} + \frac{- 3200}{0.05}$

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

Combine the numerators over the common denominator.

$A = \frac{\pm e^{\frac{t + C}{20}}}{0.05} + \frac{- 3200}{0.05}$

Step 3.4.4.3.1.2

Multiply by $1$ .

$A = \frac{1 (\pm e^{\frac{t + C}{20}})}{0.05} + \frac{- 3200}{0.05}$

Step 3.4.4.3.1.3

Factor $0.05$ out of $0.05$ .

$A = \frac{1 (\pm e^{\frac{t + C}{20}})}{0.05 (1)} + \frac{- 3200}{0.05}$

Step 3.4.4.3.1.4

Separate fractions.

$A = \frac{1}{0.05} \cdot \frac{\pm e^{\frac{t + C}{20}}}{1} + \frac{- 3200}{0.05}$

Step 3.4.4.3.1.5

Divide $1$ by $0.05$ .

$A = 20 \frac{\pm e^{\frac{t + C}{20}}}{1} + \frac{- 3200}{0.05}$

Step 3.4.4.3.1.6

Divide $\pm e^{\frac{t + C}{20}}$ by $1$ .

$A = 20 (\pm e^{\frac{t + C}{20}}) + \frac{- 3200}{0.05}$

Step 3.4.4.3.1.7

Divide $- 3200$ by $0.05$ .

$A = 20 (\pm e^{\frac{t + C}{20}}) - 64000$