Calculus Examples

$\frac{d A}{d t} = 0.07 A$

Step 1

Separate the variables.

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Step 1.1

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

$\frac{1}{A} \frac{d A}{d t} = \frac{1}{A} (0.07 A)$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{A} \frac{d A}{d t} = 0.07 \frac{1}{A} A$

Step 1.2.2

Combine $0.07$ and $\frac{1}{A}$ .

$\frac{1}{A} \frac{d A}{d t} = \frac{0.07}{A} A$

Step 1.2.3

Cancel the common factor of $A$ .

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Step 1.2.3.1

Cancel the common factor.

$\frac{1}{A} \frac{d A}{d t} = \frac{0.07}{A} A$

Step 1.2.3.2

Rewrite the expression.

$\frac{1}{A} \frac{d A}{d t} = 0.07$

Step 1.3

Rewrite the equation.

$\frac{1}{A} d A = 0.07 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{A} d A = \int 0.07 d t$

Step 2.2

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

$\ln (| A |) + C_{1} = \int 0.07 d t$

Step 2.3

Apply the constant rule.

$\ln (| A |) + C_{1} = 0.07 t + C_{2}$

Step 2.4

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

$\ln (| A |) = 0.07 t + C$

Step 3

Solve for $A$ .

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Step 3.1

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

$e^{\ln (| A |)} = e^{0.07 t + C}$

Step 3.2

Rewrite $\ln (| A |) = 0.07 t + C$ 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 $b^{y} = x$ .

$e^{0.07 t + C} = | A |$

Step 3.3

Solve for $A$ .

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Step 3.3.1

Rewrite the equation as $| A | = e^{0.07 t + C}$ .

$| A | = e^{0.07 t + C}$

Step 3.3.2

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

$A = \pm e^{0.07 t + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{0.07 t + C}$ as $e^{0.07 t} e^{C}$ .

$A = \pm e^{0.07 t} e^{C}$

Step 4.2

Reorder $e^{0.07 t}$ and $e^{C}$ .

$A = \pm e^{C} e^{0.07 t}$

Step 4.3

Combine constants with the plus or minus.

$A = C e^{0.07 t}$