Calculus Examples

$- 5 e^{- 5 \cdot 0.007} \sin (250 π (0.007)) + 250 π (0.007) + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1

Simplify each term.

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Step 1.1

Multiply $- 5$ by $0.007$ .

$- 5 e^{- 0.035} \sin (250 π (0.007)) + 250 π (0.007) + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1.2

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

$- 5 \frac{1}{e^{0.035}} \sin (250 π (0.007)) + 250 π (0.007) + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1.3

Combine $- 5$ and $\frac{1}{e^{0.035}}$ .

$\frac{- 5}{e^{0.035}} \sin (250 π (0.007)) + 250 π (0.007) + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1.4

Move the negative in front of the fraction.

$- \frac{5}{e^{0.035}} \sin (250 π (0.007)) + 250 π (0.007) + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1.5

Replace $e$ with an approximation.

$- \frac{5}{{2.71828182}^{0.035}} \sin (250 π (0.007)) + 250 π (0.007) + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1.6

Raise $2.71828182$ to the power of $0.035$ .

$- \frac{5}{1.0356197} \sin (250 π (0.007)) + 250 π (0.007) + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1.7

Divide $5$ by $1.0356197$ .

$- 1 \cdot 4.82802708 \sin (250 π (0.007)) + 250 π (0.007) + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1.8

Multiply $- 1$ by $4.82802708$ .

$- 4.82802708 \sin (250 π (0.007)) + 250 π (0.007) + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1.9

Multiply $250 π (0.007)$ .

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Step 1.9.1

Multiply $0.007$ by $250$ .

$- 4.82802708 \sin (1.75 π) + 250 π (0.007) + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1.9.2

Multiply $1.75$ by $π$ .

$- 4.82802708 \sin (5.49778714) + 250 π (0.007) + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1.10

Multiply $- 4.82802708$ by $\sin (5.49778714)$ .

$3.41393068 + 250 π (0.007) + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1.11

Multiply $250 π (0.007)$ .

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Step 1.11.1

Multiply $0.007$ by $250$ .

$3.41393068 + 1.75 π + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1.11.2

Multiply $1.75$ by $π$ .

$3.41393068 + 5.49778714 + 250 π e^{- 5 \cdot 0.007} \cos (250 π (0.007))$

Step 1.12

Multiply $- 5$ by $0.007$ .

$3.41393068 + 5.49778714 + 250 π e^{- 0.035} \cos (250 π (0.007))$

Step 1.13

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

$3.41393068 + 5.49778714 + 250 π \frac{1}{e^{0.035}} \cos (250 π (0.007))$

Step 1.14

Multiply $250 π \frac{1}{e^{0.035}}$ .

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Step 1.14.1

Combine $\frac{1}{e^{0.035}}$ and $250$ .

$3.41393068 + 5.49778714 + \frac{250}{e^{0.035}} π \cos (250 π (0.007))$

Step 1.14.2

Combine $\frac{250}{e^{0.035}}$ and $π$ .

$3.41393068 + 5.49778714 + \frac{250 π}{e^{0.035}} \cos (250 π (0.007))$

Step 1.15

Multiply $250 π (0.007)$ .

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Step 1.15.1

Multiply $0.007$ by $250$ .

$3.41393068 + 5.49778714 + \frac{250 π}{e^{0.035}} \cos (1.75 π)$

Step 1.15.2

Multiply $1.75$ by $π$ .

$3.41393068 + 5.49778714 + \frac{250 π}{e^{0.035}} \cos (5.49778714)$

Step 1.16

Combine $\frac{250 π}{e^{0.035}}$ and $\cos (5.49778714)$ .

$3.41393068 + 5.49778714 + \frac{250 π \cos (5.49778714)}{e^{0.035}}$

Step 1.17

Simplify the numerator.

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Step 1.17.1

Evaluate $\cos (5.49778714)$ .

$3.41393068 + 5.49778714 + \frac{250 π \frac{\sqrt{2}}{2}}{e^{0.035}}$

Step 1.17.2

Combine exponents.

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Step 1.17.2.1

Combine $\frac{\sqrt{2}}{2}$ and $250$ .

$3.41393068 + 5.49778714 + \frac{\frac{\sqrt{2} \cdot 250}{2} π}{e^{0.035}}$

Step 1.17.2.2

Combine $\frac{\sqrt{2} \cdot 250}{2}$ and $π$ .

$3.41393068 + 5.49778714 + \frac{\frac{\sqrt{2} \cdot 250 π}{2}}{e^{0.035}}$

Step 1.17.3

Cancel the common factor of $250$ and $2$ .

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Step 1.17.3.1

Factor $2$ out of $\sqrt{2} \cdot 250 π$ .

$3.41393068 + 5.49778714 + \frac{\frac{2 (\sqrt{2} \cdot 125 π)}{2}}{e^{0.035}}$

Step 1.17.3.2

Cancel the common factors.

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Step 1.17.3.2.1

Factor $2$ out of $2$ .

$3.41393068 + 5.49778714 + \frac{\frac{2 (\sqrt{2} \cdot 125 π)}{2 (1)}}{e^{0.035}}$

Step 1.17.3.2.2

Cancel the common factor.

$3.41393068 + 5.49778714 + \frac{\frac{2 (\sqrt{2} \cdot 125 π)}{2 \cdot 1}}{e^{0.035}}$

Step 1.17.3.2.3

Rewrite the expression.

$3.41393068 + 5.49778714 + \frac{\frac{\sqrt{2} \cdot 125 π}{1}}{e^{0.035}}$

Step 1.17.3.2.4

Divide $\sqrt{2} \cdot 125 π$ by $1$ .

$3.41393068 + 5.49778714 + \frac{\sqrt{2} \cdot 125 π}{e^{0.035}}$

Step 1.17.4

Move $125$ to the left of $\sqrt{2}$ .

$3.41393068 + 5.49778714 + \frac{125 \sqrt{2} π}{e^{0.035}}$

Step 2

Simplify by adding terms.

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Step 2.1

Add $3.41393068$ and $5.49778714$ .

$8.91171783 + \frac{125 \sqrt{2} π}{e^{0.035}}$

Step 2.2

Add $8.91171783$ and $\frac{125 \sqrt{2} π}{e^{0.035}}$ .

$545.17069644$