Calculus Examples

$y = x^{2} \sin (4 x)$

Step 1

Set $x^{2} \sin (4 x)$ equal to $0$ .

$x^{2} \sin (4 x) = 0$

Step 2

Solve for $x$ .

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Step 2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$\sin (4 x) = 0$

Step 2.2

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 2.2.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.2.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.2.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 2.2.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.2.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.2.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.3

Set $\sin (4 x)$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $\sin (4 x)$ equal to $0$ .

$\sin (4 x) = 0$

Step 2.3.2

Solve $\sin (4 x) = 0$ for $x$ .

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Step 2.3.2.1

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$4 x = \arcsin (0)$

Step 2.3.2.2

Simplify the right side.

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Step 2.3.2.2.1

The exact value of $\arcsin (0)$ is $0$ .

$4 x = 0$

Step 2.3.2.3

Divide each term in $4 x = 0$ by $4$ and simplify.

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Step 2.3.2.3.1

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.3.2.3.2

Simplify the left side.

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Step 2.3.2.3.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.3.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Step 2.3.2.3.3

Simplify the right side.

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Step 2.3.2.3.3.1

Divide $0$ by $4$ .

$x = 0$

Step 2.3.2.4

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$4 x = π - 0$

Step 2.3.2.5

Solve for $x$ .

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Step 2.3.2.5.1

Simplify.

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Step 2.3.2.5.1.1

Multiply $- 1$ by $0$ .

$4 x = π + 0$

Step 2.3.2.5.1.2

Add $π$ and $0$ .

$4 x = π$

Step 2.3.2.5.2

Divide each term in $4 x = π$ by $4$ and simplify.

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Step 2.3.2.5.2.1

Divide each term in $4 x = π$ by $4$ .

$\frac{4 x}{4} = \frac{π}{4}$

Step 2.3.2.5.2.2

Simplify the left side.

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Step 2.3.2.5.2.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.5.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{π}{4}$

Step 2.3.2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{4}$

Step 2.3.2.6

Find the period of $\sin (4 x)$ .

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Step 2.3.2.6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 2.3.2.6.2

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 2.3.2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Step 2.3.2.6.4

Cancel the common factor of $2$ and $4$ .

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Step 2.3.2.6.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 2.3.2.6.4.2

Cancel the common factors.

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Step 2.3.2.6.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 2.3.2.6.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 2.3.2.6.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 2.3.2.7

The period of the $\sin (4 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{π n}{2}, \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$

Step 2.4

The final solution is all the values that make $x^{2} \sin (4 x) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = 0$ (Multiplicity of $2$ )

$x = \frac{π n}{2}$ (Multiplicity of $1$ )

$x = \frac{π}{4} + \frac{π n}{2}$ (Multiplicity of $1$ )

Step 2.5

Consolidate the answers.

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Step 2.5.1

Consolidate $0$ and $\frac{π n}{2}$ to $\frac{π n}{2}$ .

$x = \frac{π n}{2}, \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$

Step 2.5.2

Consolidate the answers.

$x = \frac{π n}{4}$ (Multiplicity of $1$ )

Step 3