Calculus Examples

$\sin (\frac{x}{2})$

Step 1

Write $\sin (\frac{x}{2})$ as a function.

$f (x) = \sin (\frac{x}{2})$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = \frac{x}{2}$ .

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Step 2.1.1.1.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [\frac{x}{2}]$

Step 2.1.1.1.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$\cos (u) \frac{d}{d x} [\frac{x}{2}]$

Step 2.1.1.1.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

$\cos (\frac{x}{2}) \frac{d}{d x} [\frac{x}{2}]$

Step 2.1.1.2

Differentiate.

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Step 2.1.1.2.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$\cos (\frac{x}{2}) (\frac{1}{2} \frac{d}{d x} [x])$

Step 2.1.1.2.2

Combine $\frac{1}{2}$ and $\cos (\frac{x}{2})$ .

$\frac{\cos (\frac{x}{2})}{2} \frac{d}{d x} [x]$

Step 2.1.1.2.3

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

$\frac{\cos (\frac{x}{2})}{2} \cdot 1$

Step 2.1.1.2.4

Multiply $\frac{\cos (\frac{x}{2})}{2}$ by $1$ .

$f' (x) = \frac{\cos (\frac{x}{2})}{2}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{\cos (\frac{x}{2})}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\cos (\frac{x}{2})]$ .

$\frac{1}{2} \frac{d}{d x} [\cos (\frac{x}{2})]$

Step 2.1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = \frac{x}{2}$ .

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Step 2.1.2.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

$\frac{1}{2} (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{x}{2}])$

Step 2.1.2.2.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$\frac{1}{2} (- \sin (u) \frac{d}{d x} [\frac{x}{2}])$

Step 2.1.2.2.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

$\frac{1}{2} (- \sin (\frac{x}{2}) \frac{d}{d x} [\frac{x}{2}])$

Step 2.1.2.3

Differentiate.

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Step 2.1.2.3.1

Combine $\frac{1}{2}$ and $\sin (\frac{x}{2})$ .

$- \frac{\sin (\frac{x}{2})}{2} \frac{d}{d x} [\frac{x}{2}]$

Step 2.1.2.3.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$- \frac{\sin (\frac{x}{2})}{2} (\frac{1}{2} \frac{d}{d x} [x])$

Step 2.1.2.3.3

Combine fractions.

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Step 2.1.2.3.3.1

Multiply $\frac{1}{2}$ by $\frac{\sin (\frac{x}{2})}{2}$ .

$- \frac{\sin (\frac{x}{2})}{2 \cdot 2} \frac{d}{d x} [x]$

Step 2.1.2.3.3.2

Multiply $2$ by $2$ .

$- \frac{\sin (\frac{x}{2})}{4} \frac{d}{d x} [x]$

Step 2.1.2.3.4

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

$- \frac{\sin (\frac{x}{2})}{4} \cdot 1$

Step 2.1.2.3.5

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{\sin (\frac{x}{2})}{4}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{\sin (\frac{x}{2})}{4}$ .

$- \frac{\sin (\frac{x}{2})}{4}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $- \frac{\sin (\frac{x}{2})}{4} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

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

Step 2.2.2

Set the numerator equal to zero.

$\sin (\frac{x}{2}) = 0$

Step 2.2.3

Solve the equation for $x$ .

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Step 2.2.3.1

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

$\frac{x}{2} = \arcsin (0)$

Step 2.2.3.2

Simplify the right side.

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Step 2.2.3.2.1

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

$\frac{x}{2} = 0$

Step 2.2.3.3

Set the numerator equal to zero.

$x = 0$

Step 2.2.3.4

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.

$\frac{x}{2} = π - 0$

Step 2.2.3.5

Solve for $x$ .

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Step 2.2.3.5.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π - 0)$

Step 2.2.3.5.2

Simplify both sides of the equation.

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Step 2.2.3.5.2.1

Simplify the left side.

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Step 2.2.3.5.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.3.5.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π - 0)$

Step 2.2.3.5.2.1.1.2

Rewrite the expression.

$x = 2 (π - 0)$

Step 2.2.3.5.2.2

Simplify the right side.

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Step 2.2.3.5.2.2.1

Subtract $0$ from $π$ .

$x = 2 π$

Step 2.2.3.6

Find the period of $\sin (\frac{x}{2})$ .

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Step 2.2.3.6.1

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

$\frac{2 π}{| b |}$

Step 2.2.3.6.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 2.2.3.6.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 2.2.3.6.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 2.2.3.6.5

Multiply $2$ by $2$ .

$4 π$

Step 2.2.3.7

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

$x = 4 π n, 2 π + 4 π n$ , for any integer $n$

Step 2.2.4

Consolidate the answers.

$x = 2 π n$ , for any integer $n$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, \infty)$

Step 5

Substitute any number from the interval $(- \infty, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f'' (0) = - \frac{\sin (\frac{0}{2})}{4}$

Step 5.2

Simplify the result.

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Step 5.2.1

Cancel the common factor of $0$ and $2$ .

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Step 5.2.1.1

Factor $2$ out of $0$ .

$f'' (0) = - \frac{\sin (\frac{2 (0)}{2})}{4}$

Step 5.2.1.2

Cancel the common factors.

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Step 5.2.1.2.1

Factor $2$ out of $2$ .

$f'' (0) = - \frac{\sin (\frac{2 \cdot 0}{2 \cdot 1})}{4}$

Step 5.2.1.2.2

Cancel the common factor.

$f'' (0) = - \frac{\sin (\frac{2 \cdot 0}{2 \cdot 1})}{4}$

Step 5.2.1.2.3

Rewrite the expression.

$f'' (0) = - \frac{\sin (\frac{0}{1})}{4}$

Step 5.2.1.2.4

Divide $0$ by $1$ .

$f'' (0) = - \frac{\sin (0)}{4}$

Step 5.2.2

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

$f'' (0) = - \frac{0}{4}$

Step 5.2.3

Divide $0$ by $4$ .

$f'' (0) = - 0$

Step 5.2.4

Multiply $- 1$ by $0$ .

$f'' (0) = 0$

Step 5.2.5

The final answer is $0$ .

$0$

Step 5.3

The graph is concave up on the interval $(- \infty, \infty)$ because $f'' (0)$ is positive.

Concave up on $(- \infty, \infty)$ since $f'' (x)$ is positive

Step 6