Calculus Examples

$\sin^{2} (x)$

Step 1

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

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

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 $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

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Step 2.1.1.1.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

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

Step 2.1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

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

Step 2.1.1.1.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 2.1.1.2

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

$2 \sin (x) \cos (x)$

Step 2.1.1.3

Simplify.

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Step 2.1.1.3.1

Reorder the factors of $2 \sin (x) \cos (x)$ .

$2 \cos (x) \sin (x)$

Step 2.1.1.3.2

Reorder $2 \cos (x)$ and $\sin (x)$ .

$\sin (x) (2 \cos (x))$

Step 2.1.1.3.3

Reorder $\sin (x)$ and $2$ .

$2 \cdot \sin (x) \cos (x)$

Step 2.1.1.3.4

Apply the sine double-angle identity.

$f' (x) = \sin (2 x)$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

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 $g (x) = 2 x$ .

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Step 2.1.2.1.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 2.1.2.1.2

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

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

Step 2.1.2.1.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 2.1.2.2

Differentiate.

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Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

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

$\cos (2 x) (2 \cdot 1)$

Step 2.1.2.2.3

Simplify the expression.

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Step 2.1.2.2.3.1

Multiply $2$ by $1$ .

$\cos (2 x) \cdot 2$

Step 2.1.2.2.3.2

Move $2$ to the left of $\cos (2 x)$ .

$f'' (x) = 2 \cos (2 x)$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $2 \cos (2 x)$ .

$2 \cos (2 x)$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $2 \cos (2 x) = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$2 \cos (2 x) = 0$

Step 2.2.2

Divide each term in $2 \cos (2 x) = 0$ by $2$ and simplify.

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Step 2.2.2.1

Divide each term in $2 \cos (2 x) = 0$ by $2$ .

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

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.2.1.2

Divide $\cos (2 x)$ by $1$ .

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

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $0$ by $2$ .

$\cos (2 x) = 0$

Step 2.2.3

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

$2 x = \arccos (0)$

Step 2.2.4

Simplify the right side.

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Step 2.2.4.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$2 x = \frac{π}{2}$

Step 2.2.5

Divide each term in $2 x = \frac{π}{2}$ by $2$ and simplify.

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Step 2.2.5.1

Divide each term in $2 x = \frac{π}{2}$ by $2$ .

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

Step 2.2.5.2

Simplify the left side.

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Step 2.2.5.2.1

Cancel the common factor of $2$ .

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Step 2.2.5.2.1.1

Cancel the common factor.

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

Step 2.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 2.2.5.3

Simplify the right side.

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Step 2.2.5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 2.2.5.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 2.2.5.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

$x = \frac{π}{2 \cdot 2}$

Step 2.2.5.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 2.2.6

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$2 x = 2 π - \frac{π}{2}$

Step 2.2.7

Solve for $x$ .

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Step 2.2.7.1

Simplify.

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Step 2.2.7.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 2.2.7.1.2

Combine $2 π$ and $\frac{2}{2}$ .

$2 x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 2.2.7.1.3

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 2 - π}{2}$

Step 2.2.7.1.4

Multiply $2$ by $2$ .

$2 x = \frac{4 π - π}{2}$

Step 2.2.7.1.5

Subtract $π$ from $4 π$ .

$2 x = \frac{3 π}{2}$

Step 2.2.7.2

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ and simplify.

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Step 2.2.7.2.1

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2}$

Step 2.2.7.2.2

Simplify the left side.

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Step 2.2.7.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.7.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2}$

Step 2.2.7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{2}$

Step 2.2.7.2.3

Simplify the right side.

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Step 2.2.7.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 2.2.7.2.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{2}$ .

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Step 2.2.7.2.3.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{2}$ .

$x = \frac{3 π}{2 \cdot 2}$

Step 2.2.7.2.3.2.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 2.2.8

Find the period of $\cos (2 x)$ .

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Step 2.2.8.1

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

$\frac{2 π}{| b |}$

Step 2.2.8.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 2.2.8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 2.2.8.4

Cancel the common factor of $2$ .

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Step 2.2.8.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 2.2.8.4.2

Divide $π$ by $1$ .

$π$

Step 2.2.9

The period of the $\cos (2 x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 2.2.10

Consolidate the answers.

$x = \frac{π}{4} + \frac{π n}{2}$ , 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.

$f'' (0) = 2 \cos (2 (0))$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $2$ by $0$ .

$f'' (0) = 2 \cos (0)$

Step 5.2.2

The exact value of $\cos (0)$ is $1$ .

$f'' (0) = 2 \cdot 1$

Step 5.2.3

Multiply $2$ by $1$ .

$f'' (0) = 2$

Step 5.2.4

The final answer is $2$ .

$2$

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