Calculus Examples

$f (x) = \cos (π x)$

Step 1

Find the first derivative of the function.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

$- \sin (u) \frac{d}{d x} [π x]$

Step 1.1.3

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

$- \sin (π x) \frac{d}{d x} [π x]$

Step 1.2

Differentiate.

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Step 1.2.1

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

$- \sin (π x) (π \frac{d}{d x} [x])$

Step 1.2.2

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

$- \sin (π x) (π \cdot 1)$

Step 1.2.3

Simplify the expression.

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Step 1.2.3.1

Multiply $π$ by $1$ .

$- \sin (π x) π$

Step 1.2.3.2

Reorder the factors of $- \sin (π x) π$ .

$- π \sin (π x)$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- π$ is constant with respect to $x$ , the derivative of $- π \sin (π x)$ with respect to $x$ is $- π \frac{d}{d x} [\sin (π x)]$ .

$f'' (x) = - π \frac{d}{d x} \sin (π x)$

Step 2.2

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

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Step 2.2.1

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

$f'' (x) = - π (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (π x))$

Step 2.2.2

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

$f'' (x) = - π (\cos (u) \frac{d}{d x} (π x))$

Step 2.2.3

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

$f'' (x) = - π (\cos (π x) \frac{d}{d x} (π x))$

Step 2.3

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

$f'' (x) = - π \cos (π x) (π \frac{d}{d x} (x))$

Step 2.4

Raise $π$ to the power of $1$ .

$f'' (x) = - π π \cos (π x) \frac{d}{d x} x$

Step 2.5

Raise $π$ to the power of $1$ .

$f'' (x) = - π π \cos (π x) \frac{d}{d x} x$

Step 2.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - π^{1 + 1} \cos (π x) \frac{d}{d x} x$

Step 2.7

Add $1$ and $1$ .

$f'' (x) = - π^{2} \cos (π x) \frac{d}{d x} x$

Step 2.8

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

$f'' (x) = - π^{2} \cos (π x) \cdot 1$

Step 2.9

Multiply $- 1$ by $1$ .

$f'' (x) = - π^{2} \cos (π x)$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- π \sin (π x) = 0$

Step 4

Divide each term in $- π \sin (π x) = 0$ by $- π$ and simplify.

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Step 4.1

Divide each term in $- π \sin (π x) = 0$ by $- π$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

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

Step 4.2.2

Cancel the common factor of $π$ .

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Step 4.2.2.1

Cancel the common factor.

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

Step 4.2.2.2

Divide $\sin (π x)$ by $1$ .

$\sin (π x) = \frac{0}{- π}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- π$ .

$\sin (π x) = 0$

Step 5

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

$π x = \arcsin (0)$

Step 6

Simplify the right side.

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Step 6.1

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

$π x = 0$

Step 7

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

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Step 7.1

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

$\frac{π x}{π} = \frac{0}{π}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $π$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{0}{π}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{π}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $0$ by $π$ .

$x = 0$

Step 8

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.

$π x = π - 0$

Step 9

Solve for $x$ .

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Step 9.1

Simplify.

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Step 9.1.1

Multiply $- 1$ by $0$ .

$π x = π + 0$

Step 9.1.2

Add $π$ and $0$ .

$π x = π$

Step 9.2

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

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Step 9.2.1

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

$\frac{π x}{π} = \frac{π}{π}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $π$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{π}{π}$

Step 9.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{π}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Cancel the common factor of $π$ .

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Step 9.2.3.1.1

Cancel the common factor.

$x = \frac{π}{π}$

Step 9.2.3.1.2

Rewrite the expression.

$x = 1$

Step 10

The solution to the equation $π x = 0$ .

$x = 0, 1$

Step 11

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- π^{2} \cos (π (0))$

Step 12

Evaluate the second derivative.

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Step 12.1

Multiply $π$ by $0$ .

$- π^{2} \cos (0)$

Step 12.2

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

$- π^{2} \cdot 1$

Step 12.3

Multiply $- 1$ by $1$ .

$- π^{2}$

Step 13

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 14

Find the y-value when $x = 0$ .

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Step 14.1

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

$f (0) = \cos (π (0))$

Step 14.2

Simplify the result.

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Step 14.2.1

Multiply $π$ by $0$ .

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

Step 14.2.2

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

$f (0) = 1$

Step 14.2.3

The final answer is $1$ .

$y = 1$

Step 15

Evaluate the second derivative at $x = 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- π^{2} \cos (π (1))$

Step 16

Evaluate the second derivative.

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Step 16.1

Multiply $π$ by $1$ .

$- π^{2} \cos (π)$

Step 16.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- π^{2} (- \cos (0))$

Step 16.3

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

$- π^{2} (- 1 \cdot 1)$

Step 16.4

Multiply $- 1$ by $1$ .

$- π^{2} \cdot - 1$

Step 16.5

Multiply $- π^{2} \cdot - 1$ .

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Step 16.5.1

Multiply $- 1$ by $- 1$ .

$1 π^{2}$

Step 16.5.2

Multiply $π^{2}$ by $1$ .

$π^{2}$

Step 17

$x = 1$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 1$ is a local minimum

Step 18

Find the y-value when $x = 1$ .

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Step 18.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \cos (π (1))$

Step 18.2

Simplify the result.

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Step 18.2.1

Multiply $π$ by $1$ .

$f (1) = \cos (π)$

Step 18.2.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$f (1) = - \cos (0)$

Step 18.2.3

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

$f (1) = - 1 \cdot 1$

Step 18.2.4

Multiply $- 1$ by $1$ .

$f (1) = - 1$

Step 18.2.5

The final answer is $- 1$ .

$y = - 1$

Step 19

These are the local extrema for $f (x) = \cos (π x)$ .

$(0, 1)$ is a local maxima

$(1, - 1)$ is a local minima

Step 20