Calculus Examples

$f (x) = 10 x - 10 \cos (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $10 x - 10 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [10 x] + \frac{d}{d x} [- 10 \cos (x)]$ .

$\frac{d}{d x} [10 x] + \frac{d}{d x} [- 10 \cos (x)]$

Step 1.2

Evaluate $\frac{d}{d x} [10 x]$ .

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Step 1.2.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$10 \frac{d}{d x} [x] + \frac{d}{d x} [- 10 \cos (x)]$

Step 1.2.2

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

$10 \cdot 1 + \frac{d}{d x} [- 10 \cos (x)]$

Step 1.2.3

Multiply $10$ by $1$ .

$10 + \frac{d}{d x} [- 10 \cos (x)]$

Step 1.3

Evaluate $\frac{d}{d x} [- 10 \cos (x)]$ .

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Step 1.3.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 \cos (x)$ with respect to $x$ is $- 10 \frac{d}{d x} [\cos (x)]$ .

$10 - 10 \frac{d}{d x} [\cos (x)]$

Step 1.3.2

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

$10 - 10 (- \sin (x))$

Step 1.3.3

Multiply $- 1$ by $- 10$ .

$10 + 10 \sin (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $10 + 10 \sin (x)$ with respect to $x$ is $\frac{d}{d x} [10] + \frac{d}{d x} [10 \sin (x)]$ .

$f'' (x) = \frac{d}{d x} (10) + \frac{d}{d x} (10 \sin (x))$

Step 2.1.2

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$f'' (x) = 0 + \frac{d}{d x} (10 \sin (x))$

Step 2.2

Evaluate $\frac{d}{d x} [10 \sin (x)]$ .

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Step 2.2.1

Since $10$ is constant with respect to $x$ , the derivative of $10 \sin (x)$ with respect to $x$ is $10 \frac{d}{d x} [\sin (x)]$ .

$f'' (x) = 0 + 10 \frac{d}{d x} (\sin (x))$

Step 2.2.2

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

$f'' (x) = 0 + 10 \cos (x)$

Step 2.3

Add $0$ and $10 \cos (x)$ .

$f'' (x) = 10 \cos (x)$

Step 3

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

$10 + 10 \sin (x) = 0$

Step 4

Subtract $10$ from both sides of the equation.

$10 \sin (x) = - 10$

Step 5

Divide each term in $10 \sin (x) = - 10$ by $10$ and simplify.

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Step 5.1

Divide each term in $10 \sin (x) = - 10$ by $10$ .

$\frac{10 \sin (x)}{10} = \frac{- 10}{10}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $10$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{10 \sin (x)}{10} = \frac{- 10}{10}$

Step 5.2.1.2

Divide $\sin (x)$ by $1$ .

$\sin (x) = \frac{- 10}{10}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $- 10$ by $10$ .

$\sin (x) = - 1$

Step 6

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

$x = \arcsin (- 1)$

Step 7

Simplify the right side.

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Step 7.1

The exact value of $\arcsin (- 1)$ is $- \frac{π}{2}$ .

$x = - \frac{π}{2}$

Step 8

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{2} + π$

Step 9

Simplify the expression to find the second solution.

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Step 9.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 9.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$x = \frac{3 π}{2}$

Step 10

The solution to the equation $x = - \frac{π}{2}$ .

$x = - \frac{π}{2}, \frac{3 π}{2}$

Step 11

Evaluate the second derivative at $x = - \frac{π}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$10 \cos (- \frac{π}{2})$

Step 12

Evaluate the second derivative.

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Step 12.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$10 \cos (\frac{3 π}{2})$

Step 12.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$10 \cos (\frac{π}{2})$

Step 12.3

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

$10 \cdot 0$

Step 12.4

Multiply $10$ by $0$ .

$0$

Step 13

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 13.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, - \frac{π}{2}) \cup (- \frac{π}{2}, \frac{3 π}{2}) \cup (\frac{3 π}{2}, \infty)$

Step 13.2

Substitute any number, such as $- 4$ , from the interval $(- \infty, - \frac{π}{2})$ in the first derivative $10 + 10 \sin (x)$ to check if the result is negative or positive.

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Step 13.2.1

Replace the variable $x$ with $- 4$ in the expression.

$f' (- 4) = 10 + 10 \sin (- 4)$

Step 13.2.2

Simplify the result.

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Step 13.2.2.1

Simplify each term.

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Step 13.2.2.1.1

Evaluate $\sin (- 4)$ .

$f' (- 4) = 10 + 10 \cdot 0.75680249$

Step 13.2.2.1.2

Multiply $10$ by $0.75680249$ .

$f' (- 4) = 10 + 7.56802495$

Step 13.2.2.2

Add $10$ and $7.56802495$ .

$f' (- 4) = 17.56802495$

Step 13.2.2.3

The final answer is $17.56802495$ .

$17.56802495$

Step 13.3

Substitute any number, such as $0$ , from the interval $(- \frac{π}{2}, \frac{3 π}{2})$ in the first derivative $10 + 10 \sin (x)$ to check if the result is negative or positive.

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Step 13.3.1

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

$f' (0) = 10 + 10 \sin (0)$

Step 13.3.2

Simplify the result.

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Step 13.3.2.1

Simplify each term.

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Step 13.3.2.1.1

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

$f' (0) = 10 + 10 \cdot 0$

Step 13.3.2.1.2

Multiply $10$ by $0$ .

$f' (0) = 10 + 0$

Step 13.3.2.2

Add $10$ and $0$ .

$f' (0) = 10$

Step 13.3.2.3

The final answer is $10$ .

$10$

Step 13.4

Substitute any number, such as $7$ , from the interval $(\frac{3 π}{2}, \infty)$ in the first derivative $10 + 10 \sin (x)$ to check if the result is negative or positive.

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Step 13.4.1

Replace the variable $x$ with $7$ in the expression.

$f' (7) = 10 + 10 \sin (7)$

Step 13.4.2

Simplify the result.

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Step 13.4.2.1

Simplify each term.

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Step 13.4.2.1.1

Evaluate $\sin (7)$ .

$f' (7) = 10 + 10 \cdot 0.65698659$

Step 13.4.2.1.2

Multiply $10$ by $0.65698659$ .

$f' (7) = 10 + 6.56986598$

Step 13.4.2.2

Add $10$ and $6.56986598$ .

$f' (7) = 16.56986598$

Step 13.4.2.3

The final answer is $16.56986598$ .

$16.56986598$

Step 13.5

Since the first derivative did not change signs around $x = - \frac{π}{2}$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 13.6

No local maxima or minima found for $f (x) = 10 x - 10 \cos (x)$ .

No local maxima or minima

Step 14