Calculus Examples

$y = 180 x - 0.3 x^{3}$

Step 1

Write $y = 180 x - 0.3 x^{3}$ as a function.

$f (x) = 180 x - 0.3 x^{3}$

Step 2

Find the first derivative of the function.

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By the Sum Rule, the derivative of $180 x - 0.3 x^{3}$ with respect to $x$ is $\frac{d}{d x} [180 x] + \frac{d}{d x} [- 0.3 x^{3}]$ .

$\frac{d}{d x} [180 x] + \frac{d}{d x} [- 0.3 x^{3}]$

Evaluate $\frac{d}{d x} [180 x]$ .

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Since $180$ is constant with respect to $x$ , the derivative of $180 x$ with respect to $x$ is $180 \frac{d}{d x} [x]$ .

$180 \frac{d}{d x} [x] + \frac{d}{d x} [- 0.3 x^{3}]$

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

$180 \cdot 1 + \frac{d}{d x} [- 0.3 x^{3}]$

Multiply $180$ by $1$ .

$180 + \frac{d}{d x} [- 0.3 x^{3}]$

Evaluate $\frac{d}{d x} [- 0.3 x^{3}]$ .

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Since $- 0.3$ is constant with respect to $x$ , the derivative of $- 0.3 x^{3}$ with respect to $x$ is $- 0.3 \frac{d}{d x} [x^{3}]$ .

$180 - 0.3 \frac{d}{d x} [x^{3}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$180 - 0.3 (3 x^{2})$

Multiply $3$ by $- 0.3$ .

$180 - 0.9 x^{2}$

Reorder terms.

$- 0.9 x^{2} + 180$

Step 3

Find the second derivative of the function.

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By the Sum Rule, the derivative of $- 0.9 x^{2} + 180$ with respect to $x$ is $\frac{d}{d x} [- 0.9 x^{2}] + \frac{d}{d x} [180]$ .

$f'' (x) = \frac{d}{d x} (- 0.9 x^{2}) + \frac{d}{d x} (180)$

Evaluate $\frac{d}{d x} [- 0.9 x^{2}]$ .

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Since $- 0.9$ is constant with respect to $x$ , the derivative of $- 0.9 x^{2}$ with respect to $x$ is $- 0.9 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = - 0.9 \frac{d}{d x} x^{2} + \frac{d}{d x} (180)$

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

$f'' (x) = - 0.9 (2 x) + \frac{d}{d x} (180)$

Multiply $2$ by $- 0.9$ .

$f'' (x) = - 1.8 x + \frac{d}{d x} (180)$

Differentiate using the Constant Rule.

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Since $180$ is constant with respect to $x$ , the derivative of $180$ with respect to $x$ is $0$ .

$f'' (x) = - 1.8 x + 0$

Add $- 1.8 x$ and $0$ .

$f'' (x) = - 1.8 x$

Step 4

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

$- 0.9 x^{2} + 180 = 0$

Step 5

Find the first derivative.

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Find the first derivative.

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By the Sum Rule, the derivative of $180 x - 0.3 x^{3}$ with respect to $x$ is $\frac{d}{d x} [180 x] + \frac{d}{d x} [- 0.3 x^{3}]$ .

$f' (x) = \frac{d}{d x} (180 x) + \frac{d}{d x} (- 0.3 x^{3})$

Evaluate $\frac{d}{d x} [180 x]$ .

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Since $180$ is constant with respect to $x$ , the derivative of $180 x$ with respect to $x$ is $180 \frac{d}{d x} [x]$ .

$f' (x) = 180 \frac{d}{d x} (x) + \frac{d}{d x} (- 0.3 x^{3})$

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

$f' (x) = 180 \cdot 1 + \frac{d}{d x} (- 0.3 x^{3})$

Multiply $180$ by $1$ .

$f' (x) = 180 + \frac{d}{d x} (- 0.3 x^{3})$

Evaluate $\frac{d}{d x} [- 0.3 x^{3}]$ .

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Since $- 0.3$ is constant with respect to $x$ , the derivative of $- 0.3 x^{3}$ with respect to $x$ is $- 0.3 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 180 - 0.3 \frac{d}{d x} x^{3}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = 180 - 0.3 (3 x^{2})$

Multiply $3$ by $- 0.3$ .

$f' (x) = 180 - 0.9 x^{2}$

Reorder terms.

$f' (x) = - 0.9 x^{2} + 180$

The first derivative of $f (x)$ with respect to $x$ is $- 0.9 x^{2} + 180$ .

$- 0.9 x^{2} + 180$

Step 6

Set the first derivative equal to $0$ then solve the equation $- 0.9 x^{2} + 180 = 0$ .

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Set the first derivative equal to $0$ .

$- 0.9 x^{2} + 180 = 0$

Subtract $180$ from both sides of the equation.

$- 0.9 x^{2} = - 180$

Divide each term in $- 0.9 x^{2} = - 180$ by $- 0.9$ and simplify.

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Divide each term in $- 0.9 x^{2} = - 180$ by $- 0.9$ .

$\frac{- 0.9 x^{2}}{- 0.9} = \frac{- 180}{- 0.9}$

Simplify the left side.

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Cancel the common factor of $- 0.9$ .

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Cancel the common factor.

$\frac{- 0.9 x^{2}}{- 0.9} = \frac{- 180}{- 0.9}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 180}{- 0.9}$

Simplify the right side.

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Divide $- 180$ by $- 0.9$ .

$x^{2} = 200$

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

$x = \pm \sqrt{200}$

Simplify $\pm \sqrt{200}$ .

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Rewrite $200$ as $10^{2} \cdot 2$ .

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Factor $100$ out of $200$ .

$x = \pm \sqrt{100 (2)}$

Rewrite $100$ as $10^{2}$ .

$x = \pm \sqrt{10^{2} \cdot 2}$

Pull terms out from under the radical.

$x = \pm 10 \sqrt{2}$

The complete solution is the result of both the positive and negative portions of the solution.

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First, use the positive value of the $\pm$ to find the first solution.

$x = 10 \sqrt{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 10 \sqrt{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 10 \sqrt{2}, - 10 \sqrt{2}$

Step 7

Find the values where the derivative is undefined.

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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.

Step 8

Critical points to evaluate.

$x = 10 \sqrt{2}, - 10 \sqrt{2}$

Step 9

Evaluate the second derivative at $x = 10 \sqrt{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 1.8 (10 \sqrt{2})$

Step 10

Multiply $- 1.8 (10 \sqrt{2})$ .

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Multiply $10$ by $- 1.8$ .

$- 18 \sqrt{2}$

Multiply $- 18$ by $\sqrt{2}$ .

$- 25.45584412$

Step 11

$x = 10 \sqrt{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 10 \sqrt{2}$ is a local maximum

Step 12

Find the y-value when $x = 10 \sqrt{2}$ .

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Replace the variable $x$ with $10 \sqrt{2}$ in the expression.

$f (10 \sqrt{2}) = 180 (10 \sqrt{2}) - 0.3 {(10 \sqrt{2})}^{3}$

Simplify the result.

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Simplify each term.

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Multiply $10$ by $180$ .

$f (10 \sqrt{2}) = 1800 \sqrt{2} - 0.3 {(10 \sqrt{2})}^{3}$

Apply the product rule to $10 \sqrt{2}$ .

$f (10 \sqrt{2}) = 1800 \sqrt{2} - 0.3 (10^{3} {\sqrt{2}}^{3})$

Raise $10$ to the power of $3$ .

$f (10 \sqrt{2}) = 1800 \sqrt{2} - 0.3 (1000 {\sqrt{2}}^{3})$

Rewrite ${\sqrt{2}}^{3}$ as ${(2^{3})}^{\frac{1}{2}}$ .

$f (10 \sqrt{2}) = 1800 \sqrt{2} - 0.3 (1000 \sqrt{2^{3}})$

Raise $2$ to the power of $3$ .

$f (10 \sqrt{2}) = 1800 \sqrt{2} - 0.3 (1000 \sqrt{8})$

Rewrite $8$ as $2^{2} \cdot 2$ .

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Factor $4$ out of $8$ .

$f (10 \sqrt{2}) = 1800 \sqrt{2} - 0.3 (1000 \sqrt{4 (2)})$

Rewrite $4$ as $2^{2}$ .

$f (10 \sqrt{2}) = 1800 \sqrt{2} - 0.3 (1000 \sqrt{2^{2} \cdot 2})$

Pull terms out from under the radical.

$f (10 \sqrt{2}) = 1800 \sqrt{2} - 0.3 (1000 (2 \sqrt{2}))$

Multiply $2$ by $1000$ .

$f (10 \sqrt{2}) = 1800 \sqrt{2} - 0.3 (2000 \sqrt{2})$

Multiply $- 0.3 (2000 \sqrt{2})$ .

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Multiply $2000$ by $- 0.3$ .

$f (10 \sqrt{2}) = 1800 \sqrt{2} - 600 \sqrt{2}$

Multiply $- 600$ by $\sqrt{2}$ .

$f (10 \sqrt{2}) = 1800 \sqrt{2} - 848.52813742$

Subtract $848.52813742$ from $1800 \sqrt{2}$ .

$f (10 \sqrt{2}) = 1697.05627484$

The final answer is $1697.05627484$ .

$y = 1697.05627484$

Step 13

Evaluate the second derivative at $x = - 10 \sqrt{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 1.8 (- 10 \sqrt{2})$

Step 14

Multiply $- 1.8 (- 10 \sqrt{2})$ .

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Multiply $- 10$ by $- 1.8$ .

$18 \sqrt{2}$

Multiply $18$ by $\sqrt{2}$ .

$25.45584412$

Step 15

$x = - 10 \sqrt{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 10 \sqrt{2}$ is a local minimum

Step 16

Find the y-value when $x = - 10 \sqrt{2}$ .

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Replace the variable $x$ with $- 10 \sqrt{2}$ in the expression.

$f (- 10 \sqrt{2}) = 180 (- 10 \sqrt{2}) - 0.3 {(- 10 \sqrt{2})}^{3}$

Simplify the result.

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Simplify each term.

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Multiply $- 10$ by $180$ .

$f (- 10 \sqrt{2}) = - 1800 \sqrt{2} - 0.3 {(- 10 \sqrt{2})}^{3}$

Apply the product rule to $- 10 \sqrt{2}$ .

$f (- 10 \sqrt{2}) = - 1800 \sqrt{2} - 0.3 ({(- 10)}^{3} {\sqrt{2}}^{3})$

Raise $- 10$ to the power of $3$ .

$f (- 10 \sqrt{2}) = - 1800 \sqrt{2} - 0.3 (- 1000 {\sqrt{2}}^{3})$

Rewrite ${\sqrt{2}}^{3}$ as ${(2^{3})}^{\frac{1}{2}}$ .

$f (- 10 \sqrt{2}) = - 1800 \sqrt{2} - 0.3 (- 1000 \sqrt{2^{3}})$

Raise $2$ to the power of $3$ .

$f (- 10 \sqrt{2}) = - 1800 \sqrt{2} - 0.3 (- 1000 \sqrt{8})$

Rewrite $8$ as $2^{2} \cdot 2$ .

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Factor $4$ out of $8$ .

$f (- 10 \sqrt{2}) = - 1800 \sqrt{2} - 0.3 (- 1000 \sqrt{4 (2)})$

Rewrite $4$ as $2^{2}$ .

$f (- 10 \sqrt{2}) = - 1800 \sqrt{2} - 0.3 (- 1000 \sqrt{2^{2} \cdot 2})$

Pull terms out from under the radical.

$f (- 10 \sqrt{2}) = - 1800 \sqrt{2} - 0.3 (- 1000 (2 \sqrt{2}))$

Multiply $2$ by $- 1000$ .

$f (- 10 \sqrt{2}) = - 1800 \sqrt{2} - 0.3 (- 2000 \sqrt{2})$

Multiply $- 0.3 (- 2000 \sqrt{2})$ .

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Multiply $- 2000$ by $- 0.3$ .

$f (- 10 \sqrt{2}) = - 1800 \sqrt{2} + 600 \sqrt{2}$

Multiply $600$ by $\sqrt{2}$ .

$f (- 10 \sqrt{2}) = - 1800 \sqrt{2} + 848.52813742$

Add $- 1800 \sqrt{2}$ and $848.52813742$ .

$f (- 10 \sqrt{2}) = - 1697.05627484$

The final answer is $- 1697.05627484$ .

$y = - 1697.05627484$

Step 17

These are the local extrema for $f (x) = 180 x - 0.3 x^{3}$ .

$(10 \sqrt{2}, 1697.05627484)$ is a local maxima

$(- 10 \sqrt{2}, - 1697.05627484)$ is a local minima

Step 18