Calculus Examples

$2.32 x - \frac{x^{2}}{24000} - 3500$

Step 1

Write $2.32 x - \frac{x^{2}}{24000} - 3500$ as a function.

$f (x) = 2.32 x - \frac{x^{2}}{24000} - 3500$

Step 2

Find the first derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $2.32 x - \frac{x^{2}}{24000} - 3500$ with respect to $x$ is $\frac{d}{d x} [2.32 x] + \frac{d}{d x} [- \frac{x^{2}}{24000}] + \frac{d}{d x} [- 3500]$ .

$\frac{d}{d x} [2.32 x] + \frac{d}{d x} [- \frac{x^{2}}{24000}] + \frac{d}{d x} [- 3500]$

Step 2.1.2

Evaluate $\frac{d}{d x} [2.32 x]$ .

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Step 2.1.2.1

Since $2.32$ is constant with respect to $x$ , the derivative of $2.32 x$ with respect to $x$ is $2.32 \frac{d}{d x} [x]$ .

$2.32 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{x^{2}}{24000}] + \frac{d}{d x} [- 3500]$

Step 2.1.2.2

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

$2.32 \cdot 1 + \frac{d}{d x} [- \frac{x^{2}}{24000}] + \frac{d}{d x} [- 3500]$

Step 2.1.2.3

Multiply $2.32$ by $1$ .

$2.32 + \frac{d}{d x} [- \frac{x^{2}}{24000}] + \frac{d}{d x} [- 3500]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- \frac{x^{2}}{24000}]$ .

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Step 2.1.3.1

Since $- \frac{1}{24000}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{24000}$ with respect to $x$ is $- \frac{1}{24000} \frac{d}{d x} [x^{2}]$ .

$2.32 - \frac{1}{24000} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3500]$

Step 2.1.3.2

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

$2.32 - \frac{1}{24000} (2 x) + \frac{d}{d x} [- 3500]$

Step 2.1.3.3

Multiply $2$ by $- 1$ .

$2.32 - 2 (\frac{1}{24000}) x + \frac{d}{d x} [- 3500]$

Step 2.1.3.4

Combine $- 2$ and $\frac{1}{24000}$ .

$2.32 + \frac{- 2}{24000} x + \frac{d}{d x} [- 3500]$

Step 2.1.3.5

Combine $\frac{- 2}{24000}$ and $x$ .

$2.32 + \frac{- 2 x}{24000} + \frac{d}{d x} [- 3500]$

Step 2.1.3.6

Cancel the common factor of $- 2$ and $24000$ .

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Step 2.1.3.6.1

Factor $2$ out of $- 2 x$ .

$2.32 + \frac{2 (- x)}{24000} + \frac{d}{d x} [- 3500]$

Step 2.1.3.6.2

Cancel the common factors.

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Step 2.1.3.6.2.1

Factor $2$ out of $24000$ .

$2.32 + \frac{2 (- x)}{2 (12000)} + \frac{d}{d x} [- 3500]$

Step 2.1.3.6.2.2

Cancel the common factor.

$2.32 + \frac{2 (- x)}{2 \cdot 12000} + \frac{d}{d x} [- 3500]$

Step 2.1.3.6.2.3

Rewrite the expression.

$2.32 + \frac{- x}{12000} + \frac{d}{d x} [- 3500]$

Step 2.1.3.7

Move the negative in front of the fraction.

$2.32 - \frac{x}{12000} + \frac{d}{d x} [- 3500]$

Step 2.1.4

Since $- 3500$ is constant with respect to $x$ , the derivative of $- 3500$ with respect to $x$ is $0$ .

$2.32 - \frac{x}{12000} + 0$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Add $2.32 - \frac{x}{12000}$ and $0$ .

$2.32 - \frac{x}{12000}$

Step 2.1.5.2

Reorder terms.

$f' (x) = - \frac{x}{12000} + 2.32$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{x}{12000} + 2.32$ .

$- \frac{x}{12000} + 2.32$

Step 3

Set the first derivative equal to $0$ then solve the equation $- \frac{x}{12000} + 2.32 = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$- \frac{x}{12000} + 2.32 = 0$

Step 3.2

Subtract $2.32$ from both sides of the equation.

$- \frac{x}{12000} = - 2.32$

Step 3.3

Multiply both sides of the equation by $- 12000$ .

$- 12000 (- \frac{x}{12000}) = - 12000 \cdot - 2.32$

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify $- 12000 (- \frac{x}{12000})$ .

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Step 3.4.1.1.1

Cancel the common factor of $12000$ .

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Step 3.4.1.1.1.1

Move the leading negative in $- \frac{x}{12000}$ into the numerator.

$- 12000 \frac{- x}{12000} = - 12000 \cdot - 2.32$

Step 3.4.1.1.1.2

Factor $12000$ out of $- 12000$ .

$12000 (- 1) \frac{- x}{12000} = - 12000 \cdot - 2.32$

Step 3.4.1.1.1.3

Cancel the common factor.

$12000 \cdot - 1 \frac{- x}{12000} = - 12000 \cdot - 2.32$

Step 3.4.1.1.1.4

Rewrite the expression.

$- - x = - 12000 \cdot - 2.32$

Step 3.4.1.1.2

Multiply.

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Step 3.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x = - 12000 \cdot - 2.32$

Step 3.4.1.1.2.2

Multiply $x$ by $1$ .

$x = - 12000 \cdot - 2.32$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Multiply $- 12000$ by $- 2.32$ .

$x = 27840$

Step 4

The values which make the derivative equal to $0$ are $27840$ .

$27840$

Step 5

After finding the point that makes the derivative $f' (x) = - \frac{x}{12000} + 2.32$ equal to $0$ or undefined, the interval to check where $f (x) = 2.32 x - \frac{x^{2}}{24000} - 3500$ is increasing and where it is decreasing is $(- \infty, 27840) \cup (27840, \infty)$ .

$(- \infty, 27840) \cup (27840, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 27840)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $27839$ in the expression.

$f' (27839) = - \frac{27839}{12000} + 2.32$

Step 6.2

Simplify the result.

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Step 6.2.1

Cancel the common factor of $27839$ and $12000$ .

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Step 6.2.1.1

Rewrite $27839$ as $1 (27839)$ .

$f' (27839) = - \frac{1 (27839)}{12000} + 2.32$

Step 6.2.1.2

Cancel the common factors.

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Step 6.2.1.2.1

Rewrite $12000$ as $1 (12000)$ .

$f' (27839) = - \frac{1 \cdot 27839}{1 \cdot 12000} + 2.32$

Step 6.2.1.2.2

Cancel the common factor.

$f' (27839) = - \frac{1 \cdot 27839}{1 \cdot 12000} + 2.32$

Step 6.2.1.2.3

Rewrite the expression.

$f' (27839) = - \frac{27839}{12000} + 2.32$

Step 6.2.2

To write $2.32$ as a fraction with a common denominator, multiply by $\frac{12000}{12000}$ .

$f' (27839) = - \frac{27839}{12000} + 2.32 \cdot \frac{12000}{12000}$

Step 6.2.3

Combine $2.32$ and $\frac{12000}{12000}$ .

$f' (27839) = - \frac{27839}{12000} + \frac{2.32 \cdot 12000}{12000}$

Step 6.2.4

Combine the numerators over the common denominator.

$f' (27839) = \frac{- 27839 + 2.32 \cdot 12000}{12000}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $2.32$ by $12000$ .

$f' (27839) = \frac{- 27839 + 27840}{12000}$

Step 6.2.5.2

Add $- 27839$ and $27840$ .

$f' (27839) = \frac{1}{12000}$

Step 6.2.6

Cancel the common factor of $1$ and $12000$ .

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Step 6.2.6.1

Rewrite $1$ as $1 (1)$ .

$f' (27839) = \frac{1 (1)}{12000}$

Step 6.2.6.2

Cancel the common factors.

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Step 6.2.6.2.1

Rewrite $12000$ as $1 (12000)$ .

$f' (27839) = \frac{1 \cdot 1}{1 \cdot 12000}$

Step 6.2.6.2.2

Cancel the common factor.

$f' (27839) = \frac{1 \cdot 1}{1 \cdot 12000}$

Step 6.2.6.2.3

Rewrite the expression.

$f' (27839) = \frac{1}{12000}$

Step 6.2.7

The final answer is $\frac{1}{12000}$ .

$\frac{1}{12000}$

Step 6.3

At $x = 27839$ the derivative is $\frac{1}{12000}$ . Since this is positive, the function is increasing on $(- \infty, 27840)$ .

Increasing on $(- \infty, 27840)$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(27840, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $27841$ in the expression.

$f' (27841) = - \frac{27841}{12000} + 2.32$

Step 7.2

Simplify the result.

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Step 7.2.1

Cancel the common factor of $27841$ and $12000$ .

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Step 7.2.1.1

Rewrite $27841$ as $1 (27841)$ .

$f' (27841) = - \frac{1 (27841)}{12000} + 2.32$

Step 7.2.1.2

Cancel the common factors.

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Step 7.2.1.2.1

Rewrite $12000$ as $1 (12000)$ .

$f' (27841) = - \frac{1 \cdot 27841}{1 \cdot 12000} + 2.32$

Step 7.2.1.2.2

Cancel the common factor.

$f' (27841) = - \frac{1 \cdot 27841}{1 \cdot 12000} + 2.32$

Step 7.2.1.2.3

Rewrite the expression.

$f' (27841) = - \frac{27841}{12000} + 2.32$

Step 7.2.2

To write $2.32$ as a fraction with a common denominator, multiply by $\frac{12000}{12000}$ .

$f' (27841) = - \frac{27841}{12000} + 2.32 \cdot \frac{12000}{12000}$

Step 7.2.3

Combine $2.32$ and $\frac{12000}{12000}$ .

$f' (27841) = - \frac{27841}{12000} + \frac{2.32 \cdot 12000}{12000}$

Step 7.2.4

Combine the numerators over the common denominator.

$f' (27841) = \frac{- 27841 + 2.32 \cdot 12000}{12000}$

Step 7.2.5

Simplify the numerator.

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Step 7.2.5.1

Multiply $2.32$ by $12000$ .

$f' (27841) = \frac{- 27841 + 27840}{12000}$

Step 7.2.5.2

Add $- 27841$ and $27840$ .

$f' (27841) = \frac{- 1}{12000}$

Step 7.2.6

Cancel the common factor of $- 1$ and $12000$ .

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Step 7.2.6.1

Rewrite $- 1$ as $1 (- 1)$ .

$f' (27841) = \frac{1 (- 1)}{12000}$

Step 7.2.6.2

Cancel the common factors.

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Step 7.2.6.2.1

Rewrite $12000$ as $1 (12000)$ .

$f' (27841) = \frac{1 \cdot - 1}{1 \cdot 12000}$

Step 7.2.6.2.2

Cancel the common factor.

$f' (27841) = \frac{1 \cdot - 1}{1 \cdot 12000}$

Step 7.2.6.2.3

Rewrite the expression.

$f' (27841) = \frac{- 1}{12000}$

Step 7.2.7

Move the negative in front of the fraction.

$f' (27841) = - \frac{1}{12000}$

Step 7.2.8

The final answer is $- \frac{1}{12000}$ .

$- \frac{1}{12000}$

Step 7.3

At $x = 27841$ the derivative is $- \frac{1}{12000}$ . Since this is negative, the function is decreasing on $(27840, \infty)$ .

Decreasing on $(27840, \infty)$ since $f' (x) < 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, 27840)$

Decreasing on: $(27840, \infty)$

Step 9