Calculus Examples

$R (q) = (\frac{5000 - q}{100})$

Step 1

Find the first derivative of the function.

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Step 1.1

Since $\frac{1}{100}$ is constant with respect to $q$ , the derivative of $\frac{5000 - q}{100}$ with respect to $q$ is $\frac{1}{100} \frac{d}{d q} [5000 - q]$ .

$\frac{1}{100} \frac{d}{d q} [5000 - q]$

Step 1.2

By the Sum Rule, the derivative of $5000 - q$ with respect to $q$ is $\frac{d}{d q} [5000] + \frac{d}{d q} [- q]$ .

$\frac{1}{100} (\frac{d}{d q} [5000] + \frac{d}{d q} [- q])$

Step 1.3

Since $5000$ is constant with respect to $q$ , the derivative of $5000$ with respect to $q$ is $0$ .

$\frac{1}{100} (0 + \frac{d}{d q} [- q])$

Step 1.4

Add $0$ and $\frac{d}{d q} [- q]$ .

$\frac{1}{100} \frac{d}{d q} [- q]$

Step 1.5

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

$\frac{1}{100} (- \frac{d}{d q} [q])$

Step 1.6

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

$\frac{1}{100} (- 1 \cdot 1)$

Step 1.7

Combine fractions.

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Step 1.7.1

Multiply $- 1$ by $1$ .

$\frac{1}{100} \cdot - 1$

Step 1.7.2

Combine $\frac{1}{100}$ and $- 1$ .

$\frac{- 1}{100}$

Step 1.7.3

Move the negative in front of the fraction.

$- \frac{1}{100}$

Step 2

Since $- \frac{1}{100}$ is constant with respect to $q$ , the derivative of $- \frac{1}{100}$ with respect to $q$ is $0$ .

$R'' (q) = 0$

Step 3

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

$- \frac{1}{100} = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema