Calculus Examples

$C' (q) = 0.008 {(190)}^{2} - q + 46$

Step 1

The function $C (q)$ can be found by evaluating the indefinite integral of the derivative $C' (q)$ .

$C (q) = \int C' (q) d q$

Step 2

Simplify.

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Step 2.1

Raise $190$ to the power of $2$ .

$\int 0.008 \cdot 36100 - q + 46 d q$

Step 2.2

Multiply $0.008$ by $36100$ .

$\int 288.8 - q + 46 d q$

Step 2.3

Add $288.8$ and $46$ .

$\int - q + 334.8 d q$

Step 3

Split the single integral into multiple integrals.

$\int - q d q + \int 334.8 d q$

Step 4

Since $- 1$ is constant with respect to $q$ , move $- 1$ out of the integral.

$- \int q d q + \int 334.8 d q$

Step 5

By the Power Rule, the integral of $q$ with respect to $q$ is $\frac{1}{2} q^{2}$ .

$- (\frac{1}{2} q^{2} + C) + \int 334.8 d q$

Step 6

Apply the constant rule.

$- (\frac{1}{2} q^{2} + C) + 334.8 q + C$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{2}$ and $q^{2}$ .

$- (\frac{q^{2}}{2} + C) + 334.8 q + C$

Step 7.2

Simplify.

$- \frac{1}{2} q^{2} + 334.8 q + C$

Step 8

The function $C$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$C (q) = - \frac{1}{2} q^{2} + 334.8 q + C$