Calculus Examples

$f (x) = 9 x^{3} + 9$ , $[- 3, 3]$

Step 1

Find the derivative of $f (x) = 9 x^{3} + 9$ .

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $9 x^{3} + 9$ with respect to $x$ is $\frac{d}{d x} [9 x^{3}] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [9 x^{3}] + \frac{d}{d x} [9]$

Step 1.1.2

Evaluate $\frac{d}{d x} [9 x^{3}]$ .

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Step 1.1.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{3}$ with respect to $x$ is $9 \frac{d}{d x} [x^{3}]$ .

$9 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [9]$

Step 1.1.2.2

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

$9 (3 x^{2}) + \frac{d}{d x} [9]$

Step 1.1.2.3

Multiply $3$ by $9$ .

$27 x^{2} + \frac{d}{d x} [9]$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$27 x^{2} + 0$

Step 1.1.3.2

Add $27 x^{2}$ and $0$ .

$f' (x) = 27 x^{2}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $27 x^{2}$ .

$27 x^{2}$

Step 2

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

$f' (x)$ is continuous on $[- 3, 3]$ .

$f' (x)$ is continuous

Step 4

The average value of function $f'$ over the interval $[a, b]$ is defined as $A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

$A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

Step 5

Substitute the actual values into the formula for the average value of a function.

$A (x) = \frac{1}{3 + 3} (\int_{- 3}^{3} 27 x^{2} d x)$

Step 6

Since $27$ is constant with respect to $x$ , move $27$ out of the integral.

$A (x) = \frac{1}{3 + 3} (27 \int_{- 3}^{3} x^{2} d x)$

Step 7

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$A (x) = \frac{1}{3 + 3} (27 (\frac{1}{3} x^{3}]_{- 3}^{3}))$

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$A (x) = \frac{1}{3 + 3} (27 (\frac{x^{3}}{3}]_{- 3}^{3}))$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{x^{3}}{3}$ at $3$ and at $- 3$ .

$A (x) = \frac{1}{3 + 3} (27 ((\frac{3^{3}}{3}) - \frac{{(- 3)}^{3}}{3}))$

Step 8.2.2

Simplify.

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Step 8.2.2.1

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

$A (x) = \frac{1}{3 + 3} (27 (\frac{27}{3} - \frac{{(- 3)}^{3}}{3}))$

Step 8.2.2.2

Cancel the common factor of $27$ and $3$ .

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Step 8.2.2.2.1

Factor $3$ out of $27$ .

$A (x) = \frac{1}{3 + 3} (27 (\frac{3 \cdot 9}{3} - \frac{{(- 3)}^{3}}{3}))$

Step 8.2.2.2.2

Cancel the common factors.

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Step 8.2.2.2.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{3 + 3} (27 (\frac{3 \cdot 9}{3 (1)} - \frac{{(- 3)}^{3}}{3}))$

Step 8.2.2.2.2.2

Cancel the common factor.

$A (x) = \frac{1}{3 + 3} (27 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{{(- 3)}^{3}}{3}))$

Step 8.2.2.2.2.3

Rewrite the expression.

$A (x) = \frac{1}{3 + 3} (27 (\frac{9}{1} - \frac{{(- 3)}^{3}}{3}))$

Step 8.2.2.2.2.4

Divide $9$ by $1$ .

$A (x) = \frac{1}{3 + 3} (27 (9 - \frac{{(- 3)}^{3}}{3}))$

Step 8.2.2.3

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

$A (x) = \frac{1}{3 + 3} (27 (9 - \frac{- 27}{3}))$

Step 8.2.2.4

Cancel the common factor of $- 27$ and $3$ .

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Step 8.2.2.4.1

Factor $3$ out of $- 27$ .

$A (x) = \frac{1}{3 + 3} (27 (9 - \frac{3 \cdot - 9}{3}))$

Step 8.2.2.4.2

Cancel the common factors.

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Step 8.2.2.4.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{3 + 3} (27 (9 - \frac{3 \cdot - 9}{3 (1)}))$

Step 8.2.2.4.2.2

Cancel the common factor.

$A (x) = \frac{1}{3 + 3} (27 (9 - \frac{3 \cdot - 9}{3 \cdot 1}))$

Step 8.2.2.4.2.3

Rewrite the expression.

$A (x) = \frac{1}{3 + 3} (27 (9 - \frac{- 9}{1}))$

Step 8.2.2.4.2.4

Divide $- 9$ by $1$ .

$A (x) = \frac{1}{3 + 3} (27 (9 + 9))$

Step 8.2.2.5

Multiply $- 1$ by $- 9$ .

$A (x) = \frac{1}{3 + 3} (27 (9 + 9))$

Step 8.2.2.6

Add $9$ and $9$ .

$A (x) = \frac{1}{3 + 3} (27 \cdot 18)$

Step 8.2.2.7

Multiply $27$ by $18$ .

$A (x) = \frac{1}{3 + 3} (486)$

Step 9

Add $3$ and $3$ .

$A (x) = \frac{1}{6} \cdot 486$

Step 10

Cancel the common factor of $6$ .

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Step 10.1

Factor $6$ out of $486$ .

$A (x) = \frac{1}{6} \cdot (6 (81))$

Step 10.2

Cancel the common factor.

$A (x) = \frac{1}{6} \cdot (6 \cdot 81)$

Step 10.3

Rewrite the expression.

$A (x) = 81$

Step 11