Calculus Examples

$f (x) = x^{6} - 3 x^{4}$ , $[0, 2]$

Step 1

Find the derivative of $f (x) = x^{6} - 3 x^{4}$ .

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{6} - 3 x^{4}$ with respect to $x$ is $\frac{d}{d x} [x^{6}] + \frac{d}{d x} [- 3 x^{4}]$ .

$\frac{d}{d x} [x^{6}] + \frac{d}{d x} [- 3 x^{4}]$

Step 1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$6 x^{5} + \frac{d}{d x} [- 3 x^{4}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 3 x^{4}]$ .

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Step 1.1.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{4}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{4}]$ .

$6 x^{5} - 3 \frac{d}{d x} [x^{4}]$

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 $n = 4$ .

$6 x^{5} - 3 (4 x^{3})$

Step 1.1.2.3

Multiply $4$ by $- 3$ .

$f' (x) = 6 x^{5} - 12 x^{3}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $6 x^{5} - 12 x^{3}$ .

$6 x^{5} - 12 x^{3}$

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 $[0, 2]$ .

$f' (x)$ 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.

$A (x) = \frac{1}{2 - 0} (\int_{0}^{2} 6 x^{5} - 12 x^{3} d x)$

Step 6

Split the single integral into multiple integrals.

$A (x) = \frac{1}{2 - 0} (\int_{0}^{2} 6 x^{5} d x + \int_{0}^{2} - 12 x^{3} d x)$

Step 7

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$A (x) = \frac{1}{2 - 0} (6 \int_{0}^{2} x^{5} d x + \int_{0}^{2} - 12 x^{3} d x)$

Step 8

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{1}{6} x^{6}]_{0}^{2}) + \int_{0}^{2} - 12 x^{3} d x)$

Step 9

Combine $\frac{1}{6}$ and $x^{6}$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{x^{6}}{6}]_{0}^{2}) + \int_{0}^{2} - 12 x^{3} d x)$

Step 10

Since $- 12$ is constant with respect to $x$ , move $- 12$ out of the integral.

$A (x) = \frac{1}{2 - 0} (6 (\frac{x^{6}}{6}]_{0}^{2}) - 12 \int_{0}^{2} x^{3} d x)$

Step 11

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{x^{6}}{6}]_{0}^{2}) - 12 (\frac{1}{4} x^{4}]_{0}^{2}))$

Step 12

Simplify the answer.

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Step 12.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{x^{6}}{6}]_{0}^{2}) - 12 (\frac{x^{4}}{4}]_{0}^{2}))$

Step 12.2

Substitute and simplify.

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Step 12.2.1

Evaluate $\frac{x^{6}}{6}$ at $2$ and at $0$ .

$A (x) = \frac{1}{2 - 0} (6 ((\frac{2^{6}}{6}) - \frac{0^{6}}{6}) - 12 (\frac{x^{4}}{4}]_{0}^{2}))$

Step 12.2.2

Evaluate $\frac{x^{4}}{4}$ at $2$ and at $0$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{2^{6}}{6} - \frac{0^{6}}{6}) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3

Simplify.

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Step 12.2.3.1

Raise $2$ to the power of $6$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{64}{6} - \frac{0^{6}}{6}) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.2

Cancel the common factor of $64$ and $6$ .

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Step 12.2.3.2.1

Factor $2$ out of $64$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{2 (32)}{6} - \frac{0^{6}}{6}) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.2.2

Cancel the common factors.

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Step 12.2.3.2.2.1

Factor $2$ out of $6$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{2 \cdot 32}{2 \cdot 3} - \frac{0^{6}}{6}) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.2.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 0} (6 (\frac{2 \cdot 32}{2 \cdot 3} - \frac{0^{6}}{6}) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.2.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 0} (6 (\frac{32}{3} - \frac{0^{6}}{6}) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.3

Raising $0$ to any positive power yields $0$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{32}{3} - \frac{0}{6}) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.4

Cancel the common factor of $0$ and $6$ .

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Step 12.2.3.4.1

Factor $6$ out of $0$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{32}{3} - \frac{6 (0)}{6}) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.4.2

Cancel the common factors.

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Step 12.2.3.4.2.1

Factor $6$ out of $6$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{32}{3} - \frac{6 \cdot 0}{6 \cdot 1}) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.4.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 0} (6 (\frac{32}{3} - \frac{6 \cdot 0}{6 \cdot 1}) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.4.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 0} (6 (\frac{32}{3} - \frac{0}{1}) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.4.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{32}{3} - 0) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.5

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{32}{3} + 0) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.6

Add $\frac{32}{3}$ and $0$ .

$A (x) = \frac{1}{2 - 0} (6 (\frac{32}{3}) - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.7

Combine $6$ and $\frac{32}{3}$ .

$A (x) = \frac{1}{2 - 0} (\frac{6 \cdot 32}{3} - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.8

Multiply $6$ by $32$ .

$A (x) = \frac{1}{2 - 0} (\frac{192}{3} - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.9

Cancel the common factor of $192$ and $3$ .

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Step 12.2.3.9.1

Factor $3$ out of $192$ .

$A (x) = \frac{1}{2 - 0} (\frac{3 \cdot 64}{3} - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.9.2

Cancel the common factors.

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Step 12.2.3.9.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{2 - 0} (\frac{3 \cdot 64}{3 (1)} - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.9.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 0} (\frac{3 \cdot 64}{3 \cdot 1} - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.9.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 0} (\frac{64}{1} - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.9.2.4

Divide $64$ by $1$ .

$A (x) = \frac{1}{2 - 0} (64 - 12 (\frac{2^{4}}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.10

Raise $2$ to the power of $4$ .

$A (x) = \frac{1}{2 - 0} (64 - 12 (\frac{16}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.11

Cancel the common factor of $16$ and $4$ .

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Step 12.2.3.11.1

Factor $4$ out of $16$ .

$A (x) = \frac{1}{2 - 0} (64 - 12 (\frac{4 \cdot 4}{4} - \frac{0^{4}}{4}))$

Step 12.2.3.11.2

Cancel the common factors.

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Step 12.2.3.11.2.1

Factor $4$ out of $4$ .

$A (x) = \frac{1}{2 - 0} (64 - 12 (\frac{4 \cdot 4}{4 (1)} - \frac{0^{4}}{4}))$

Step 12.2.3.11.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 0} (64 - 12 (\frac{4 \cdot 4}{4 \cdot 1} - \frac{0^{4}}{4}))$

Step 12.2.3.11.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 0} (64 - 12 (\frac{4}{1} - \frac{0^{4}}{4}))$

Step 12.2.3.11.2.4

Divide $4$ by $1$ .

$A (x) = \frac{1}{2 - 0} (64 - 12 (4 - \frac{0^{4}}{4}))$

Step 12.2.3.12

Raising $0$ to any positive power yields $0$ .

$A (x) = \frac{1}{2 - 0} (64 - 12 (4 - \frac{0}{4}))$

Step 12.2.3.13

Cancel the common factor of $0$ and $4$ .

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Step 12.2.3.13.1

Factor $4$ out of $0$ .

$A (x) = \frac{1}{2 - 0} (64 - 12 (4 - \frac{4 (0)}{4}))$

Step 12.2.3.13.2

Cancel the common factors.

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Step 12.2.3.13.2.1

Factor $4$ out of $4$ .

$A (x) = \frac{1}{2 - 0} (64 - 12 (4 - \frac{4 \cdot 0}{4 \cdot 1}))$

Step 12.2.3.13.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 0} (64 - 12 (4 - \frac{4 \cdot 0}{4 \cdot 1}))$

Step 12.2.3.13.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 0} (64 - 12 (4 - \frac{0}{1}))$

Step 12.2.3.13.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{2 - 0} (64 - 12 (4 - 0))$

Step 12.2.3.14

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{2 - 0} (64 - 12 (4 + 0))$

Step 12.2.3.15

Add $4$ and $0$ .

$A (x) = \frac{1}{2 - 0} (64 - 12 \cdot 4)$

Step 12.2.3.16

Multiply $- 12$ by $4$ .

$A (x) = \frac{1}{2 - 0} (64 - 48)$

Step 12.2.3.17

Subtract $48$ from $64$ .

$A (x) = \frac{1}{2 - 0} (16)$

Step 13

Simplify the denominator.

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Step 13.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{2 + 0} \cdot 16$

Step 13.2

Add $2$ and $0$ .

$A (x) = \frac{1}{2} \cdot 16$

Step 14

Cancel the common factor of $2$ .

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Step 14.1

Factor $2$ out of $16$ .

$A (x) = \frac{1}{2} \cdot (2 (8))$

Step 14.2

Cancel the common factor.

$A (x) = \frac{1}{2} \cdot (2 \cdot 8)$

Step 14.3

Rewrite the expression.

$A (x) = 8$

Step 15

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