Calculus Examples

$f (x) = 4 x^{3} - 3 x^{2}$ , $[0, 1]$

Step 1

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 2

$f (x)$ is continuous on $[0, 1]$ .

$f (x)$ is continuous

Step 3

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 4

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

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

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{1 - 0} (\int_{0}^{1} 4 x^{3} d x + \int_{0}^{1} - 3 x^{2} d x)$

Step 6

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$A (x) = \frac{1}{1 - 0} (4 \int_{0}^{1} x^{3} d x + \int_{0}^{1} - 3 x^{2} d x)$

Step 7

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

$A (x) = \frac{1}{1 - 0} (4 (\frac{1}{4} x^{4}]_{0}^{1}) + \int_{0}^{1} - 3 x^{2} d x)$

Step 8

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

$A (x) = \frac{1}{1 - 0} (4 (\frac{x^{4}}{4}]_{0}^{1}) + \int_{0}^{1} - 3 x^{2} d x)$

Step 9

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$A (x) = \frac{1}{1 - 0} (4 (\frac{x^{4}}{4}]_{0}^{1}) - 3 \int_{0}^{1} x^{2} d x)$

Step 10

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

$A (x) = \frac{1}{1 - 0} (4 (\frac{x^{4}}{4}]_{0}^{1}) - 3 (\frac{1}{3} x^{3}]_{0}^{1}))$

Step 11

Simplify the answer.

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Step 11.1

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

$A (x) = \frac{1}{1 - 0} (4 (\frac{x^{4}}{4}]_{0}^{1}) - 3 (\frac{x^{3}}{3}]_{0}^{1}))$

Step 11.2

Substitute and simplify.

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Step 11.2.1

Evaluate $\frac{x^{4}}{4}$ at $1$ and at $0$ .

$A (x) = \frac{1}{1 - 0} (4 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}) - 3 (\frac{x^{3}}{3}]_{0}^{1}))$

Step 11.2.2

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $0$ .

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

Step 11.2.3

Simplify.

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Step 11.2.3.1

One to any power is one.

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

Step 11.2.3.2

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

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

Step 11.2.3.3

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

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Step 11.2.3.3.1

Factor $4$ out of $0$ .

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

Step 11.2.3.3.2

Cancel the common factors.

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Step 11.2.3.3.2.1

Factor $4$ out of $4$ .

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

Step 11.2.3.3.2.2

Cancel the common factor.

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

Step 11.2.3.3.2.3

Rewrite the expression.

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

Step 11.2.3.3.2.4

Divide $0$ by $1$ .

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

Step 11.2.3.4

Multiply $- 1$ by $0$ .

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

Step 11.2.3.5

Add $\frac{1}{4}$ and $0$ .

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

Step 11.2.3.6

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

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

Step 11.2.3.7

Cancel the common factor of $4$ .

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Step 11.2.3.7.1

Cancel the common factor.

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

Step 11.2.3.7.2

Rewrite the expression.

$A (x) = \frac{1}{1 - 0} (1 - 3 (\frac{1^{3}}{3} - \frac{0^{3}}{3}))$

Step 11.2.3.8

One to any power is one.

$A (x) = \frac{1}{1 - 0} (1 - 3 (\frac{1}{3} - \frac{0^{3}}{3}))$

Step 11.2.3.9

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

$A (x) = \frac{1}{1 - 0} (1 - 3 (\frac{1}{3} - \frac{0}{3}))$

Step 11.2.3.10

Cancel the common factor of $0$ and $3$ .

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Step 11.2.3.10.1

Factor $3$ out of $0$ .

$A (x) = \frac{1}{1 - 0} (1 - 3 (\frac{1}{3} - \frac{3 (0)}{3}))$

Step 11.2.3.10.2

Cancel the common factors.

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Step 11.2.3.10.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{1 - 0} (1 - 3 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 11.2.3.10.2.2

Cancel the common factor.

$A (x) = \frac{1}{1 - 0} (1 - 3 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 11.2.3.10.2.3

Rewrite the expression.

$A (x) = \frac{1}{1 - 0} (1 - 3 (\frac{1}{3} - \frac{0}{1}))$

Step 11.2.3.10.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{1 - 0} (1 - 3 (\frac{1}{3} - 0))$

Step 11.2.3.11

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{1 - 0} (1 - 3 (\frac{1}{3} + 0))$

Step 11.2.3.12

Add $\frac{1}{3}$ and $0$ .

$A (x) = \frac{1}{1 - 0} (1 - 3 (\frac{1}{3}))$

Step 11.2.3.13

Combine $- 3$ and $\frac{1}{3}$ .

$A (x) = \frac{1}{1 - 0} (1 + \frac{- 3}{3})$

Step 11.2.3.14

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

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Step 11.2.3.14.1

Factor $3$ out of $- 3$ .

$A (x) = \frac{1}{1 - 0} (1 + \frac{3 \cdot - 1}{3})$

Step 11.2.3.14.2

Cancel the common factors.

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Step 11.2.3.14.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{1 - 0} (1 + \frac{3 \cdot - 1}{3 (1)})$

Step 11.2.3.14.2.2

Cancel the common factor.

$A (x) = \frac{1}{1 - 0} (1 + \frac{3 \cdot - 1}{3 \cdot 1})$

Step 11.2.3.14.2.3

Rewrite the expression.

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

Step 11.2.3.14.2.4

Divide $- 1$ by $1$ .

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

Step 11.2.3.15

Subtract $1$ from $1$ .

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

Step 12

Simplify the denominator.

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Step 12.1

Multiply $- 1$ by $0$ .

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

Step 12.2

Add $1$ and $0$ .

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

Step 13

Reduce the expression by cancelling the common factors.

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Step 13.1

Cancel the common factor of $1$ .

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Step 13.1.1

Cancel the common factor.

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

Step 13.1.2

Rewrite the expression.

$A (x) = 1 \cdot 0$

Step 13.2

Multiply $0$ by $1$ .

$A (x) = 0$

Step 14