Calculus Examples

$y = 3 x^{2} + 3 x$ , $(- 5, 1)$

Step 1

Write $y = 3 x^{2} + 3 x$ as a function.

$f (x) = 3 x^{2} + 3 x$

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 $[- 5, 1]$ .

$f (x)$ 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.

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

$A (x) = \frac{1}{1 + 5} (3 (\frac{1}{3} x^{3}]_{- 5}^{1}) + \int_{- 5}^{1} 3 x d x)$

Step 9

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

$A (x) = \frac{1}{1 + 5} (3 (\frac{x^{3}}{3}]_{- 5}^{1}) + \int_{- 5}^{1} 3 x d x)$

Step 10

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

$A (x) = \frac{1}{1 + 5} (3 (\frac{x^{3}}{3}]_{- 5}^{1}) + 3 \int_{- 5}^{1} x d x)$

Step 11

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

$A (x) = \frac{1}{1 + 5} (3 (\frac{x^{3}}{3}]_{- 5}^{1}) + 3 (\frac{1}{2} x^{2}]_{- 5}^{1}))$

Step 12

Simplify the answer.

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Step 12.1

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

$A (x) = \frac{1}{1 + 5} (3 (\frac{x^{3}}{3}]_{- 5}^{1}) + 3 (\frac{x^{2}}{2}]_{- 5}^{1}))$

Step 12.2

Substitute and simplify.

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Step 12.2.1

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $- 5$ .

$A (x) = \frac{1}{1 + 5} (3 ((\frac{1^{3}}{3}) - \frac{{(- 5)}^{3}}{3}) + 3 (\frac{x^{2}}{2}]_{- 5}^{1}))$

Step 12.2.2

Evaluate $\frac{x^{2}}{2}$ at $1$ and at $- 5$ .

$A (x) = \frac{1}{1 + 5} (3 (\frac{1^{3}}{3} - \frac{{(- 5)}^{3}}{3}) + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3

Simplify.

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Step 12.2.3.1

One to any power is one.

$A (x) = \frac{1}{1 + 5} (3 (\frac{1}{3} - \frac{{(- 5)}^{3}}{3}) + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.2

Raise $- 5$ to the power of $3$ .

$A (x) = \frac{1}{1 + 5} (3 (\frac{1}{3} - \frac{- 125}{3}) + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.3

Move the negative in front of the fraction.

$A (x) = \frac{1}{1 + 5} (3 (\frac{1}{3} + \frac{125}{3}) + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.4

Multiply $- 1$ by $- 1$ .

$A (x) = \frac{1}{1 + 5} (3 (\frac{1}{3} + 1 (\frac{125}{3})) + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.5

Multiply $\frac{125}{3}$ by $1$ .

$A (x) = \frac{1}{1 + 5} (3 (\frac{1}{3} + \frac{125}{3}) + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.6

Combine the numerators over the common denominator.

$A (x) = \frac{1}{1 + 5} (3 (\frac{1 + 125}{3}) + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.7

Add $1$ and $125$ .

$A (x) = \frac{1}{1 + 5} (3 (\frac{126}{3}) + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.8

Cancel the common factor of $126$ and $3$ .

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Step 12.2.3.8.1

Factor $3$ out of $126$ .

$A (x) = \frac{1}{1 + 5} (3 (\frac{3 \cdot 42}{3}) + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.8.2

Cancel the common factors.

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Step 12.2.3.8.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{1 + 5} (3 (\frac{3 \cdot 42}{3 (1)}) + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.8.2.2

Cancel the common factor.

$A (x) = \frac{1}{1 + 5} (3 (\frac{3 \cdot 42}{3 \cdot 1}) + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.8.2.3

Rewrite the expression.

$A (x) = \frac{1}{1 + 5} (3 (\frac{42}{1}) + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.8.2.4

Divide $42$ by $1$ .

$A (x) = \frac{1}{1 + 5} (3 \cdot 42 + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.9

Multiply $3$ by $42$ .

$A (x) = \frac{1}{1 + 5} (126 + 3 (\frac{1^{2}}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.10

One to any power is one.

$A (x) = \frac{1}{1 + 5} (126 + 3 (\frac{1}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 12.2.3.11

Raise $- 5$ to the power of $2$ .

$A (x) = \frac{1}{1 + 5} (126 + 3 (\frac{1}{2} - \frac{25}{2}))$

Step 12.2.3.12

Combine the numerators over the common denominator.

$A (x) = \frac{1}{1 + 5} (126 + 3 (\frac{1 - 25}{2}))$

Step 12.2.3.13

Subtract $25$ from $1$ .

$A (x) = \frac{1}{1 + 5} (126 + 3 (\frac{- 24}{2}))$

Step 12.2.3.14

Cancel the common factor of $- 24$ and $2$ .

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Step 12.2.3.14.1

Factor $2$ out of $- 24$ .

$A (x) = \frac{1}{1 + 5} (126 + 3 (\frac{2 \cdot - 12}{2}))$

Step 12.2.3.14.2

Cancel the common factors.

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Step 12.2.3.14.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{1 + 5} (126 + 3 (\frac{2 \cdot - 12}{2 (1)}))$

Step 12.2.3.14.2.2

Cancel the common factor.

$A (x) = \frac{1}{1 + 5} (126 + 3 (\frac{2 \cdot - 12}{2 \cdot 1}))$

Step 12.2.3.14.2.3

Rewrite the expression.

$A (x) = \frac{1}{1 + 5} (126 + 3 (\frac{- 12}{1}))$

Step 12.2.3.14.2.4

Divide $- 12$ by $1$ .

$A (x) = \frac{1}{1 + 5} (126 + 3 \cdot - 12)$

Step 12.2.3.15

Multiply $3$ by $- 12$ .

$A (x) = \frac{1}{1 + 5} (126 - 36)$

Step 12.2.3.16

Subtract $36$ from $126$ .

$A (x) = \frac{1}{1 + 5} (90)$

Step 13

Add $1$ and $5$ .

$A (x) = \frac{1}{6} \cdot 90$

Step 14

Cancel the common factor of $6$ .

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Step 14.1

Factor $6$ out of $90$ .

$A (x) = \frac{1}{6} \cdot (6 (15))$

Step 14.2

Cancel the common factor.

$A (x) = \frac{1}{6} \cdot (6 \cdot 15)$

Step 14.3

Rewrite the expression.

$A (x) = 15$

Step 15

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