Calculus Examples

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

Step 1

Find the derivative of $f (x) = x^{2} + 2 x - 3$ .

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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 $x^{2} + 2 x - 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]$

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 $n = 2$ .

$2 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]$

Step 1.1.2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

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 $n = 1$ .

$2 x + 2 \cdot 1 + \frac{d}{d x} [- 3]$

Step 1.1.2.3

Multiply $2$ by $1$ .

$2 x + 2 + \frac{d}{d x} [- 3]$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$2 x + 2 + 0$

Step 1.1.3.2

Add $2 x + 2$ and $0$ .

$f' (x) = 2 x + 2$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2 x + 2$ .

$2 x + 2$

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 $[0, 6]$ .

$f' (x)$ 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.

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 8

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

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

Step 9

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

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

Step 10

Apply the constant rule.

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

Step 11

Substitute and simplify.

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Step 11.1

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

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

Step 11.2

Evaluate $2 x$ at $6$ and at $0$ .

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

Step 11.3

Simplify.

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Step 11.3.1

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

$A (x) = \frac{1}{6 - 0} (2 (\frac{36}{2} - \frac{0^{2}}{2}) + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.2

Cancel the common factor of $36$ and $2$ .

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Step 11.3.2.1

Factor $2$ out of $36$ .

$A (x) = \frac{1}{6 - 0} (2 (\frac{2 \cdot 18}{2} - \frac{0^{2}}{2}) + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.2.2

Cancel the common factors.

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Step 11.3.2.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{6 - 0} (2 (\frac{2 \cdot 18}{2 (1)} - \frac{0^{2}}{2}) + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.2.2.2

Cancel the common factor.

$A (x) = \frac{1}{6 - 0} (2 (\frac{2 \cdot 18}{2 \cdot 1} - \frac{0^{2}}{2}) + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.2.2.3

Rewrite the expression.

$A (x) = \frac{1}{6 - 0} (2 (\frac{18}{1} - \frac{0^{2}}{2}) + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.2.2.4

Divide $18$ by $1$ .

$A (x) = \frac{1}{6 - 0} (2 (18 - \frac{0^{2}}{2}) + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.3

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

$A (x) = \frac{1}{6 - 0} (2 (18 - \frac{0}{2}) + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.4

Cancel the common factor of $0$ and $2$ .

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Step 11.3.4.1

Factor $2$ out of $0$ .

$A (x) = \frac{1}{6 - 0} (2 (18 - \frac{2 (0)}{2}) + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.4.2

Cancel the common factors.

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Step 11.3.4.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{6 - 0} (2 (18 - \frac{2 \cdot 0}{2 \cdot 1}) + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.4.2.2

Cancel the common factor.

$A (x) = \frac{1}{6 - 0} (2 (18 - \frac{2 \cdot 0}{2 \cdot 1}) + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.4.2.3

Rewrite the expression.

$A (x) = \frac{1}{6 - 0} (2 (18 - \frac{0}{1}) + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.4.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{6 - 0} (2 (18 - 0) + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.5

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{6 - 0} (2 (18 + 0) + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.6

Add $18$ and $0$ .

$A (x) = \frac{1}{6 - 0} (2 \cdot 18 + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.7

Multiply $2$ by $18$ .

$A (x) = \frac{1}{6 - 0} (36 + 2 \cdot 6 - 2 \cdot 0)$

Step 11.3.8

Multiply $2$ by $6$ .

$A (x) = \frac{1}{6 - 0} (36 + 12 - 2 \cdot 0)$

Step 11.3.9

Multiply $- 2$ by $0$ .

$A (x) = \frac{1}{6 - 0} (36 + 12 + 0)$

Step 11.3.10

Add $12$ and $0$ .

$A (x) = \frac{1}{6 - 0} (36 + 12)$

Step 11.3.11

Add $36$ and $12$ .

$A (x) = \frac{1}{6 - 0} (48)$

Step 12

Simplify the denominator.

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Step 12.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{6 + 0} \cdot 48$

Step 12.2

Add $6$ and $0$ .

$A (x) = \frac{1}{6} \cdot 48$

Step 13

Cancel the common factor of $6$ .

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Step 13.1

Factor $6$ out of $48$ .

$A (x) = \frac{1}{6} \cdot (6 (8))$

Step 13.2

Cancel the common factor.

$A (x) = \frac{1}{6} \cdot (6 \cdot 8)$

Step 13.3

Rewrite the expression.

$A (x) = 8$

Step 14

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