Calculus Examples

$f (x) = x^{2} + 2 x$ , $[0, 7]$

Step 1

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

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Differentiate.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of $x^{2} + 2 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x]$ .

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

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 $n = 2$ .

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

Step 1.1.2

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

Tap for more steps...

Step 1.1.2.1

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

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

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 $n = 1$ .

$2 x + 2 \cdot 1$

Step 1.1.2.3

Multiply $2$ by $1$ .

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

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 $[0, 7]$ .

$f' (x)$ 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.

$A (x) = \frac{1}{7 - 0} (\int_{0}^{7} 2 x + 2 d x)$

Step 6

Split the single integral into multiple integrals.

$A (x) = \frac{1}{7 - 0} (\int_{0}^{7} 2 x d x + \int_{0}^{7} 2 d x)$

Step 7

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

$A (x) = \frac{1}{7 - 0} (2 \int_{0}^{7} x d x + \int_{0}^{7} 2 d x)$

Step 8

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

$A (x) = \frac{1}{7 - 0} (2 (\frac{1}{2} x^{2}]_{0}^{7}) + \int_{0}^{7} 2 d x)$

Step 9

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

$A (x) = \frac{1}{7 - 0} (2 (\frac{x^{2}}{2}]_{0}^{7}) + \int_{0}^{7} 2 d x)$

Step 10

Apply the constant rule.

$A (x) = \frac{1}{7 - 0} (2 (\frac{x^{2}}{2}]_{0}^{7}) + 2 x]_{0}^{7})$

Step 11

Substitute and simplify.

Tap for more steps...

Step 11.1

Evaluate $\frac{x^{2}}{2}$ at $7$ and at $0$ .

$A (x) = \frac{1}{7 - 0} (2 ((\frac{7^{2}}{2}) - \frac{0^{2}}{2}) + 2 x]_{0}^{7})$

Step 11.2

Evaluate $2 x$ at $7$ and at $0$ .

$A (x) = \frac{1}{7 - 0} (2 (\frac{7^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3

Simplify.

Tap for more steps...

Step 11.3.1

Raise $7$ to the power of $2$ .

$A (x) = \frac{1}{7 - 0} (2 (\frac{49}{2} - \frac{0^{2}}{2}) + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.2

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

$A (x) = \frac{1}{7 - 0} (2 (\frac{49}{2} - \frac{0}{2}) + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.3

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

Tap for more steps...

Step 11.3.3.1

Factor $2$ out of $0$ .

$A (x) = \frac{1}{7 - 0} (2 (\frac{49}{2} - \frac{2 (0)}{2}) + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.3.2

Cancel the common factors.

Tap for more steps...

Step 11.3.3.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{7 - 0} (2 (\frac{49}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.3.2.2

Cancel the common factor.

$A (x) = \frac{1}{7 - 0} (2 (\frac{49}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.3.2.3

Rewrite the expression.

$A (x) = \frac{1}{7 - 0} (2 (\frac{49}{2} - \frac{0}{1}) + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.3.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{7 - 0} (2 (\frac{49}{2} - 0) + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.4

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{7 - 0} (2 (\frac{49}{2} + 0) + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.5

Add $\frac{49}{2}$ and $0$ .

$A (x) = \frac{1}{7 - 0} (2 (\frac{49}{2}) + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.6

Combine $2$ and $\frac{49}{2}$ .

$A (x) = \frac{1}{7 - 0} (\frac{2 \cdot 49}{2} + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.7

Multiply $2$ by $49$ .

$A (x) = \frac{1}{7 - 0} (\frac{98}{2} + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.8

Cancel the common factor of $98$ and $2$ .

Tap for more steps...

Step 11.3.8.1

Factor $2$ out of $98$ .

$A (x) = \frac{1}{7 - 0} (\frac{2 \cdot 49}{2} + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.8.2

Cancel the common factors.

Tap for more steps...

Step 11.3.8.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{7 - 0} (\frac{2 \cdot 49}{2 (1)} + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.8.2.2

Cancel the common factor.

$A (x) = \frac{1}{7 - 0} (\frac{2 \cdot 49}{2 \cdot 1} + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.8.2.3

Rewrite the expression.

$A (x) = \frac{1}{7 - 0} (\frac{49}{1} + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.8.2.4

Divide $49$ by $1$ .

$A (x) = \frac{1}{7 - 0} (49 + 2 \cdot 7 - 2 \cdot 0)$

Step 11.3.9

Multiply $2$ by $7$ .

$A (x) = \frac{1}{7 - 0} (49 + 14 - 2 \cdot 0)$

Step 11.3.10

Multiply $- 2$ by $0$ .

$A (x) = \frac{1}{7 - 0} (49 + 14 + 0)$

Step 11.3.11

Add $14$ and $0$ .

$A (x) = \frac{1}{7 - 0} (49 + 14)$

Step 11.3.12

Add $49$ and $14$ .

$A (x) = \frac{1}{7 - 0} (63)$

Step 12

Simplify the denominator.

Tap for more steps...

Step 12.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{7 + 0} \cdot 63$

Step 12.2

Add $7$ and $0$ .

$A (x) = \frac{1}{7} \cdot 63$

Step 13

Cancel the common factor of $7$ .

Tap for more steps...

Step 13.1

Factor $7$ out of $63$ .

$A (x) = \frac{1}{7} \cdot (7 (9))$

Step 13.2

Cancel the common factor.

$A (x) = \frac{1}{7} \cdot (7 \cdot 9)$

Step 13.3

Rewrite the expression.

$A (x) = 9$

Step 14