Calculus Examples

$f (x) = 5 x + 4$ , $[2, 5]$

Step 1

Find the derivative of $f (x) = 5 x + 4$ .

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $5 x + 4$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [5 x] + \frac{d}{d x} [4]$

Step 1.1.2

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 1.1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x] + \frac{d}{d x} [4]$

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 $n = 1$ .

$5 \cdot 1 + \frac{d}{d x} [4]$

Step 1.1.2.3

Multiply $5$ by $1$ .

$5 + \frac{d}{d x} [4]$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$5 + 0$

Step 1.1.3.2

Add $5$ and $0$ .

$f' (x) = 5$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $5$ .

$5$

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 $[2, 5]$ .

$f' (x)$ 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.

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

Step 6

Apply the constant rule.

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

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $5 x$ at $5$ and at $2$ .

$A (x) = \frac{1}{5 - 2} ((5 \cdot 5) - 5 \cdot 2)$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $5$ by $5$ .

$A (x) = \frac{1}{5 - 2} (25 - 5 \cdot 2)$

Step 7.2.2

Multiply $- 5$ by $2$ .

$A (x) = \frac{1}{5 - 2} (25 - 10)$

Step 7.2.3

Subtract $10$ from $25$ .

$A (x) = \frac{1}{5 - 2} (15)$

Step 8

Subtract $2$ from $5$ .

$A (x) = \frac{1}{3} \cdot 15$

Step 9

Cancel the common factor of $3$ .

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Step 9.1

Factor $3$ out of $15$ .

$A (x) = \frac{1}{3} \cdot (3 (5))$

Step 9.2

Cancel the common factor.

$A (x) = \frac{1}{3} \cdot (3 \cdot 5)$

Step 9.3

Rewrite the expression.

$A (x) = 5$

Step 10