Calculus Examples

$y = 2 x$ , $[3, 5]$

Step 1

Write $y = 2 x$ as a function.

$f (x) = 2 x$

Step 2

Find the derivative of $f (x) = 2 x$ .

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Step 2.1

Find the first derivative.

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Step 2.1.1

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

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

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 2.1.3

Multiply $2$ by $1$ .

$f' (x) = 2$

Step 2.2

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

$2$

Step 3

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 4

$f' (x)$ is continuous on $[3, 5]$ .

$f' (x)$ is continuous

Step 5

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 6

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

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

Step 7

Apply the constant rule.

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

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $2 x$ at $5$ and at $3$ .

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

Step 8.2

Simplify.

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Step 8.2.1

Multiply $2$ by $5$ .

$A (x) = \frac{1}{5 - 3} (10 - 2 \cdot 3)$

Step 8.2.2

Multiply $- 2$ by $3$ .

$A (x) = \frac{1}{5 - 3} (10 - 6)$

Step 8.2.3

Subtract $6$ from $10$ .

$A (x) = \frac{1}{5 - 3} (4)$

Step 9

Subtract $3$ from $5$ .

$A (x) = \frac{1}{2} \cdot 4$

Step 10

Cancel the common factor of $2$ .

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Step 10.1

Factor $2$ out of $4$ .

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

Step 10.2

Cancel the common factor.

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

Step 10.3

Rewrite the expression.

$A (x) = 2$

Step 11