Calculus Examples

$f (x) = x^{3} + 1$ , $[2, 3]$

Step 1

Find the derivative of $f (x) = x^{3} + 1$ .

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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 $x^{3} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [1]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 1.1.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$3 x^{2} + 0$

Step 1.1.4

Add $3 x^{2}$ and $0$ .

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2}$ .

$3 x^{2}$

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 $[2, 3]$ .

$f' (x)$ 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.

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the answer.

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Step 8.1

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

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

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{x^{3}}{3}$ at $3$ and at $2$ .

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

Step 8.2.2

Simplify.

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Step 8.2.2.1

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

$A (x) = \frac{1}{3 - 2} (3 (\frac{27}{3} - \frac{2^{3}}{3}))$

Step 8.2.2.2

Cancel the common factor of $27$ and $3$ .

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Step 8.2.2.2.1

Factor $3$ out of $27$ .

$A (x) = \frac{1}{3 - 2} (3 (\frac{3 \cdot 9}{3} - \frac{2^{3}}{3}))$

Step 8.2.2.2.2

Cancel the common factors.

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Step 8.2.2.2.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{3 - 2} (3 (\frac{3 \cdot 9}{3 (1)} - \frac{2^{3}}{3}))$

Step 8.2.2.2.2.2

Cancel the common factor.

$A (x) = \frac{1}{3 - 2} (3 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{2^{3}}{3}))$

Step 8.2.2.2.2.3

Rewrite the expression.

$A (x) = \frac{1}{3 - 2} (3 (\frac{9}{1} - \frac{2^{3}}{3}))$

Step 8.2.2.2.2.4

Divide $9$ by $1$ .

$A (x) = \frac{1}{3 - 2} (3 (9 - \frac{2^{3}}{3}))$

Step 8.2.2.3

Raise $2$ to the power of $3$ .

$A (x) = \frac{1}{3 - 2} (3 (9 - \frac{8}{3}))$

Step 8.2.2.4

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A (x) = \frac{1}{3 - 2} (3 (9 \cdot \frac{3}{3} - \frac{8}{3}))$

Step 8.2.2.5

Combine $9$ and $\frac{3}{3}$ .

$A (x) = \frac{1}{3 - 2} (3 (\frac{9 \cdot 3}{3} - \frac{8}{3}))$

Step 8.2.2.6

Combine the numerators over the common denominator.

$A (x) = \frac{1}{3 - 2} (3 (\frac{9 \cdot 3 - 8}{3}))$

Step 8.2.2.7

Simplify the numerator.

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Step 8.2.2.7.1

Multiply $9$ by $3$ .

$A (x) = \frac{1}{3 - 2} (3 (\frac{27 - 8}{3}))$

Step 8.2.2.7.2

Subtract $8$ from $27$ .

$A (x) = \frac{1}{3 - 2} (3 (\frac{19}{3}))$

Step 8.2.2.8

Combine $3$ and $\frac{19}{3}$ .

$A (x) = \frac{1}{3 - 2} (\frac{3 \cdot 19}{3})$

Step 8.2.2.9

Multiply $3$ by $19$ .

$A (x) = \frac{1}{3 - 2} (\frac{57}{3})$

Step 8.2.2.10

Cancel the common factor of $57$ and $3$ .

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Step 8.2.2.10.1

Factor $3$ out of $57$ .

$A (x) = \frac{1}{3 - 2} (\frac{3 \cdot 19}{3})$

Step 8.2.2.10.2

Cancel the common factors.

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Step 8.2.2.10.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{3 - 2} (\frac{3 \cdot 19}{3 (1)})$

Step 8.2.2.10.2.2

Cancel the common factor.

$A (x) = \frac{1}{3 - 2} (\frac{3 \cdot 19}{3 \cdot 1})$

Step 8.2.2.10.2.3

Rewrite the expression.

$A (x) = \frac{1}{3 - 2} (\frac{19}{1})$

Step 8.2.2.10.2.4

Divide $19$ by $1$ .

$A (x) = \frac{1}{3 - 2} (19)$

Step 9

Subtract $2$ from $3$ .

$A (x) = \frac{1}{1} \cdot 19$

Step 10

Cancel the common factor of $1$ .

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Step 10.1

Cancel the common factor.

$A (x) = \frac{1}{1} \cdot 19$

Step 10.2

Rewrite the expression.

$A (x) = 1 \cdot 19$

Step 11

Multiply $19$ by $1$ .

$A (x) = 19$

Step 12