Calculus Examples

$f (x) = x^{3} - x^{2} + 2 x - 1$ , $[0, 2]$

Step 1

Find the derivative of $f (x) = x^{3} - x^{2} + 2 x - 1$ .

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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 $x^{3} - x^{2} + 2 x - 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

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

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 $n = 3$ .

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

Step 1.1.2

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

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Step 1.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

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 $n = 2$ .

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

Step 1.1.2.3

Multiply $2$ by $- 1$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply $2$ by $1$ .

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

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$3 x^{2} - 2 x + 2 + 0$

Step 1.1.4.2

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

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

Step 1.2

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

$3 x^{2} - 2 x + 2$

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 $[0, 2]$ .

$f' (x)$ 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.

$A (x) = \frac{1}{2 - 0} (\int_{0}^{2} 3 x^{2} - 2 x + 2 d x)$

Step 6

Split the single integral into multiple integrals.

$A (x) = \frac{1}{2 - 0} (\int_{0}^{2} 3 x^{2} d x + \int_{0}^{2} - 2 x d x + \int_{0}^{2} 2 d x)$

Step 7

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

$A (x) = \frac{1}{2 - 0} (3 \int_{0}^{2} x^{2} d x + \int_{0}^{2} - 2 x d x + \int_{0}^{2} 2 d x)$

Step 8

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

$A (x) = \frac{1}{2 - 0} (3 (\frac{1}{3} x^{3}]_{0}^{2}) + \int_{0}^{2} - 2 x d x + \int_{0}^{2} 2 d x)$

Step 9

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

$A (x) = \frac{1}{2 - 0} (3 (\frac{x^{3}}{3}]_{0}^{2}) + \int_{0}^{2} - 2 x d x + \int_{0}^{2} 2 d x)$

Step 10

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

$A (x) = \frac{1}{2 - 0} (3 (\frac{x^{3}}{3}]_{0}^{2}) - 2 \int_{0}^{2} x d x + \int_{0}^{2} 2 d x)$

Step 11

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

$A (x) = \frac{1}{2 - 0} (3 (\frac{x^{3}}{3}]_{0}^{2}) - 2 (\frac{1}{2} x^{2}]_{0}^{2}) + \int_{0}^{2} 2 d x)$

Step 12

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

$A (x) = \frac{1}{2 - 0} (3 (\frac{x^{3}}{3}]_{0}^{2}) - 2 (\frac{x^{2}}{2}]_{0}^{2}) + \int_{0}^{2} 2 d x)$

Step 13

Apply the constant rule.

$A (x) = \frac{1}{2 - 0} (3 (\frac{x^{3}}{3}]_{0}^{2}) - 2 (\frac{x^{2}}{2}]_{0}^{2}) + 2 x]_{0}^{2})$

Step 14

Substitute and simplify.

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Step 14.1

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $0$ .

$A (x) = \frac{1}{2 - 0} (3 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3}) - 2 (\frac{x^{2}}{2}]_{0}^{2}) + 2 x]_{0}^{2})$

Step 14.2

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

$A (x) = \frac{1}{2 - 0} (3 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 x]_{0}^{2})$

Step 14.3

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

$A (x) = \frac{1}{2 - 0} (3 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4

Simplify.

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Step 14.4.1

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

$A (x) = \frac{1}{2 - 0} (3 (\frac{8}{3} - \frac{0^{3}}{3}) - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.2

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

$A (x) = \frac{1}{2 - 0} (3 (\frac{8}{3} - \frac{0}{3}) - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.3

Cancel the common factor of $0$ and $3$ .

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Step 14.4.3.1

Factor $3$ out of $0$ .

$A (x) = \frac{1}{2 - 0} (3 (\frac{8}{3} - \frac{3 (0)}{3}) - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.3.2

Cancel the common factors.

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Step 14.4.3.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{2 - 0} (3 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.3.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 0} (3 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.3.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 0} (3 (\frac{8}{3} - \frac{0}{1}) - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.3.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{2 - 0} (3 (\frac{8}{3} - 0) - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.4

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{2 - 0} (3 (\frac{8}{3} + 0) - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.5

Add $\frac{8}{3}$ and $0$ .

$A (x) = \frac{1}{2 - 0} (3 (\frac{8}{3}) - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.6

Combine $3$ and $\frac{8}{3}$ .

$A (x) = \frac{1}{2 - 0} (\frac{3 \cdot 8}{3} - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.7

Multiply $3$ by $8$ .

$A (x) = \frac{1}{2 - 0} (\frac{24}{3} - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.8

Cancel the common factor of $24$ and $3$ .

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Step 14.4.8.1

Factor $3$ out of $24$ .

$A (x) = \frac{1}{2 - 0} (\frac{3 \cdot 8}{3} - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.8.2

Cancel the common factors.

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Step 14.4.8.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{2 - 0} (\frac{3 \cdot 8}{3 (1)} - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.8.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 0} (\frac{3 \cdot 8}{3 \cdot 1} - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.8.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 0} (\frac{8}{1} - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.8.2.4

Divide $8$ by $1$ .

$A (x) = \frac{1}{2 - 0} (8 - 2 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.9

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

$A (x) = \frac{1}{2 - 0} (8 - 2 (\frac{4}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.10

Cancel the common factor of $4$ and $2$ .

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Step 14.4.10.1

Factor $2$ out of $4$ .

$A (x) = \frac{1}{2 - 0} (8 - 2 (\frac{2 \cdot 2}{2} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.10.2

Cancel the common factors.

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Step 14.4.10.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{2 - 0} (8 - 2 (\frac{2 \cdot 2}{2 (1)} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.10.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 0} (8 - 2 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.10.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 0} (8 - 2 (\frac{2}{1} - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.10.2.4

Divide $2$ by $1$ .

$A (x) = \frac{1}{2 - 0} (8 - 2 (2 - \frac{0^{2}}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.11

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

$A (x) = \frac{1}{2 - 0} (8 - 2 (2 - \frac{0}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.12

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

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Step 14.4.12.1

Factor $2$ out of $0$ .

$A (x) = \frac{1}{2 - 0} (8 - 2 (2 - \frac{2 (0)}{2}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.12.2

Cancel the common factors.

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Step 14.4.12.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{2 - 0} (8 - 2 (2 - \frac{2 \cdot 0}{2 \cdot 1}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.12.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 0} (8 - 2 (2 - \frac{2 \cdot 0}{2 \cdot 1}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.12.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 0} (8 - 2 (2 - \frac{0}{1}) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.12.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{2 - 0} (8 - 2 (2 - 0) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.13

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{2 - 0} (8 - 2 (2 + 0) + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.14

Add $2$ and $0$ .

$A (x) = \frac{1}{2 - 0} (8 - 2 \cdot 2 + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.15

Multiply $- 2$ by $2$ .

$A (x) = \frac{1}{2 - 0} (8 - 4 + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.16

Subtract $4$ from $8$ .

$A (x) = \frac{1}{2 - 0} (4 + 2 \cdot 2 - 2 \cdot 0)$

Step 14.4.17

Multiply $2$ by $2$ .

$A (x) = \frac{1}{2 - 0} (4 + 4 - 2 \cdot 0)$

Step 14.4.18

Multiply $- 2$ by $0$ .

$A (x) = \frac{1}{2 - 0} (4 + 4 + 0)$

Step 14.4.19

Add $4$ and $0$ .

$A (x) = \frac{1}{2 - 0} (4 + 4)$

Step 14.4.20

Add $4$ and $4$ .

$A (x) = \frac{1}{2 - 0} (8)$

Step 15

Simplify the denominator.

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Step 15.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{2 + 0} \cdot 8$

Step 15.2

Add $2$ and $0$ .

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

Step 16

Cancel the common factor of $2$ .

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Step 16.1

Factor $2$ out of $8$ .

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

Step 16.2

Cancel the common factor.

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

Step 16.3

Rewrite the expression.

$A (x) = 4$

Step 17

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