Calculus Examples

$y = x^{2} - 3 x + 1$ , $[0, 2]$

Step 1

Write $y = x^{2} - 3 x + 1$ as a function.

$f (x) = x^{2} - 3 x + 1$

Step 2

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

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate.

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Step 2.1.1.1

By the Sum Rule, the derivative of $x^{2} - 3 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1]$ .

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

Step 2.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 2.1.2

Evaluate $\frac{d}{d x} [- 3 x]$ .

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Step 2.1.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Multiply $- 3$ by $1$ .

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

Step 2.1.3

Differentiate using the Constant Rule.

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Step 2.1.3.1

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

$2 x - 3 + 0$

Step 2.1.3.2

Add $2 x - 3$ and $0$ .

$f' (x) = 2 x - 3$

Step 2.2

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

$2 x - 3$

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 $[0, 2]$ .

$f' (x)$ 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.

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Apply the constant rule.

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

Step 12

Substitute and simplify.

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Step 12.1

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

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

Step 12.2

Evaluate $- 3 x$ at $2$ and at $0$ .

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

Step 12.3

Simplify.

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Step 12.3.1

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

$A (x) = \frac{1}{2 - 0} (2 (\frac{4}{2} - \frac{0^{2}}{2}) - 3 \cdot 2 + 3 \cdot 0)$

Step 12.3.2

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

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Step 12.3.2.1

Factor $2$ out of $4$ .

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

Step 12.3.2.2

Cancel the common factors.

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Step 12.3.2.2.1

Factor $2$ out of $2$ .

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

Step 12.3.2.2.2

Cancel the common factor.

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

Step 12.3.2.2.3

Rewrite the expression.

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

Step 12.3.2.2.4

Divide $2$ by $1$ .

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

Step 12.3.3

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

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

Step 12.3.4

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

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Step 12.3.4.1

Factor $2$ out of $0$ .

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

Step 12.3.4.2

Cancel the common factors.

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Step 12.3.4.2.1

Factor $2$ out of $2$ .

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

Step 12.3.4.2.2

Cancel the common factor.

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

Step 12.3.4.2.3

Rewrite the expression.

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

Step 12.3.4.2.4

Divide $0$ by $1$ .

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

Step 12.3.5

Multiply $- 1$ by $0$ .

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

Step 12.3.6

Add $2$ and $0$ .

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

Step 12.3.7

Multiply $2$ by $2$ .

$A (x) = \frac{1}{2 - 0} (4 - 3 \cdot 2 + 3 \cdot 0)$

Step 12.3.8

Multiply $- 3$ by $2$ .

$A (x) = \frac{1}{2 - 0} (4 - 6 + 3 \cdot 0)$

Step 12.3.9

Multiply $3$ by $0$ .

$A (x) = \frac{1}{2 - 0} (4 - 6 + 0)$

Step 12.3.10

Add $- 6$ and $0$ .

$A (x) = \frac{1}{2 - 0} (4 - 6)$

Step 12.3.11

Subtract $6$ from $4$ .

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

Step 13

Simplify the denominator.

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Step 13.1

Multiply $- 1$ by $0$ .

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

Step 13.2

Add $2$ and $0$ .

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

Step 14

Cancel the common factor of $2$ .

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Step 14.1

Factor $2$ out of $- 2$ .

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

Step 14.2

Cancel the common factor.

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

Step 14.3

Rewrite the expression.

$A (x) = - 1$

Step 15