Calculus Examples

$f (x) = x^{3} - 4 x^{2} + 5 x$ , $[1, 2]$

Step 1

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 2

$f (x)$ is continuous on $[1, 2]$ .

$f (x)$ is continuous

Step 3

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 4

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

$A (x) = \frac{1}{2 - 1} (\int_{1}^{2} x^{3} - 4 x^{2} + 5 x d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{2 - 1} (\int_{1}^{2} x^{3} d x + \int_{1}^{2} - 4 x^{2} d x + \int_{1}^{2} 5 x d x)$

Step 6

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$A (x) = \frac{1}{2 - 1} (\frac{1}{4} x^{4}]_{1}^{2} + \int_{1}^{2} - 4 x^{2} d x + \int_{1}^{2} 5 x d x)$

Step 7

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$A (x) = \frac{1}{2 - 1} (\frac{1}{4} x^{4}]_{1}^{2} - 4 \int_{1}^{2} x^{2} d x + \int_{1}^{2} 5 x d x)$

Step 8

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

$A (x) = \frac{1}{2 - 1} (\frac{1}{4} x^{4}]_{1}^{2} - 4 (\frac{1}{3} x^{3}]_{1}^{2}) + \int_{1}^{2} 5 x d x)$

Step 9

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

$A (x) = \frac{1}{2 - 1} (\frac{1}{4} x^{4}]_{1}^{2} - 4 (\frac{x^{3}}{3}]_{1}^{2}) + \int_{1}^{2} 5 x d x)$

Step 10

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$A (x) = \frac{1}{2 - 1} (\frac{1}{4} x^{4}]_{1}^{2} - 4 (\frac{x^{3}}{3}]_{1}^{2}) + 5 \int_{1}^{2} x d x)$

Step 11

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

$A (x) = \frac{1}{2 - 1} (\frac{1}{4} x^{4}]_{1}^{2} - 4 (\frac{x^{3}}{3}]_{1}^{2}) + 5 (\frac{1}{2} x^{2}]_{1}^{2}))$

Step 12

Simplify the answer.

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Step 12.1

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

$A (x) = \frac{1}{2 - 1} (\frac{1}{4} x^{4}]_{1}^{2} - 4 (\frac{x^{3}}{3}]_{1}^{2}) + 5 (\frac{x^{2}}{2}]_{1}^{2}))$

Step 12.2

Substitute and simplify.

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Step 12.2.1

Evaluate $\frac{1}{4} x^{4}$ at $2$ and at $1$ .

$A (x) = \frac{1}{2 - 1} ((\frac{1}{4} \cdot 2^{4}) - \frac{1}{4} \cdot 1^{4} - 4 (\frac{x^{3}}{3}]_{1}^{2}) + 5 (\frac{x^{2}}{2}]_{1}^{2}))$

Step 12.2.2

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $1$ .

$A (x) = \frac{1}{2 - 1} (\frac{1}{4} \cdot 2^{4} - \frac{1}{4} \cdot 1^{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 (\frac{x^{2}}{2}]_{1}^{2}))$

Step 12.2.3

Evaluate $\frac{x^{2}}{2}$ at $2$ and at $1$ .

$A (x) = \frac{1}{2 - 1} (\frac{1}{4} \cdot 2^{4} - \frac{1}{4} \cdot 1^{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4

Simplify.

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Step 12.2.4.1

Raise $2$ to the power of $4$ .

$A (x) = \frac{1}{2 - 1} (\frac{1}{4} \cdot 16 - \frac{1}{4} \cdot 1^{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.2

Combine $\frac{1}{4}$ and $16$ .

$A (x) = \frac{1}{2 - 1} (\frac{16}{4} - \frac{1}{4} \cdot 1^{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.3

Cancel the common factor of $16$ and $4$ .

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Step 12.2.4.3.1

Factor $4$ out of $16$ .

$A (x) = \frac{1}{2 - 1} (\frac{4 \cdot 4}{4} - \frac{1}{4} \cdot 1^{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.3.2

Cancel the common factors.

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Step 12.2.4.3.2.1

Factor $4$ out of $4$ .

$A (x) = \frac{1}{2 - 1} (\frac{4 \cdot 4}{4 (1)} - \frac{1}{4} \cdot 1^{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.3.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 1} (\frac{4 \cdot 4}{4 \cdot 1} - \frac{1}{4} \cdot 1^{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.3.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 1} (\frac{4}{1} - \frac{1}{4} \cdot 1^{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.3.2.4

Divide $4$ by $1$ .

$A (x) = \frac{1}{2 - 1} (4 - \frac{1}{4} \cdot 1^{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.4

One to any power is one.

$A (x) = \frac{1}{2 - 1} (4 - \frac{1}{4} \cdot 1 - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.5

Multiply $- 1$ by $1$ .

$A (x) = \frac{1}{2 - 1} (4 - \frac{1}{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.6

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$A (x) = \frac{1}{2 - 1} (4 \cdot \frac{4}{4} - \frac{1}{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.7

Combine $4$ and $\frac{4}{4}$ .

$A (x) = \frac{1}{2 - 1} (\frac{4 \cdot 4}{4} - \frac{1}{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.8

Combine the numerators over the common denominator.

$A (x) = \frac{1}{2 - 1} (\frac{4 \cdot 4 - 1}{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.9

Simplify the numerator.

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Step 12.2.4.9.1

Multiply $4$ by $4$ .

$A (x) = \frac{1}{2 - 1} (\frac{16 - 1}{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.9.2

Subtract $1$ from $16$ .

$A (x) = \frac{1}{2 - 1} (\frac{15}{4} - 4 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.10

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

$A (x) = \frac{1}{2 - 1} (\frac{15}{4} - 4 (\frac{8}{3} - \frac{1^{3}}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.11

One to any power is one.

$A (x) = \frac{1}{2 - 1} (\frac{15}{4} - 4 (\frac{8}{3} - \frac{1}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.12

Combine the numerators over the common denominator.

$A (x) = \frac{1}{2 - 1} (\frac{15}{4} - 4 \frac{8 - 1}{3} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.13

Subtract $1$ from $8$ .

$A (x) = \frac{1}{2 - 1} (\frac{15}{4} - 4 (\frac{7}{3}) + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.14

Combine $- 4$ and $\frac{7}{3}$ .

$A (x) = \frac{1}{2 - 1} (\frac{15}{4} + \frac{- 4 \cdot 7}{3} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.15

Multiply $- 4$ by $7$ .

$A (x) = \frac{1}{2 - 1} (\frac{15}{4} + \frac{- 28}{3} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.16

Move the negative in front of the fraction.

$A (x) = \frac{1}{2 - 1} (\frac{15}{4} - \frac{28}{3} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.17

To write $\frac{15}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A (x) = \frac{1}{2 - 1} (\frac{15}{4} \cdot \frac{3}{3} - \frac{28}{3} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.18

To write $- \frac{28}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$A (x) = \frac{1}{2 - 1} (\frac{15}{4} \cdot \frac{3}{3} - \frac{28}{3} \cdot \frac{4}{4} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.19

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.2.4.19.1

Multiply $\frac{15}{4}$ by $\frac{3}{3}$ .

$A (x) = \frac{1}{2 - 1} (\frac{15 \cdot 3}{4 \cdot 3} - \frac{28}{3} \cdot \frac{4}{4} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.19.2

Multiply $4$ by $3$ .

$A (x) = \frac{1}{2 - 1} (\frac{15 \cdot 3}{12} - \frac{28}{3} \cdot \frac{4}{4} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.19.3

Multiply $\frac{28}{3}$ by $\frac{4}{4}$ .

$A (x) = \frac{1}{2 - 1} (\frac{15 \cdot 3}{12} - \frac{28 \cdot 4}{3 \cdot 4} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.19.4

Multiply $3$ by $4$ .

$A (x) = \frac{1}{2 - 1} (\frac{15 \cdot 3}{12} - \frac{28 \cdot 4}{12} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.20

Combine the numerators over the common denominator.

$A (x) = \frac{1}{2 - 1} (\frac{15 \cdot 3 - 28 \cdot 4}{12} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.21

Simplify the numerator.

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Step 12.2.4.21.1

Multiply $15$ by $3$ .

$A (x) = \frac{1}{2 - 1} (\frac{45 - 28 \cdot 4}{12} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.21.2

Multiply $- 28$ by $4$ .

$A (x) = \frac{1}{2 - 1} (\frac{45 - 112}{12} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.21.3

Subtract $112$ from $45$ .

$A (x) = \frac{1}{2 - 1} (\frac{- 67}{12} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.22

Move the negative in front of the fraction.

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + 5 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}))$

Step 12.2.4.23

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

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + 5 (\frac{4}{2} - \frac{1^{2}}{2}))$

Step 12.2.4.24

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

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Step 12.2.4.24.1

Factor $2$ out of $4$ .

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + 5 (\frac{2 \cdot 2}{2} - \frac{1^{2}}{2}))$

Step 12.2.4.24.2

Cancel the common factors.

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Step 12.2.4.24.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + 5 (\frac{2 \cdot 2}{2 (1)} - \frac{1^{2}}{2}))$

Step 12.2.4.24.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + 5 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{1^{2}}{2}))$

Step 12.2.4.24.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + 5 (\frac{2}{1} - \frac{1^{2}}{2}))$

Step 12.2.4.24.2.4

Divide $2$ by $1$ .

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + 5 (2 - \frac{1^{2}}{2}))$

Step 12.2.4.25

One to any power is one.

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + 5 (2 - \frac{1}{2}))$

Step 12.2.4.26

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + 5 (2 \cdot \frac{2}{2} - \frac{1}{2}))$

Step 12.2.4.27

Combine $2$ and $\frac{2}{2}$ .

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + 5 (\frac{2 \cdot 2}{2} - \frac{1}{2}))$

Step 12.2.4.28

Combine the numerators over the common denominator.

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + 5 (\frac{2 \cdot 2 - 1}{2}))$

Step 12.2.4.29

Simplify the numerator.

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Step 12.2.4.29.1

Multiply $2$ by $2$ .

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + 5 (\frac{4 - 1}{2}))$

Step 12.2.4.29.2

Subtract $1$ from $4$ .

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + 5 (\frac{3}{2}))$

Step 12.2.4.30

Combine $5$ and $\frac{3}{2}$ .

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + \frac{5 \cdot 3}{2})$

Step 12.2.4.31

Multiply $5$ by $3$ .

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + \frac{15}{2})$

Step 12.2.4.32

To write $\frac{15}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + \frac{15}{2} \cdot \frac{6}{6})$

Step 12.2.4.33

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.2.4.33.1

Multiply $\frac{15}{2}$ by $\frac{6}{6}$ .

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + \frac{15 \cdot 6}{2 \cdot 6})$

Step 12.2.4.33.2

Multiply $2$ by $6$ .

$A (x) = \frac{1}{2 - 1} (- \frac{67}{12} + \frac{15 \cdot 6}{12})$

Step 12.2.4.34

Combine the numerators over the common denominator.

$A (x) = \frac{1}{2 - 1} (\frac{- 67 + 15 \cdot 6}{12})$

Step 12.2.4.35

Simplify the numerator.

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Step 12.2.4.35.1

Multiply $15$ by $6$ .

$A (x) = \frac{1}{2 - 1} (\frac{- 67 + 90}{12})$

Step 12.2.4.35.2

Add $- 67$ and $90$ .

$A (x) = \frac{1}{2 - 1} (\frac{23}{12})$

Step 13

Subtract $1$ from $2$ .

$A (x) = \frac{1}{1} \cdot \frac{23}{12}$

Step 14

Cancel the common factor of $1$ .

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Step 14.1

Cancel the common factor.

$A (x) = \frac{1}{1} \cdot \frac{23}{12}$

Step 14.2

Rewrite the expression.

$A (x) = 1 \cdot \frac{23}{12}$

Step 15

Multiply $\frac{23}{12}$ by $1$ .

$A (x) = \frac{23}{12}$

Step 16