Calculus Examples

$y = 3 x^{\frac{2}{3}} - 2 x$ , $[- 1, 1]$

Step 1

Write $y = 3 x^{\frac{2}{3}} - 2 x$ as a function.

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

Step 2

To find the average value of a function, the function should be continuous on the closed interval $[a, b]$ . To find whether $f (x)$ is continuous on $[- 1, 1]$ or not, find the domain of $f (x) = 3 x^{\frac{2}{3}} - 2 x$ .

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Step 2.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$3 \sqrt[3]{x^{2}} - 2 x$

Step 2.2

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:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

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

$f (x)$ 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.

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

By the Power Rule, the integral of $x^{\frac{2}{3}}$ with respect to $x$ is $\frac{3}{5} x^{\frac{5}{3}}$ .

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

Step 9

Combine $\frac{3}{5}$ and $x^{\frac{5}{3}}$ .

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

Step 10

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

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

Step 11

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

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

Step 12

Simplify the answer.

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Step 12.1

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

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

Step 12.2

Substitute and simplify.

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Step 12.2.1

Evaluate $\frac{3 x^{\frac{5}{3}}}{5}$ at $1$ and at $- 1$ .

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

Step 12.2.2

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

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

Step 12.2.3

Simplify.

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Step 12.2.3.1

One to any power is one.

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

Step 12.2.3.2

Multiply $3$ by $1$ .

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

Step 12.2.3.3

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 12.2.3.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 12.2.3.5

Cancel the common factor of $3$ .

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Step 12.2.3.5.1

Cancel the common factor.

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

Step 12.2.3.5.2

Rewrite the expression.

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

Step 12.2.3.6

Raise $- 1$ to the power of $5$ .

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

Step 12.2.3.7

Multiply $3$ by $- 1$ .

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

Step 12.2.3.8

Move the negative in front of the fraction.

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

Step 12.2.3.9

Multiply $- 1$ by $- 1$ .

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

Step 12.2.3.10

Multiply $\frac{3}{5}$ by $1$ .

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

Step 12.2.3.11

Combine the numerators over the common denominator.

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

Step 12.2.3.12

Add $3$ and $3$ .

$A (x) = \frac{1}{1 + 1} (3 (\frac{6}{5}) - 2 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 12.2.3.13

Combine $3$ and $\frac{6}{5}$ .

$A (x) = \frac{1}{1 + 1} (\frac{3 \cdot 6}{5} - 2 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 12.2.3.14

Multiply $3$ by $6$ .

$A (x) = \frac{1}{1 + 1} (\frac{18}{5} - 2 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 12.2.3.15

One to any power is one.

$A (x) = \frac{1}{1 + 1} (\frac{18}{5} - 2 (\frac{1}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 12.2.3.16

Raise $- 1$ to the power of $2$ .

$A (x) = \frac{1}{1 + 1} (\frac{18}{5} - 2 (\frac{1}{2} - \frac{1}{2}))$

Step 12.2.3.17

Combine the numerators over the common denominator.

$A (x) = \frac{1}{1 + 1} (\frac{18}{5} - 2 \frac{1 - 1}{2})$

Step 12.2.3.18

Subtract $1$ from $1$ .

$A (x) = \frac{1}{1 + 1} (\frac{18}{5} - 2 (\frac{0}{2}))$

Step 12.2.3.19

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

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Step 12.2.3.19.1

Factor $2$ out of $0$ .

$A (x) = \frac{1}{1 + 1} (\frac{18}{5} - 2 \frac{2 (0)}{2})$

Step 12.2.3.19.2

Cancel the common factors.

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Step 12.2.3.19.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{1 + 1} (\frac{18}{5} - 2 \frac{2 \cdot 0}{2 \cdot 1})$

Step 12.2.3.19.2.2

Cancel the common factor.

$A (x) = \frac{1}{1 + 1} (\frac{18}{5} - 2 \frac{2 \cdot 0}{2 \cdot 1})$

Step 12.2.3.19.2.3

Rewrite the expression.

$A (x) = \frac{1}{1 + 1} (\frac{18}{5} - 2 (\frac{0}{1}))$

Step 12.2.3.19.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{1 + 1} (\frac{18}{5} - 2 \cdot 0)$

Step 12.2.3.20

Multiply $- 2$ by $0$ .

$A (x) = \frac{1}{1 + 1} (\frac{18}{5} + 0)$

Step 12.2.3.21

Add $\frac{18}{5}$ and $0$ .

$A (x) = \frac{1}{1 + 1} (\frac{18}{5})$

Step 13

Add $1$ and $1$ .

$A (x) = \frac{1}{2} \cdot \frac{18}{5}$

Step 14

Cancel the common factor of $2$ .

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Step 14.1

Factor $2$ out of $18$ .

$A (x) = \frac{1}{2} \cdot \frac{2 (9)}{5}$

Step 14.2

Cancel the common factor.

$A (x) = \frac{1}{2} \cdot \frac{2 \cdot 9}{5}$

Step 14.3

Rewrite the expression.

$A (x) = \frac{9}{5}$

Step 15