Calculus Examples

$y = x^{3}$ , $[0, 5]$

Step 1

Write $y = x^{3}$ as a function.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

$3 x^{2}$

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 $[0, 5]$ .

$f' (x)$ 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.

$A (x) = \frac{1}{5 - 0} (\int_{0}^{5} 3 x^{2} d x)$

Step 7

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

$A (x) = \frac{1}{5 - 0} (3 \int_{0}^{5} x^{2} d x)$

Step 8

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

$A (x) = \frac{1}{5 - 0} (3 (\frac{1}{3} x^{3}]_{0}^{5}))$

Step 9

Simplify the answer.

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Step 9.1

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

$A (x) = \frac{1}{5 - 0} (3 (\frac{x^{3}}{3}]_{0}^{5}))$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $\frac{x^{3}}{3}$ at $5$ and at $0$ .

$A (x) = \frac{1}{5 - 0} (3 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3}))$

Step 9.2.2

Simplify.

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Step 9.2.2.1

Raise $5$ to the power of $3$ .

$A (x) = \frac{1}{5 - 0} (3 (\frac{125}{3} - \frac{0^{3}}{3}))$

Step 9.2.2.2

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

$A (x) = \frac{1}{5 - 0} (3 (\frac{125}{3} - \frac{0}{3}))$

Step 9.2.2.3

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

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Step 9.2.2.3.1

Factor $3$ out of $0$ .

$A (x) = \frac{1}{5 - 0} (3 (\frac{125}{3} - \frac{3 (0)}{3}))$

Step 9.2.2.3.2

Cancel the common factors.

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Step 9.2.2.3.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{5 - 0} (3 (\frac{125}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 9.2.2.3.2.2

Cancel the common factor.

$A (x) = \frac{1}{5 - 0} (3 (\frac{125}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 9.2.2.3.2.3

Rewrite the expression.

$A (x) = \frac{1}{5 - 0} (3 (\frac{125}{3} - \frac{0}{1}))$

Step 9.2.2.3.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{5 - 0} (3 (\frac{125}{3} - 0))$

Step 9.2.2.4

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{5 - 0} (3 (\frac{125}{3} + 0))$

Step 9.2.2.5

Add $\frac{125}{3}$ and $0$ .

$A (x) = \frac{1}{5 - 0} (3 (\frac{125}{3}))$

Step 9.2.2.6

Combine $3$ and $\frac{125}{3}$ .

$A (x) = \frac{1}{5 - 0} (\frac{3 \cdot 125}{3})$

Step 9.2.2.7

Multiply $3$ by $125$ .

$A (x) = \frac{1}{5 - 0} (\frac{375}{3})$

Step 9.2.2.8

Cancel the common factor of $375$ and $3$ .

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Step 9.2.2.8.1

Factor $3$ out of $375$ .

$A (x) = \frac{1}{5 - 0} (\frac{3 \cdot 125}{3})$

Step 9.2.2.8.2

Cancel the common factors.

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Step 9.2.2.8.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{5 - 0} (\frac{3 \cdot 125}{3 (1)})$

Step 9.2.2.8.2.2

Cancel the common factor.

$A (x) = \frac{1}{5 - 0} (\frac{3 \cdot 125}{3 \cdot 1})$

Step 9.2.2.8.2.3

Rewrite the expression.

$A (x) = \frac{1}{5 - 0} (\frac{125}{1})$

Step 9.2.2.8.2.4

Divide $125$ by $1$ .

$A (x) = \frac{1}{5 - 0} (125)$

Step 10

Simplify the denominator.

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Step 10.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{5 + 0} \cdot 125$

Step 10.2

Add $5$ and $0$ .

$A (x) = \frac{1}{5} \cdot 125$

Step 11

Cancel the common factor of $5$ .

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Step 11.1

Factor $5$ out of $125$ .

$A (x) = \frac{1}{5} \cdot (5 (25))$

Step 11.2

Cancel the common factor.

$A (x) = \frac{1}{5} \cdot (5 \cdot 25)$

Step 11.3

Rewrite the expression.

$A (x) = 25$

Step 12