Calculus Examples

$y = \frac{x^{3}}{3}$

Step 1

Find the point at $x = 0$ .

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Step 1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{{(0)}^{3}}{3}$

Step 1.2

Simplify the result.

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Step 1.2.1

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

$f (0) = \frac{0}{3}$

Step 1.2.2

Divide $0$ by $3$ .

$f (0) = 0$

Step 1.2.3

The final answer is $0$ .

$0$

Step 1.3

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = 3$ .

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Step 2.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{{(3)}^{3}}{3}$

Step 2.2

Simplify the result.

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Step 2.2.1

Cancel the common factor of ${(3)}^{3}$ and $3$ .

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Step 2.2.1.1

Factor $3$ out of ${(3)}^{3}$ .

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

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

Factor $3$ out of $3$ .

$f (3) = \frac{3 \cdot 3^{2}}{3 (1)}$

Step 2.2.1.2.2

Cancel the common factor.

$f (3) = \frac{3 \cdot 3^{2}}{3 \cdot 1}$

Step 2.2.1.2.3

Rewrite the expression.

$f (3) = \frac{3^{2}}{1}$

Step 2.2.1.2.4

Divide $3^{2}$ by $1$ .

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

Step 2.2.2

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

$f (3) = 9$

Step 2.2.3

The final answer is $9$ .

$9$

Step 2.3

Convert $9$ to decimal.

$y = 9$

Step 3

Find the point at $x = - 2$ .

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Step 3.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 3.2

Simplify the result.

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Step 3.2.1

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

$f (- 2) = \frac{- 8}{3}$

Step 3.2.2

Move the negative in front of the fraction.

$f (- 2) = - \frac{8}{3}$

Step 3.2.3

The final answer is $- \frac{8}{3}$ .

$- \frac{8}{3}$

Step 3.3

Convert $- \frac{8}{3}$ to decimal.

$y = - 2.\overline{6}$

Step 4

Find the point at $x = - 1$ .

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Step 4.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = \frac{{(- 1)}^{3}}{3}$

Step 4.2

Simplify the result.

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Step 4.2.1

Raise $- 1$ to the power of $3$ .

$f (- 1) = \frac{- 1}{3}$

Step 4.2.2

Move the negative in front of the fraction.

$f (- 1) = - \frac{1}{3}$

Step 4.2.3

The final answer is $- \frac{1}{3}$ .

$- \frac{1}{3}$

Step 4.3

Convert $- \frac{1}{3}$ to decimal.

$y = - 0.\overline{3}$

Step 5

Find the point at $x = 1$ .

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Step 5.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{{(1)}^{3}}{3}$

Step 5.2

Simplify the result.

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Step 5.2.1

One to any power is one.

$f (1) = \frac{1}{3}$

Step 5.2.2

The final answer is $\frac{1}{3}$ .

$\frac{1}{3}$

Step 5.3

Convert $\frac{1}{3}$ to decimal.

$y = 0.\overline{3}$

Step 6

The cubic function can be graphed using the function behavior and the points.

$\begin{array}{c} x & y \\ - 2 & - 2.667 \\ - 1 & - 0.333 \\ 0 & 0 \\ 1 & 0.333 \\ 3 & 9 \end{array}$

Step 7

The cubic function can be graphed using the function behavior and the selected points.

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 2 & - 2.667 \\ - 1 & - 0.333 \\ 0 & 0 \\ 1 & 0.333 \\ 3 & 9 \end{array}$

Step 8