Calculus Examples

$f (x) = 3 x^{3} - 36 x$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

$f (- 2) = 3 {(- 2)}^{3} - 36 \cdot - 2$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

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

$f (- 2) = 3 \cdot - 8 - 36 \cdot - 2$

Step 1.2.1.2

Multiply $3$ by $- 8$ .

$f (- 2) = - 24 - 36 \cdot - 2$

Step 1.2.1.3

Multiply $- 36$ by $- 2$ .

$f (- 2) = - 24 + 72$

Step 1.2.2

Add $- 24$ and $72$ .

$f (- 2) = 48$

Step 1.2.3

The final answer is $48$ .

$48$

Step 1.3

Convert $48$ to decimal.

$y = 48$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

$f (- 1) = 3 {(- 1)}^{3} - 36 \cdot - 1$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

$f (- 1) = 3 \cdot - 1 - 36 \cdot - 1$

Step 2.2.1.2

Multiply $3$ by $- 1$ .

$f (- 1) = - 3 - 36 \cdot - 1$

Step 2.2.1.3

Multiply $- 36$ by $- 1$ .

$f (- 1) = - 3 + 36$

Step 2.2.2

Add $- 3$ and $36$ .

$f (- 1) = 33$

Step 2.2.3

The final answer is $33$ .

$33$

Step 2.3

Convert $33$ to decimal.

$y = 33$

Step 3

Find the point at $x = 0$ .

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Step 3.1

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

$f (0) = 3 {(0)}^{3} - 36 \cdot 0$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

$f (0) = 3 \cdot 0 - 36 \cdot 0$

Step 3.2.1.2

Multiply $3$ by $0$ .

$f (0) = 0 - 36 \cdot 0$

Step 3.2.1.3

Multiply $- 36$ by $0$ .

$f (0) = 0 + 0$

Step 3.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 3.2.3

The final answer is $0$ .

$0$

Step 3.3

Convert $0$ to decimal.

$y = 0$

Step 4

Find the point at $x = 1$ .

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Step 4.1

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

$f (1) = 3 {(1)}^{3} - 36 \cdot 1$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

One to any power is one.

$f (1) = 3 \cdot 1 - 36 \cdot 1$

Step 4.2.1.2

Multiply $3$ by $1$ .

$f (1) = 3 - 36 \cdot 1$

Step 4.2.1.3

Multiply $- 36$ by $1$ .

$f (1) = 3 - 36$

Step 4.2.2

Subtract $36$ from $3$ .

$f (1) = - 33$

Step 4.2.3

The final answer is $- 33$ .

$- 33$

Step 4.3

Convert $- 33$ to decimal.

$y = - 33$

Step 5

Find the point at $x = 2$ .

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Step 5.1

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

$f (2) = 3 {(2)}^{3} - 36 \cdot 2$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f (2) = 3 \cdot 8 - 36 \cdot 2$

Step 5.2.1.2

Multiply $3$ by $8$ .

$f (2) = 24 - 36 \cdot 2$

Step 5.2.1.3

Multiply $- 36$ by $2$ .

$f (2) = 24 - 72$

Step 5.2.2

Subtract $72$ from $24$ .

$f (2) = - 48$

Step 5.2.3

The final answer is $- 48$ .

$- 48$

Step 5.3

Convert $- 48$ to decimal.

$y = - 48$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & 48 \\ - 1 & 33 \\ 0 & 0 \\ 1 & - 33 \\ 2 & - 48 \end{array}$

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

$\begin{array}{c} x & y \\ - 2 & 48 \\ - 1 & 33 \\ 0 & 0 \\ 1 & - 33 \\ 2 & - 48 \end{array}$

Step 8