Calculus Examples

$f (x) = 12 x^{2} - x^{3}$

Step 1

Find the point at $x = 2$ .

Tap for more steps...

Step 1.1

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

$f (2) = 12 {(2)}^{2} - {(2)}^{3}$

Step 1.2

Simplify the result.

Tap for more steps...

Step 1.2.1

Simplify each term.

Tap for more steps...

Step 1.2.1.1

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

$f (2) = 12 \cdot 4 - {(2)}^{3}$

Step 1.2.1.2

Multiply $12$ by $4$ .

$f (2) = 48 - {(2)}^{3}$

Step 1.2.1.3

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

$f (2) = 48 - 1 \cdot 8$

Step 1.2.1.4

Multiply $- 1$ by $8$ .

$f (2) = 48 - 8$

Step 1.2.2

Subtract $8$ from $48$ .

$f (2) = 40$

Step 1.2.3

The final answer is $40$ .

$40$

Step 1.3

Convert $40$ to decimal.

$y = 40$

Step 2

Find the point at $x = 3$ .

Tap for more steps...

Step 2.1

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

$f (3) = 12 {(3)}^{2} - {(3)}^{3}$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

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

$f (3) = 12 \cdot 9 - {(3)}^{3}$

Step 2.2.1.2

Multiply $12$ by $9$ .

$f (3) = 108 - {(3)}^{3}$

Step 2.2.1.3

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

$f (3) = 108 - 1 \cdot 27$

Step 2.2.1.4

Multiply $- 1$ by $27$ .

$f (3) = 108 - 27$

Step 2.2.2

Subtract $27$ from $108$ .

$f (3) = 81$

Step 2.2.3

The final answer is $81$ .

$81$

Step 2.3

Convert $81$ to decimal.

$y = 81$

Step 3

Find the point at $x = 4$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = 12 {(4)}^{2} - {(4)}^{3}$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

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

$f (4) = 12 \cdot 16 - {(4)}^{3}$

Step 3.2.1.2

Multiply $12$ by $16$ .

$f (4) = 192 - {(4)}^{3}$

Step 3.2.1.3

Raise $4$ to the power of $3$ .

$f (4) = 192 - 1 \cdot 64$

Step 3.2.1.4

Multiply $- 1$ by $64$ .

$f (4) = 192 - 64$

Step 3.2.2

Subtract $64$ from $192$ .

$f (4) = 128$

Step 3.2.3

The final answer is $128$ .

$128$

Step 3.3

Convert $128$ to decimal.

$y = 128$

Step 4

Find the point at $x = 5$ .

Tap for more steps...

Step 4.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = 12 {(5)}^{2} - {(5)}^{3}$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Raise $5$ to the power of $2$ .

$f (5) = 12 \cdot 25 - {(5)}^{3}$

Step 4.2.1.2

Multiply $12$ by $25$ .

$f (5) = 300 - {(5)}^{3}$

Step 4.2.1.3

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

$f (5) = 300 - 1 \cdot 125$

Step 4.2.1.4

Multiply $- 1$ by $125$ .

$f (5) = 300 - 125$

Step 4.2.2

Subtract $125$ from $300$ .

$f (5) = 175$

Step 4.2.3

The final answer is $175$ .

$175$

Step 4.3

Convert $175$ to decimal.

$y = 175$

Step 5

Find the point at $x = 6$ .

Tap for more steps...

Step 5.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = 12 {(6)}^{2} - {(6)}^{3}$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Raise $6$ to the power of $2$ .

$f (6) = 12 \cdot 36 - {(6)}^{3}$

Step 5.2.1.2

Multiply $12$ by $36$ .

$f (6) = 432 - {(6)}^{3}$

Step 5.2.1.3

Raise $6$ to the power of $3$ .

$f (6) = 432 - 1 \cdot 216$

Step 5.2.1.4

Multiply $- 1$ by $216$ .

$f (6) = 432 - 216$

Step 5.2.2

Subtract $216$ from $432$ .

$f (6) = 216$

Step 5.2.3

The final answer is $216$ .

$216$

Step 5.3

Convert $216$ to decimal.

$y = 216$

Step 6

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

$\begin{array}{c} x & y \\ 2 & 40 \\ 3 & 81 \\ 4 & 128 \\ 5 & 175 \\ 6 & 216 \end{array}$

Step 7

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

Rises to the left and falls to the right

$\begin{array}{c} x & y \\ 2 & 40 \\ 3 & 81 \\ 4 & 128 \\ 5 & 175 \\ 6 & 216 \end{array}$

Step 8