Calculus Examples

$2 x^{3} - 3 x^{2} - 36 x + 5$

Step 1

Find the point at $x = - 1$ .

Tap for more steps...

Step 1.1

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

$f (- 1) = 2 {(- 1)}^{3} - 3 {(- 1)}^{2} - 36 \cdot - 1 + 5$

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

$f (- 1) = 2 \cdot - 1 - 3 {(- 1)}^{2} - 36 \cdot - 1 + 5$

Step 1.2.1.2

Multiply $2$ by $- 1$ .

$f (- 1) = - 2 - 3 {(- 1)}^{2} - 36 \cdot - 1 + 5$

Step 1.2.1.3

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

$f (- 1) = - 2 - 3 \cdot 1 - 36 \cdot - 1 + 5$

Step 1.2.1.4

Multiply $- 3$ by $1$ .

$f (- 1) = - 2 - 3 - 36 \cdot - 1 + 5$

Step 1.2.1.5

Multiply $- 36$ by $- 1$ .

$f (- 1) = - 2 - 3 + 36 + 5$

Step 1.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 1.2.2.1

Subtract $3$ from $- 2$ .

$f (- 1) = - 5 + 36 + 5$

Step 1.2.2.2

Add $- 5$ and $36$ .

$f (- 1) = 31 + 5$

Step 1.2.2.3

Add $31$ and $5$ .

$f (- 1) = 36$

Step 1.2.3

The final answer is $36$ .

$36$

Step 1.3

Convert $36$ to decimal.

$y = 36$

Step 2

Find the point at $x = 0$ .

Tap for more steps...

Step 2.1

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

$f (0) = 2 {(0)}^{3} - 3 {(0)}^{2} - 36 \cdot 0 + 5$

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

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

$f (0) = 2 \cdot 0 - 3 {(0)}^{2} - 36 \cdot 0 + 5$

Step 2.2.1.2

Multiply $2$ by $0$ .

$f (0) = 0 - 3 {(0)}^{2} - 36 \cdot 0 + 5$

Step 2.2.1.3

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

$f (0) = 0 - 3 \cdot 0 - 36 \cdot 0 + 5$

Step 2.2.1.4

Multiply $- 3$ by $0$ .

$f (0) = 0 + 0 - 36 \cdot 0 + 5$

Step 2.2.1.5

Multiply $- 36$ by $0$ .

$f (0) = 0 + 0 + 0 + 5$

Step 2.2.2

Simplify by adding numbers.

Tap for more steps...

Step 2.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 + 5$

Step 2.2.2.2

Add $0$ and $0$ .

$f (0) = 0 + 5$

Step 2.2.2.3

Add $0$ and $5$ .

$f (0) = 5$

Step 2.2.3

The final answer is $5$ .

$5$

Step 2.3

Convert $5$ to decimal.

$y = 5$

Step 3

Find the point at $x = 1$ .

Tap for more steps...

Step 3.1

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

$f (1) = 2 {(1)}^{3} - 3 {(1)}^{2} - 36 \cdot 1 + 5$

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

One to any power is one.

$f (1) = 2 \cdot 1 - 3 {(1)}^{2} - 36 \cdot 1 + 5$

Step 3.2.1.2

Multiply $2$ by $1$ .

$f (1) = 2 - 3 {(1)}^{2} - 36 \cdot 1 + 5$

Step 3.2.1.3

One to any power is one.

$f (1) = 2 - 3 \cdot 1 - 36 \cdot 1 + 5$

Step 3.2.1.4

Multiply $- 3$ by $1$ .

$f (1) = 2 - 3 - 36 \cdot 1 + 5$

Step 3.2.1.5

Multiply $- 36$ by $1$ .

$f (1) = 2 - 3 - 36 + 5$

Step 3.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 3.2.2.1

Subtract $3$ from $2$ .

$f (1) = - 1 - 36 + 5$

Step 3.2.2.2

Subtract $36$ from $- 1$ .

$f (1) = - 37 + 5$

Step 3.2.2.3

Add $- 37$ and $5$ .

$f (1) = - 32$

Step 3.2.3

The final answer is $- 32$ .

$- 32$

Step 3.3

Convert $- 32$ to decimal.

$y = - 32$

Step 4

Find the point at $x = 2$ .

Tap for more steps...

Step 4.1

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

$f (2) = 2 {(2)}^{3} - 3 {(2)}^{2} - 36 \cdot 2 + 5$

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

Multiply $2$ by ${(2)}^{3}$ by adding the exponents.

Tap for more steps...

Step 4.2.1.1.1

Multiply $2$ by ${(2)}^{3}$ .

Tap for more steps...

Step 4.2.1.1.1.1

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

$f (2) = 2 {(2)}^{3} - 3 {(2)}^{2} - 36 \cdot 2 + 5$

Step 4.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (2) = 2^{1 + 3} - 3 {(2)}^{2} - 36 \cdot 2 + 5$

Step 4.2.1.1.2

Add $1$ and $3$ .

$f (2) = 2^{4} - 3 {(2)}^{2} - 36 \cdot 2 + 5$

Step 4.2.1.2

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

$f (2) = 16 - 3 {(2)}^{2} - 36 \cdot 2 + 5$

Step 4.2.1.3

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

$f (2) = 16 - 3 \cdot 4 - 36 \cdot 2 + 5$

Step 4.2.1.4

Multiply $- 3$ by $4$ .

$f (2) = 16 - 12 - 36 \cdot 2 + 5$

Step 4.2.1.5

Multiply $- 36$ by $2$ .

$f (2) = 16 - 12 - 72 + 5$

Step 4.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.2.1

Subtract $12$ from $16$ .

$f (2) = 4 - 72 + 5$

Step 4.2.2.2

Subtract $72$ from $4$ .

$f (2) = - 68 + 5$

Step 4.2.2.3

Add $- 68$ and $5$ .

$f (2) = - 63$

Step 4.2.3

The final answer is $- 63$ .

$- 63$

Step 4.3

Convert $- 63$ to decimal.

$y = - 63$

Step 5

Find the point at $x = 3$ .

Tap for more steps...

Step 5.1

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

$f (3) = 2 {(3)}^{3} - 3 {(3)}^{2} - 36 \cdot 3 + 5$

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 $3$ to the power of $3$ .

$f (3) = 2 \cdot 27 - 3 {(3)}^{2} - 36 \cdot 3 + 5$

Step 5.2.1.2

Multiply $2$ by $27$ .

$f (3) = 54 - 3 {(3)}^{2} - 36 \cdot 3 + 5$

Step 5.2.1.3

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

$f (3) = 54 - 3 \cdot 9 - 36 \cdot 3 + 5$

Step 5.2.1.4

Multiply $- 3$ by $9$ .

$f (3) = 54 - 27 - 36 \cdot 3 + 5$

Step 5.2.1.5

Multiply $- 36$ by $3$ .

$f (3) = 54 - 27 - 108 + 5$

Step 5.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 5.2.2.1

Subtract $27$ from $54$ .

$f (3) = 27 - 108 + 5$

Step 5.2.2.2

Subtract $108$ from $27$ .

$f (3) = - 81 + 5$

Step 5.2.2.3

Add $- 81$ and $5$ .

$f (3) = - 76$

Step 5.2.3

The final answer is $- 76$ .

$- 76$

Step 5.3

Convert $- 76$ to decimal.

$y = - 76$

Step 6

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

$\begin{array}{c} x & y \\ - 1 & 36 \\ 0 & 5 \\ 1 & - 32 \\ 2 & - 63 \\ 3 & - 76 \end{array}$

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

$\begin{array}{c} x & y \\ - 1 & 36 \\ 0 & 5 \\ 1 & - 32 \\ 2 & - 63 \\ 3 & - 76 \end{array}$

Step 8