Calculus Examples

$2 x^{3} + 3 x^{2} - 72 x$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

$f (- 2) = 2 {(- 2)}^{3} + 3 {(- 2)}^{2} - 72 \cdot - 2$

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 $3$ .

$f (- 2) = 2 \cdot - 8 + 3 {(- 2)}^{2} - 72 \cdot - 2$

Step 1.2.1.2

Multiply $2$ by $- 8$ .

$f (- 2) = - 16 + 3 {(- 2)}^{2} - 72 \cdot - 2$

Step 1.2.1.3

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

$f (- 2) = - 16 + 3 \cdot 4 - 72 \cdot - 2$

Step 1.2.1.4

Multiply $3$ by $4$ .

$f (- 2) = - 16 + 12 - 72 \cdot - 2$

Step 1.2.1.5

Multiply $- 72$ by $- 2$ .

$f (- 2) = - 16 + 12 + 144$

Step 1.2.2

Simplify by adding numbers.

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Step 1.2.2.1

Add $- 16$ and $12$ .

$f (- 2) = - 4 + 144$

Step 1.2.2.2

Add $- 4$ and $144$ .

$f (- 2) = 140$

Step 1.2.3

The final answer is $140$ .

$140$

Step 1.3

Convert $140$ to decimal.

$y = 140$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

$f (- 1) = 2 {(- 1)}^{3} + 3 {(- 1)}^{2} - 72 \cdot - 1$

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 $3$ .

$f (- 1) = 2 \cdot - 1 + 3 {(- 1)}^{2} - 72 \cdot - 1$

Step 2.2.1.2

Multiply $2$ by $- 1$ .

$f (- 1) = - 2 + 3 {(- 1)}^{2} - 72 \cdot - 1$

Step 2.2.1.3

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

$f (- 1) = - 2 + 3 \cdot 1 - 72 \cdot - 1$

Step 2.2.1.4

Multiply $3$ by $1$ .

$f (- 1) = - 2 + 3 - 72 \cdot - 1$

Step 2.2.1.5

Multiply $- 72$ by $- 1$ .

$f (- 1) = - 2 + 3 + 72$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $- 2$ and $3$ .

$f (- 1) = 1 + 72$

Step 2.2.2.2

Add $1$ and $72$ .

$f (- 1) = 73$

Step 2.2.3

The final answer is $73$ .

$73$

Step 2.3

Convert $73$ to decimal.

$y = 73$

Step 3

Find the point at $x = 0$ .

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Step 3.1

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

$f (0) = 2 {(0)}^{3} + 3 {(0)}^{2} - 72 \cdot 0$

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 $0$ .

$f (0) = 2 \cdot 0 + 3 {(0)}^{2} - 72 \cdot 0$

Step 3.2.1.2

Multiply $2$ by $0$ .

$f (0) = 0 + 3 {(0)}^{2} - 72 \cdot 0$

Step 3.2.1.3

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

$f (0) = 0 + 3 \cdot 0 - 72 \cdot 0$

Step 3.2.1.4

Multiply $3$ by $0$ .

$f (0) = 0 + 0 - 72 \cdot 0$

Step 3.2.1.5

Multiply $- 72$ by $0$ .

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

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 3.2.2.2

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.

$f (1) = 2 {(1)}^{3} + 3 {(1)}^{2} - 72 \cdot 1$

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.

$f (1) = 2 \cdot 1 + 3 {(1)}^{2} - 72 \cdot 1$

Step 4.2.1.2

Multiply $2$ by $1$ .

$f (1) = 2 + 3 {(1)}^{2} - 72 \cdot 1$

Step 4.2.1.3

One to any power is one.

$f (1) = 2 + 3 \cdot 1 - 72 \cdot 1$

Step 4.2.1.4

Multiply $3$ by $1$ .

$f (1) = 2 + 3 - 72 \cdot 1$

Step 4.2.1.5

Multiply $- 72$ by $1$ .

$f (1) = 2 + 3 - 72$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $2$ and $3$ .

$f (1) = 5 - 72$

Step 4.2.2.2

Subtract $72$ from $5$ .

$f (1) = - 67$

Step 4.2.3

The final answer is $- 67$ .

$- 67$

Step 4.3

Convert $- 67$ to decimal.

$y = - 67$

Step 5

Find the point at $x = 2$ .

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Step 5.1

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

$f (2) = 2 {(2)}^{3} + 3 {(2)}^{2} - 72 \cdot 2$

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

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

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Step 5.2.1.1.1

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

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Step 5.2.1.1.1.1

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

$f (2) = 2 {(2)}^{3} + 3 {(2)}^{2} - 72 \cdot 2$

Step 5.2.1.1.1.2

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

$f (2) = 2^{1 + 3} + 3 {(2)}^{2} - 72 \cdot 2$

Step 5.2.1.1.2

Add $1$ and $3$ .

$f (2) = 2^{4} + 3 {(2)}^{2} - 72 \cdot 2$

Step 5.2.1.2

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

$f (2) = 16 + 3 {(2)}^{2} - 72 \cdot 2$

Step 5.2.1.3

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

$f (2) = 16 + 3 \cdot 4 - 72 \cdot 2$

Step 5.2.1.4

Multiply $3$ by $4$ .

$f (2) = 16 + 12 - 72 \cdot 2$

Step 5.2.1.5

Multiply $- 72$ by $2$ .

$f (2) = 16 + 12 - 144$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $16$ and $12$ .

$f (2) = 28 - 144$

Step 5.2.2.2

Subtract $144$ from $28$ .

$f (2) = - 116$

Step 5.2.3

The final answer is $- 116$ .

$- 116$

Step 5.3

Convert $- 116$ to decimal.

$y = - 116$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & 140 \\ - 1 & 73 \\ 0 & 0 \\ 1 & - 67 \\ 2 & - 116 \end{array}$

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

$\begin{array}{c} x & y \\ - 2 & 140 \\ - 1 & 73 \\ 0 & 0 \\ 1 & - 67 \\ 2 & - 116 \end{array}$

Step 8