Calculus Examples

$f (x) = \frac{1}{3} x^{3} + 3 x^{2} + 5 x$

Step 1

Find the point at $x = - 3$ .

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Step 1.1

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

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

Step 1.2

Simplify the result.

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Step 1.2.1

Find the common denominator.

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Step 1.2.1.1

Write $3 {(- 3)}^{2}$ as a fraction with denominator $1$ .

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

Step 1.2.1.2

Multiply $\frac{3 {(- 3)}^{2}}{1}$ by $\frac{3}{3}$ .

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

Step 1.2.1.3

Multiply $\frac{3 {(- 3)}^{2}}{1}$ by $\frac{3}{3}$ .

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

Step 1.2.1.4

Write $5 (- 3)$ as a fraction with denominator $1$ .

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

Step 1.2.1.5

Multiply $\frac{5 (- 3)}{1}$ by $\frac{3}{3}$ .

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

Step 1.2.1.6

Multiply $\frac{5 (- 3)}{1}$ by $\frac{3}{3}$ .

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 1.2.3

Simplify each term.

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

$f (- 3) = \frac{- 27 + 3 \cdot 9 \cdot 3 + 5 (- 3) \cdot 3}{3}$

Step 1.2.3.3

Multiply $3 \cdot 9 \cdot 3$ .

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Step 1.2.3.3.1

Multiply $3$ by $9$ .

$f (- 3) = \frac{- 27 + 27 \cdot 3 + 5 (- 3) \cdot 3}{3}$

Step 1.2.3.3.2

Multiply $27$ by $3$ .

$f (- 3) = \frac{- 27 + 81 + 5 (- 3) \cdot 3}{3}$

Step 1.2.3.4

Multiply $5 (- 3) \cdot 3$ .

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Step 1.2.3.4.1

Multiply $5$ by $- 3$ .

$f (- 3) = \frac{- 27 + 81 - 15 \cdot 3}{3}$

Step 1.2.3.4.2

Multiply $- 15$ by $3$ .

$f (- 3) = \frac{- 27 + 81 - 45}{3}$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $- 27$ and $81$ .

$f (- 3) = \frac{54 - 45}{3}$

Step 1.2.4.2

Subtract $45$ from $54$ .

$f (- 3) = \frac{9}{3}$

Step 1.2.4.3

Divide $9$ by $3$ .

$f (- 3) = 3$

Step 1.2.5

The final answer is $3$ .

$3$

Step 1.3

Convert $3$ to decimal.

$y = 3$

Step 2

Find the point at $x = 0$ .

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Step 2.1

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

$f (0) = \frac{{(0)}^{3}}{3} + 3 {(0)}^{2} + 5 (0)$

Step 2.2

Simplify the result.

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Step 2.2.1

Find the common denominator.

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Step 2.2.1.1

Write $3 {(0)}^{2}$ as a fraction with denominator $1$ .

$f (0) = \frac{{(0)}^{3}}{3} + \frac{3 {(0)}^{2}}{1} + 5 (0)$

Step 2.2.1.2

Multiply $\frac{3 {(0)}^{2}}{1}$ by $\frac{3}{3}$ .

$f (0) = \frac{{(0)}^{3}}{3} + \frac{3 {(0)}^{2}}{1} \cdot \frac{3}{3} + 5 (0)$

Step 2.2.1.3

Multiply $\frac{3 {(0)}^{2}}{1}$ by $\frac{3}{3}$ .

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

Step 2.2.1.4

Write $5 (0)$ as a fraction with denominator $1$ .

$f (0) = \frac{{(0)}^{3}}{3} + \frac{3 {(0)}^{2} \cdot 3}{3} + \frac{5 (0)}{1}$

Step 2.2.1.5

Multiply $\frac{5 (0)}{1}$ by $\frac{3}{3}$ .

$f (0) = \frac{{(0)}^{3}}{3} + \frac{3 {(0)}^{2} \cdot 3}{3} + \frac{5 (0)}{1} \cdot \frac{3}{3}$

Step 2.2.1.6

Multiply $\frac{5 (0)}{1}$ by $\frac{3}{3}$ .

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

Step 2.2.2

Combine the numerators over the common denominator.

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

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

$f (0) = \frac{0 + 3 \cdot 0 \cdot 3 + 5 (0) \cdot 3}{3}$

Step 2.2.3.3

Multiply $3 \cdot 0 \cdot 3$ .

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Step 2.2.3.3.1

Multiply $3$ by $0$ .

$f (0) = \frac{0 + 0 \cdot 3 + 5 (0) \cdot 3}{3}$

Step 2.2.3.3.2

Multiply $0$ by $3$ .

$f (0) = \frac{0 + 0 + 5 (0) \cdot 3}{3}$

Step 2.2.3.4

Multiply $5 (0) \cdot 3$ .

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Step 2.2.3.4.1

Multiply $5$ by $0$ .

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

Step 2.2.3.4.2

Multiply $0$ by $3$ .

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

Step 2.2.4

Simplify the expression.

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Step 2.2.4.1

Add $0$ and $0$ .

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

Step 2.2.4.2

Add $0$ and $0$ .

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

Step 2.2.4.3

Divide $0$ by $3$ .

$f (0) = 0$

Step 2.2.5

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = - 5$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Find the common denominator.

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Step 3.2.1.1

Write $3 {(- 5)}^{2}$ as a fraction with denominator $1$ .

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

Step 3.2.1.2

Multiply $\frac{3 {(- 5)}^{2}}{1}$ by $\frac{3}{3}$ .

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

Step 3.2.1.3

Multiply $\frac{3 {(- 5)}^{2}}{1}$ by $\frac{3}{3}$ .

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

Step 3.2.1.4

Write $5 (- 5)$ as a fraction with denominator $1$ .

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

Step 3.2.1.5

Multiply $\frac{5 (- 5)}{1}$ by $\frac{3}{3}$ .

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

Step 3.2.1.6

Multiply $\frac{5 (- 5)}{1}$ by $\frac{3}{3}$ .

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

Step 3.2.2

Combine the numerators over the common denominator.

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

Step 3.2.3

Simplify each term.

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Step 3.2.3.1

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

$f (- 5) = \frac{- 125 + 3 {(- 5)}^{2} \cdot 3 + 5 (- 5) \cdot 3}{3}$

Step 3.2.3.2

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

$f (- 5) = \frac{- 125 + 3 \cdot 25 \cdot 3 + 5 (- 5) \cdot 3}{3}$

Step 3.2.3.3

Multiply $3 \cdot 25 \cdot 3$ .

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Step 3.2.3.3.1

Multiply $3$ by $25$ .

$f (- 5) = \frac{- 125 + 75 \cdot 3 + 5 (- 5) \cdot 3}{3}$

Step 3.2.3.3.2

Multiply $75$ by $3$ .

$f (- 5) = \frac{- 125 + 225 + 5 (- 5) \cdot 3}{3}$

Step 3.2.3.4

Multiply $5 (- 5) \cdot 3$ .

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Step 3.2.3.4.1

Multiply $5$ by $- 5$ .

$f (- 5) = \frac{- 125 + 225 - 25 \cdot 3}{3}$

Step 3.2.3.4.2

Multiply $- 25$ by $3$ .

$f (- 5) = \frac{- 125 + 225 - 75}{3}$

Step 3.2.4

Simplify by adding and subtracting.

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Step 3.2.4.1

Add $- 125$ and $225$ .

$f (- 5) = \frac{100 - 75}{3}$

Step 3.2.4.2

Subtract $75$ from $100$ .

$f (- 5) = \frac{25}{3}$

Step 3.2.5

The final answer is $\frac{25}{3}$ .

$\frac{25}{3}$

Step 3.3

Convert $\frac{25}{3}$ to decimal.

$y = 8.\overline{3}$

Step 4

Find the point at $x = - 4$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Find the common denominator.

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Step 4.2.1.1

Write $3 {(- 4)}^{2}$ as a fraction with denominator $1$ .

$f (- 4) = \frac{{(- 4)}^{3}}{3} + \frac{3 {(- 4)}^{2}}{1} + 5 (- 4)$

Step 4.2.1.2

Multiply $\frac{3 {(- 4)}^{2}}{1}$ by $\frac{3}{3}$ .

$f (- 4) = \frac{{(- 4)}^{3}}{3} + \frac{3 {(- 4)}^{2}}{1} \cdot \frac{3}{3} + 5 (- 4)$

Step 4.2.1.3

Multiply $\frac{3 {(- 4)}^{2}}{1}$ by $\frac{3}{3}$ .

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

Step 4.2.1.4

Write $5 (- 4)$ as a fraction with denominator $1$ .

$f (- 4) = \frac{{(- 4)}^{3}}{3} + \frac{3 {(- 4)}^{2} \cdot 3}{3} + \frac{5 (- 4)}{1}$

Step 4.2.1.5

Multiply $\frac{5 (- 4)}{1}$ by $\frac{3}{3}$ .

$f (- 4) = \frac{{(- 4)}^{3}}{3} + \frac{3 {(- 4)}^{2} \cdot 3}{3} + \frac{5 (- 4)}{1} \cdot \frac{3}{3}$

Step 4.2.1.6

Multiply $\frac{5 (- 4)}{1}$ by $\frac{3}{3}$ .

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

Step 4.2.2

Combine the numerators over the common denominator.

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

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

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

$f (- 4) = \frac{- 64 + 3 {(- 4)}^{2} \cdot 3 + 5 (- 4) \cdot 3}{3}$

Step 4.2.3.2

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

$f (- 4) = \frac{- 64 + 3 \cdot 16 \cdot 3 + 5 (- 4) \cdot 3}{3}$

Step 4.2.3.3

Multiply $3 \cdot 16 \cdot 3$ .

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Step 4.2.3.3.1

Multiply $3$ by $16$ .

$f (- 4) = \frac{- 64 + 48 \cdot 3 + 5 (- 4) \cdot 3}{3}$

Step 4.2.3.3.2

Multiply $48$ by $3$ .

$f (- 4) = \frac{- 64 + 144 + 5 (- 4) \cdot 3}{3}$

Step 4.2.3.4

Multiply $5 (- 4) \cdot 3$ .

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Step 4.2.3.4.1

Multiply $5$ by $- 4$ .

$f (- 4) = \frac{- 64 + 144 - 20 \cdot 3}{3}$

Step 4.2.3.4.2

Multiply $- 20$ by $3$ .

$f (- 4) = \frac{- 64 + 144 - 60}{3}$

Step 4.2.4

Simplify by adding and subtracting.

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Step 4.2.4.1

Add $- 64$ and $144$ .

$f (- 4) = \frac{80 - 60}{3}$

Step 4.2.4.2

Subtract $60$ from $80$ .

$f (- 4) = \frac{20}{3}$

Step 4.2.5

The final answer is $\frac{20}{3}$ .

$\frac{20}{3}$

Step 4.3

Convert $\frac{20}{3}$ to decimal.

$y = 6.\overline{6}$

Step 5

Find the point at $x = - 2$ .

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Find the common denominator.

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Step 5.2.1.1

Write $3 {(- 2)}^{2}$ as a fraction with denominator $1$ .

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

Step 5.2.1.2

Multiply $\frac{3 {(- 2)}^{2}}{1}$ by $\frac{3}{3}$ .

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

Step 5.2.1.3

Multiply $\frac{3 {(- 2)}^{2}}{1}$ by $\frac{3}{3}$ .

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

Step 5.2.1.4

Write $5 (- 2)$ as a fraction with denominator $1$ .

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

Step 5.2.1.5

Multiply $\frac{5 (- 2)}{1}$ by $\frac{3}{3}$ .

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

Step 5.2.1.6

Multiply $\frac{5 (- 2)}{1}$ by $\frac{3}{3}$ .

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

Step 5.2.2

Combine the numerators over the common denominator.

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

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

$f (- 2) = \frac{- 8 + 3 {(- 2)}^{2} \cdot 3 + 5 (- 2) \cdot 3}{3}$

Step 5.2.3.2

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

$f (- 2) = \frac{- 8 + 3 \cdot 4 \cdot 3 + 5 (- 2) \cdot 3}{3}$

Step 5.2.3.3

Multiply $3 \cdot 4 \cdot 3$ .

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Step 5.2.3.3.1

Multiply $3$ by $4$ .

$f (- 2) = \frac{- 8 + 12 \cdot 3 + 5 (- 2) \cdot 3}{3}$

Step 5.2.3.3.2

Multiply $12$ by $3$ .

$f (- 2) = \frac{- 8 + 36 + 5 (- 2) \cdot 3}{3}$

Step 5.2.3.4

Multiply $5 (- 2) \cdot 3$ .

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Step 5.2.3.4.1

Multiply $5$ by $- 2$ .

$f (- 2) = \frac{- 8 + 36 - 10 \cdot 3}{3}$

Step 5.2.3.4.2

Multiply $- 10$ by $3$ .

$f (- 2) = \frac{- 8 + 36 - 30}{3}$

Step 5.2.4

Simplify the expression.

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Step 5.2.4.1

Add $- 8$ and $36$ .

$f (- 2) = \frac{28 - 30}{3}$

Step 5.2.4.2

Subtract $30$ from $28$ .

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

Step 5.2.4.3

Move the negative in front of the fraction.

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

Step 5.2.5

The final answer is $- \frac{2}{3}$ .

$- \frac{2}{3}$

Step 5.3

Convert $- \frac{2}{3}$ to decimal.

$y = - 0.\overline{6}$

Step 6

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

$\begin{array}{c} x & y \\ - 5 & 8.333 \\ - 4 & 6.667 \\ - 3 & 3 \\ - 2 & - 0.667 \\ 0 & 0 \end{array}$

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

$\begin{array}{c} x & y \\ - 5 & 8.333 \\ - 4 & 6.667 \\ - 3 & 3 \\ - 2 & - 0.667 \\ 0 & 0 \end{array}$

Step 8