Calculus Examples

$s (t) = 2 t^{2} - \frac{t^{3}}{6}$

Step 1

Find the point at $x = 6$ .

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Step 1.1

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

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

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 $6$ to the power of $2$ .

$f (6) = 2 \cdot 36 - \frac{{(6)}^{3}}{6}$

Step 1.2.1.2

Multiply $2$ by $36$ .

$f (6) = 72 - \frac{{(6)}^{3}}{6}$

Step 1.2.1.3

Cancel the common factor of ${(6)}^{3}$ and $6$ .

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Step 1.2.1.3.1

Factor $6$ out of ${(6)}^{3}$ .

$f (6) = 72 - \frac{6 \cdot 6^{2}}{6}$

Step 1.2.1.3.2

Cancel the common factors.

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Step 1.2.1.3.2.1

Factor $6$ out of $6$ .

$f (6) = 72 - \frac{6 \cdot 6^{2}}{6 (1)}$

Step 1.2.1.3.2.2

Cancel the common factor.

$f (6) = 72 - \frac{6 \cdot 6^{2}}{6 \cdot 1}$

Step 1.2.1.3.2.3

Rewrite the expression.

$f (6) = 72 - \frac{6^{2}}{1}$

Step 1.2.1.3.2.4

Divide $6^{2}$ by $1$ .

$f (6) = 72 - 6^{2}$

Step 1.2.1.4

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

$f (6) = 72 - 1 \cdot 36$

Step 1.2.1.5

Multiply $- 1$ by $36$ .

$f (6) = 72 - 36$

Step 1.2.2

Subtract $36$ from $72$ .

$f (6) = 36$

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 $x = 2$ .

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Step 2.1

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

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

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

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

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Step 2.2.1.1.1

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

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Step 2.2.1.1.1.1

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

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

Step 2.2.1.1.1.2

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

$f (2) = 2^{1 + 2} - \frac{{(2)}^{3}}{6}$

Step 2.2.1.1.2

Add $1$ and $2$ .

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

Step 2.2.1.2

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

$f (2) = 8 - \frac{{(2)}^{3}}{6}$

Step 2.2.1.3

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

$f (2) = 8 - \frac{8}{6}$

Step 2.2.1.4

Cancel the common factor of $8$ and $6$ .

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Step 2.2.1.4.1

Factor $2$ out of $8$ .

$f (2) = 8 - \frac{2 (4)}{6}$

Step 2.2.1.4.2

Cancel the common factors.

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Step 2.2.1.4.2.1

Factor $2$ out of $6$ .

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

Step 2.2.1.4.2.2

Cancel the common factor.

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

Step 2.2.1.4.2.3

Rewrite the expression.

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

Step 2.2.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.2.3

Combine $8$ and $\frac{3}{3}$ .

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

Step 2.2.4

Combine the numerators over the common denominator.

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

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $8$ by $3$ .

$f (2) = \frac{24 - 4}{3}$

Step 2.2.5.2

Subtract $4$ from $24$ .

$f (2) = \frac{20}{3}$

Step 2.2.6

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

$\frac{20}{3}$

Step 2.3

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

$y = 6.\overline{6}$

Step 3

Find the point at $x = 3$ .

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Step 3.1

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

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

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

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

$f (3) = 2 \cdot 9 - \frac{{(3)}^{3}}{6}$

Step 3.2.1.2

Multiply $2$ by $9$ .

$f (3) = 18 - \frac{{(3)}^{3}}{6}$

Step 3.2.1.3

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

$f (3) = 18 - \frac{27}{6}$

Step 3.2.1.4

Cancel the common factor of $27$ and $6$ .

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Step 3.2.1.4.1

Factor $3$ out of $27$ .

$f (3) = 18 - \frac{3 (9)}{6}$

Step 3.2.1.4.2

Cancel the common factors.

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Step 3.2.1.4.2.1

Factor $3$ out of $6$ .

$f (3) = 18 - \frac{3 \cdot 9}{3 \cdot 2}$

Step 3.2.1.4.2.2

Cancel the common factor.

$f (3) = 18 - \frac{3 \cdot 9}{3 \cdot 2}$

Step 3.2.1.4.2.3

Rewrite the expression.

$f (3) = 18 - \frac{9}{2}$

Step 3.2.2

To write $18$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (3) = 18 \cdot \frac{2}{2} - \frac{9}{2}$

Step 3.2.3

Combine $18$ and $\frac{2}{2}$ .

$f (3) = \frac{18 \cdot 2}{2} - \frac{9}{2}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (3) = \frac{18 \cdot 2 - 9}{2}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $18$ by $2$ .

$f (3) = \frac{36 - 9}{2}$

Step 3.2.5.2

Subtract $9$ from $36$ .

$f (3) = \frac{27}{2}$

Step 3.2.6

The final answer is $\frac{27}{2}$ .

$\frac{27}{2}$

Step 3.3

Convert $\frac{27}{2}$ to decimal.

$y = 13.5$

Step 4

Find the point at $x = 4$ .

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Step 4.1

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

$f (4) = 2 {(4)}^{2} - \frac{{(4)}^{3}}{6}$

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

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

$f (4) = 2 \cdot 16 - \frac{{(4)}^{3}}{6}$

Step 4.2.1.2

Multiply $2$ by $16$ .

$f (4) = 32 - \frac{{(4)}^{3}}{6}$

Step 4.2.1.3

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

$f (4) = 32 - \frac{64}{6}$

Step 4.2.1.4

Cancel the common factor of $64$ and $6$ .

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Step 4.2.1.4.1

Factor $2$ out of $64$ .

$f (4) = 32 - \frac{2 (32)}{6}$

Step 4.2.1.4.2

Cancel the common factors.

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Step 4.2.1.4.2.1

Factor $2$ out of $6$ .

$f (4) = 32 - \frac{2 \cdot 32}{2 \cdot 3}$

Step 4.2.1.4.2.2

Cancel the common factor.

$f (4) = 32 - \frac{2 \cdot 32}{2 \cdot 3}$

Step 4.2.1.4.2.3

Rewrite the expression.

$f (4) = 32 - \frac{32}{3}$

Step 4.2.2

To write $32$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (4) = 32 \cdot \frac{3}{3} - \frac{32}{3}$

Step 4.2.3

Combine $32$ and $\frac{3}{3}$ .

$f (4) = \frac{32 \cdot 3}{3} - \frac{32}{3}$

Step 4.2.4

Combine the numerators over the common denominator.

$f (4) = \frac{32 \cdot 3 - 32}{3}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $32$ by $3$ .

$f (4) = \frac{96 - 32}{3}$

Step 4.2.5.2

Subtract $32$ from $96$ .

$f (4) = \frac{64}{3}$

Step 4.2.6

The final answer is $\frac{64}{3}$ .

$\frac{64}{3}$

Step 4.3

Convert $\frac{64}{3}$ to decimal.

$y = 21.\overline{3}$

Step 5

Find the point at $x = 5$ .

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Step 5.1

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

$f (5) = 2 {(5)}^{2} - \frac{{(5)}^{3}}{6}$

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 $5$ to the power of $2$ .

$f (5) = 2 \cdot 25 - \frac{{(5)}^{3}}{6}$

Step 5.2.1.2

Multiply $2$ by $25$ .

$f (5) = 50 - \frac{{(5)}^{3}}{6}$

Step 5.2.1.3

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

$f (5) = 50 - \frac{125}{6}$

Step 5.2.2

To write $50$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$f (5) = 50 \cdot \frac{6}{6} - \frac{125}{6}$

Step 5.2.3

Combine $50$ and $\frac{6}{6}$ .

$f (5) = \frac{50 \cdot 6}{6} - \frac{125}{6}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (5) = \frac{50 \cdot 6 - 125}{6}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $50$ by $6$ .

$f (5) = \frac{300 - 125}{6}$

Step 5.2.5.2

Subtract $125$ from $300$ .

$f (5) = \frac{175}{6}$

Step 5.2.6

The final answer is $\frac{175}{6}$ .

$\frac{175}{6}$

Step 5.3

Convert $\frac{175}{6}$ to decimal.

$y = 29.1\overline{6}$

Step 6

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

$\begin{array}{c} x & y \\ 2 & 6.667 \\ 3 & 13.5 \\ 4 & 21.333 \\ 5 & 29.167 \\ 6 & 36 \end{array}$

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

$\begin{array}{c} x & y \\ 2 & 6.667 \\ 3 & 13.5 \\ 4 & 21.333 \\ 5 & 29.167 \\ 6 & 36 \end{array}$

Step 8