Calculus Examples

$f (\frac{7}{3}) = x^{3} - 14 x^{2} + 49 x$

Step 1

Replace the variable $x$ with $\frac{7}{3}$ in the expression.

$f (\frac{7}{3}) = {(\frac{7}{3})}^{3} - 14 {(\frac{7}{3})}^{2} + 49 (\frac{7}{3})$

Step 2

Simplify the result.

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the product rule to $\frac{7}{3}$ .

$f (\frac{7}{3}) = \frac{7^{3}}{3^{3}} - 14 {(\frac{7}{3})}^{2} + 49 (\frac{7}{3})$

Step 2.1.2

Raise $7$ to the power of $3$ .

$f (\frac{7}{3}) = \frac{343}{3^{3}} - 14 {(\frac{7}{3})}^{2} + 49 (\frac{7}{3})$

Step 2.1.3

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

$f (\frac{7}{3}) = \frac{343}{27} - 14 {(\frac{7}{3})}^{2} + 49 (\frac{7}{3})$

Step 2.1.4

Apply the product rule to $\frac{7}{3}$ .

$f (\frac{7}{3}) = \frac{343}{27} - 14 \frac{7^{2}}{3^{2}} + 49 (\frac{7}{3})$

Step 2.1.5

Raise $7$ to the power of $2$ .

$f (\frac{7}{3}) = \frac{343}{27} - 14 \frac{49}{3^{2}} + 49 (\frac{7}{3})$

Step 2.1.6

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

$f (\frac{7}{3}) = \frac{343}{27} - 14 (\frac{49}{9}) + 49 (\frac{7}{3})$

Step 2.1.7

Multiply $- 14 (\frac{49}{9})$ .

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Step 2.1.7.1

Combine $- 14$ and $\frac{49}{9}$ .

$f (\frac{7}{3}) = \frac{343}{27} + \frac{- 14 \cdot 49}{9} + 49 (\frac{7}{3})$

Step 2.1.7.2

Multiply $- 14$ by $49$ .

$f (\frac{7}{3}) = \frac{343}{27} + \frac{- 686}{9} + 49 (\frac{7}{3})$

Step 2.1.8

Move the negative in front of the fraction.

$f (\frac{7}{3}) = \frac{343}{27} - \frac{686}{9} + 49 (\frac{7}{3})$

Step 2.1.9

Multiply $49 (\frac{7}{3})$ .

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Step 2.1.9.1

Combine $49$ and $\frac{7}{3}$ .

$f (\frac{7}{3}) = \frac{343}{27} - \frac{686}{9} + \frac{49 \cdot 7}{3}$

Step 2.1.9.2

Multiply $49$ by $7$ .

$f (\frac{7}{3}) = \frac{343}{27} - \frac{686}{9} + \frac{343}{3}$

Step 2.2

Find the common denominator.

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Step 2.2.1

Multiply $\frac{686}{9}$ by $\frac{3}{3}$ .

$f (\frac{7}{3}) = \frac{343}{27} - (\frac{686}{9} \cdot \frac{3}{3}) + \frac{343}{3}$

Step 2.2.2

Multiply $\frac{686}{9}$ by $\frac{3}{3}$ .

$f (\frac{7}{3}) = \frac{343}{27} - \frac{686 \cdot 3}{9 \cdot 3} + \frac{343}{3}$

Step 2.2.3

Multiply $\frac{343}{3}$ by $\frac{9}{9}$ .

$f (\frac{7}{3}) = \frac{343}{27} - \frac{686 \cdot 3}{9 \cdot 3} + \frac{343}{3} \cdot \frac{9}{9}$

Step 2.2.4

Multiply $\frac{343}{3}$ by $\frac{9}{9}$ .

$f (\frac{7}{3}) = \frac{343}{27} - \frac{686 \cdot 3}{9 \cdot 3} + \frac{343 \cdot 9}{3 \cdot 9}$

Step 2.2.5

Reorder the factors of $9 \cdot 3$ .

$f (\frac{7}{3}) = \frac{343}{27} - \frac{686 \cdot 3}{3 \cdot 9} + \frac{343 \cdot 9}{3 \cdot 9}$

Step 2.2.6

Multiply $3$ by $9$ .

$f (\frac{7}{3}) = \frac{343}{27} - \frac{686 \cdot 3}{27} + \frac{343 \cdot 9}{3 \cdot 9}$

Step 2.2.7

Multiply $3$ by $9$ .

$f (\frac{7}{3}) = \frac{343}{27} - \frac{686 \cdot 3}{27} + \frac{343 \cdot 9}{27}$

Step 2.3

Combine the numerators over the common denominator.

$f (\frac{7}{3}) = \frac{343 - 686 \cdot 3 + 343 \cdot 9}{27}$

Step 2.4

Simplify each term.

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Step 2.4.1

Multiply $- 686$ by $3$ .

$f (\frac{7}{3}) = \frac{343 - 2058 + 343 \cdot 9}{27}$

Step 2.4.2

Multiply $343$ by $9$ .

$f (\frac{7}{3}) = \frac{343 - 2058 + 3087}{27}$

Step 2.5

Simplify by adding and subtracting.

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Step 2.5.1

Subtract $2058$ from $343$ .

$f (\frac{7}{3}) = \frac{- 1715 + 3087}{27}$

Step 2.5.2

Add $- 1715$ and $3087$ .

$f (\frac{7}{3}) = \frac{1372}{27}$

Step 2.6

The final answer is $\frac{1372}{27}$ .

$\frac{1372}{27}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{1372}{27}$

Decimal Form:

$50.\overline{814}$

Mixed Number Form:

$50 \frac{22}{27}$

Step 4