Calculus Examples

$x^{3} - 7 x^{2} + \frac{40}{3} x$

Step 1

Find the point at $x = 3$ .

Tap for more steps...

Step 1.1

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

$f (3) = {(3)}^{3} - 7 {(3)}^{2} + \frac{40 (3)}{3}$

Step 1.2

Simplify the result.

Tap for more steps...

Step 1.2.1

Multiply $40$ by $3$ .

$f (3) = {(3)}^{3} - 7 {(3)}^{2} + \frac{120}{3}$

Step 1.2.2

Simplify each term.

Tap for more steps...

Step 1.2.2.1

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

$f (3) = 27 - 7 {(3)}^{2} + \frac{120}{3}$

Step 1.2.2.2

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

$f (3) = 27 - 7 \cdot 9 + \frac{120}{3}$

Step 1.2.2.3

Multiply $- 7$ by $9$ .

$f (3) = 27 - 63 + \frac{120}{3}$

Step 1.2.2.4

Divide $120$ by $3$ .

$f (3) = 27 - 63 + 40$

Step 1.2.3

Simplify by adding and subtracting.

Tap for more steps...

Step 1.2.3.1

Subtract $63$ from $27$ .

$f (3) = - 36 + 40$

Step 1.2.3.2

Add $- 36$ and $40$ .

$f (3) = 4$

Step 1.2.4

The final answer is $4$ .

$4$

Step 1.3

Convert $4$ to decimal.

$y = 4$

Step 2

Find the point at $x = 6$ .

Tap for more steps...

Step 2.1

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

$f (6) = {(6)}^{3} - 7 {(6)}^{2} + \frac{40 (6)}{3}$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Multiply $40$ by $6$ .

$f (6) = {(6)}^{3} - 7 {(6)}^{2} + \frac{240}{3}$

Step 2.2.2

Simplify each term.

Tap for more steps...

Step 2.2.2.1

Raise $6$ to the power of $3$ .

$f (6) = 216 - 7 {(6)}^{2} + \frac{240}{3}$

Step 2.2.2.2

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

$f (6) = 216 - 7 \cdot 36 + \frac{240}{3}$

Step 2.2.2.3

Multiply $- 7$ by $36$ .

$f (6) = 216 - 252 + \frac{240}{3}$

Step 2.2.2.4

Divide $240$ by $3$ .

$f (6) = 216 - 252 + 80$

Step 2.2.3

Simplify by adding and subtracting.

Tap for more steps...

Step 2.2.3.1

Subtract $252$ from $216$ .

$f (6) = - 36 + 80$

Step 2.2.3.2

Add $- 36$ and $80$ .

$f (6) = 44$

Step 2.2.4

The final answer is $44$ .

$44$

Step 2.3

Convert $44$ to decimal.

$y = 44$

Step 3

Find the point at $x = 1$ .

Tap for more steps...

Step 3.1

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

$f (1) = {(1)}^{3} - 7 {(1)}^{2} + \frac{40 (1)}{3}$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Multiply $40$ by $1$ .

$f (1) = {(1)}^{3} - 7 {(1)}^{2} + \frac{40}{3}$

Step 3.2.2

Simplify each term.

Tap for more steps...

Step 3.2.2.1

One to any power is one.

$f (1) = 1 - 7 {(1)}^{2} + \frac{40}{3}$

Step 3.2.2.2

One to any power is one.

$f (1) = 1 - 7 \cdot 1 + \frac{40}{3}$

Step 3.2.2.3

Multiply $- 7$ by $1$ .

$f (1) = 1 - 7 + \frac{40}{3}$

Step 3.2.3

Find the common denominator.

Tap for more steps...

Step 3.2.3.1

Write $1$ as a fraction with denominator $1$ .

$f (1) = \frac{1}{1} - 7 + \frac{40}{3}$

Step 3.2.3.2

Multiply $\frac{1}{1}$ by $\frac{3}{3}$ .

$f (1) = \frac{1}{1} \cdot \frac{3}{3} - 7 + \frac{40}{3}$

Step 3.2.3.3

Multiply $\frac{1}{1}$ by $\frac{3}{3}$ .

$f (1) = \frac{3}{3} - 7 + \frac{40}{3}$

Step 3.2.3.4

Write $- 7$ as a fraction with denominator $1$ .

$f (1) = \frac{3}{3} + \frac{- 7}{1} + \frac{40}{3}$

Step 3.2.3.5

Multiply $\frac{- 7}{1}$ by $\frac{3}{3}$ .

$f (1) = \frac{3}{3} + \frac{- 7}{1} \cdot \frac{3}{3} + \frac{40}{3}$

Step 3.2.3.6

Multiply $\frac{- 7}{1}$ by $\frac{3}{3}$ .

$f (1) = \frac{3}{3} + \frac{- 7 \cdot 3}{3} + \frac{40}{3}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (1) = \frac{3 - 7 \cdot 3 + 40}{3}$

Step 3.2.5

Simplify the expression.

Tap for more steps...

Step 3.2.5.1

Multiply $- 7$ by $3$ .

$f (1) = \frac{3 - 21 + 40}{3}$

Step 3.2.5.2

Subtract $21$ from $3$ .

$f (1) = \frac{- 18 + 40}{3}$

Step 3.2.5.3

Add $- 18$ and $40$ .

$f (1) = \frac{22}{3}$

Step 3.2.6

The final answer is $\frac{22}{3}$ .

$\frac{22}{3}$

Step 3.3

Convert $\frac{22}{3}$ to decimal.

$y = 7.\overline{3}$

Step 4

Find the point at $x = 2$ .

Tap for more steps...

Step 4.1

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

$f (2) = {(2)}^{3} - 7 {(2)}^{2} + \frac{40 (2)}{3}$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Multiply $40$ by $2$ .

$f (2) = {(2)}^{3} - 7 {(2)}^{2} + \frac{80}{3}$

Step 4.2.2

Simplify each term.

Tap for more steps...

Step 4.2.2.1

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

$f (2) = 8 - 7 {(2)}^{2} + \frac{80}{3}$

Step 4.2.2.2

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

$f (2) = 8 - 7 \cdot 4 + \frac{80}{3}$

Step 4.2.2.3

Multiply $- 7$ by $4$ .

$f (2) = 8 - 28 + \frac{80}{3}$

Step 4.2.3

Find the common denominator.

Tap for more steps...

Step 4.2.3.1

Write $8$ as a fraction with denominator $1$ .

$f (2) = \frac{8}{1} - 28 + \frac{80}{3}$

Step 4.2.3.2

Multiply $\frac{8}{1}$ by $\frac{3}{3}$ .

$f (2) = \frac{8}{1} \cdot \frac{3}{3} - 28 + \frac{80}{3}$

Step 4.2.3.3

Multiply $\frac{8}{1}$ by $\frac{3}{3}$ .

$f (2) = \frac{8 \cdot 3}{3} - 28 + \frac{80}{3}$

Step 4.2.3.4

Write $- 28$ as a fraction with denominator $1$ .

$f (2) = \frac{8 \cdot 3}{3} + \frac{- 28}{1} + \frac{80}{3}$

Step 4.2.3.5

Multiply $\frac{- 28}{1}$ by $\frac{3}{3}$ .

$f (2) = \frac{8 \cdot 3}{3} + \frac{- 28}{1} \cdot \frac{3}{3} + \frac{80}{3}$

Step 4.2.3.6

Multiply $\frac{- 28}{1}$ by $\frac{3}{3}$ .

$f (2) = \frac{8 \cdot 3}{3} + \frac{- 28 \cdot 3}{3} + \frac{80}{3}$

Step 4.2.4

Combine the numerators over the common denominator.

$f (2) = \frac{8 \cdot 3 - 28 \cdot 3 + 80}{3}$

Step 4.2.5

Simplify each term.

Tap for more steps...

Step 4.2.5.1

Multiply $8$ by $3$ .

$f (2) = \frac{24 - 28 \cdot 3 + 80}{3}$

Step 4.2.5.2

Multiply $- 28$ by $3$ .

$f (2) = \frac{24 - 84 + 80}{3}$

Step 4.2.6

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.6.1

Subtract $84$ from $24$ .

$f (2) = \frac{- 60 + 80}{3}$

Step 4.2.6.2

Add $- 60$ and $80$ .

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

Step 4.2.7

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 $x = 4$ .

Tap for more steps...

Step 5.1

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

$f (4) = {(4)}^{3} - 7 {(4)}^{2} + \frac{40 (4)}{3}$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Multiply $40$ by $4$ .

$f (4) = {(4)}^{3} - 7 {(4)}^{2} + \frac{160}{3}$

Step 5.2.2

Simplify each term.

Tap for more steps...

Step 5.2.2.1

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

$f (4) = 64 - 7 {(4)}^{2} + \frac{160}{3}$

Step 5.2.2.2

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

$f (4) = 64 - 7 \cdot 16 + \frac{160}{3}$

Step 5.2.2.3

Multiply $- 7$ by $16$ .

$f (4) = 64 - 112 + \frac{160}{3}$

Step 5.2.3

Find the common denominator.

Tap for more steps...

Step 5.2.3.1

Write $64$ as a fraction with denominator $1$ .

$f (4) = \frac{64}{1} - 112 + \frac{160}{3}$

Step 5.2.3.2

Multiply $\frac{64}{1}$ by $\frac{3}{3}$ .

$f (4) = \frac{64}{1} \cdot \frac{3}{3} - 112 + \frac{160}{3}$

Step 5.2.3.3

Multiply $\frac{64}{1}$ by $\frac{3}{3}$ .

$f (4) = \frac{64 \cdot 3}{3} - 112 + \frac{160}{3}$

Step 5.2.3.4

Write $- 112$ as a fraction with denominator $1$ .

$f (4) = \frac{64 \cdot 3}{3} + \frac{- 112}{1} + \frac{160}{3}$

Step 5.2.3.5

Multiply $\frac{- 112}{1}$ by $\frac{3}{3}$ .

$f (4) = \frac{64 \cdot 3}{3} + \frac{- 112}{1} \cdot \frac{3}{3} + \frac{160}{3}$

Step 5.2.3.6

Multiply $\frac{- 112}{1}$ by $\frac{3}{3}$ .

$f (4) = \frac{64 \cdot 3}{3} + \frac{- 112 \cdot 3}{3} + \frac{160}{3}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (4) = \frac{64 \cdot 3 - 112 \cdot 3 + 160}{3}$

Step 5.2.5

Simplify each term.

Tap for more steps...

Step 5.2.5.1

Multiply $64$ by $3$ .

$f (4) = \frac{192 - 112 \cdot 3 + 160}{3}$

Step 5.2.5.2

Multiply $- 112$ by $3$ .

$f (4) = \frac{192 - 336 + 160}{3}$

Step 5.2.6

Simplify by adding and subtracting.

Tap for more steps...

Step 5.2.6.1

Subtract $336$ from $192$ .

$f (4) = \frac{- 144 + 160}{3}$

Step 5.2.6.2

Add $- 144$ and $160$ .

$f (4) = \frac{16}{3}$

Step 5.2.7

The final answer is $\frac{16}{3}$ .

$\frac{16}{3}$

Step 5.3

Convert $\frac{16}{3}$ to decimal.

$y = 5.\overline{3}$

Step 6

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

$\begin{array}{c} x & y \\ 1 & 7.333 \\ 2 & 6.667 \\ 3 & 4 \\ 4 & 5.333 \\ 6 & 44 \end{array}$

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

$\begin{array}{c} x & y \\ 1 & 7.333 \\ 2 & 6.667 \\ 3 & 4 \\ 4 & 5.333 \\ 6 & 44 \end{array}$

Step 8