Calculus Examples

$y = x^{3} + 4 x^{2} - 11 x + 11$

Step 1

Set $y$ as a function of $x$ .

$f (x) = x^{3} + 4 x^{2} - 11 x + 11$

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{3} + 4 x^{2} - 11 x + 11$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [11]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [11]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [11]$

Step 2.2

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} + 4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [11]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$3 x^{2} + 4 (2 x) + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [11]$

Step 2.2.3

Multiply $2$ by $4$ .

$3 x^{2} + 8 x + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [11]$

Step 2.3

Evaluate $\frac{d}{d x} [- 11 x]$ .

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Step 2.3.1

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11 x$ with respect to $x$ is $- 11 \frac{d}{d x} [x]$ .

$3 x^{2} + 8 x - 11 \frac{d}{d x} [x] + \frac{d}{d x} [11]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 x^{2} + 8 x - 11 \cdot 1 + \frac{d}{d x} [11]$

Step 2.3.3

Multiply $- 11$ by $1$ .

$3 x^{2} + 8 x - 11 + \frac{d}{d x} [11]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $11$ is constant with respect to $x$ , the derivative of $11$ with respect to $x$ is $0$ .

$3 x^{2} + 8 x - 11 + 0$

Step 2.4.2

Add $3 x^{2} + 8 x - 11$ and $0$ .

$3 x^{2} + 8 x - 11$

Step 3

Set the derivative equal to $0$ then solve the equation $3 x^{2} + 8 x - 11 = 0$ .

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Step 3.1

Factor by grouping.

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Step 3.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 11 = - 33$ and whose sum is $b = 8$ .

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Step 3.1.1.1

Factor $8$ out of $8 x$ .

$3 x^{2} + 8 (x) - 11 = 0$

Step 3.1.1.2

Rewrite $8$ as $- 3$ plus $11$

$3 x^{2} + (- 3 + 11) x - 11 = 0$

Step 3.1.1.3

Apply the distributive property.

$3 x^{2} - 3 x + 11 x - 11 = 0$

Step 3.1.2

Factor out the greatest common factor from each group.

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Step 3.1.2.1

Group the first two terms and the last two terms.

$(3 x^{2} - 3 x) + 11 x - 11 = 0$

Step 3.1.2.2

Factor out the greatest common factor (GCF) from each group.

$3 x (x - 1) + 11 (x - 1) = 0$

Step 3.1.3

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$(x - 1) (3 x + 11) = 0$

Step 3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$3 x + 11 = 0$

Step 3.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 3.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.4

Set $3 x + 11$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $3 x + 11$ equal to $0$ .

$3 x + 11 = 0$

Step 3.4.2

Solve $3 x + 11 = 0$ for $x$ .

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Step 3.4.2.1

Subtract $11$ from both sides of the equation.

$3 x = - 11$

Step 3.4.2.2

Divide each term in $3 x = - 11$ by $3$ and simplify.

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Step 3.4.2.2.1

Divide each term in $3 x = - 11$ by $3$ .

$\frac{3 x}{3} = \frac{- 11}{3}$

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 11}{3}$

Step 3.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 11}{3}$

Step 3.4.2.2.3

Simplify the right side.

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Step 3.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{11}{3}$

Step 3.5

The final solution is all the values that make $(x - 1) (3 x + 11) = 0$ true.

$x = 1, - \frac{11}{3}$

Step 4

Solve the original function $f (x) = x^{3} + 4 x^{2} - 11 x + 11$ at $x = 1$ .

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Step 4.1

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

$f (1) = {(1)}^{3} + 4 {(1)}^{2} - 11 \cdot 1 + 11$

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.

$f (1) = 1 + 4 {(1)}^{2} - 11 \cdot 1 + 11$

Step 4.2.1.2

One to any power is one.

$f (1) = 1 + 4 \cdot 1 - 11 \cdot 1 + 11$

Step 4.2.1.3

Multiply $4$ by $1$ .

$f (1) = 1 + 4 - 11 \cdot 1 + 11$

Step 4.2.1.4

Multiply $- 11$ by $1$ .

$f (1) = 1 + 4 - 11 + 11$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $1$ and $4$ .

$f (1) = 5 - 11 + 11$

Step 4.2.2.2

Subtract $11$ from $5$ .

$f (1) = - 6 + 11$

Step 4.2.2.3

Add $- 6$ and $11$ .

$f (1) = 5$

Step 4.2.3

The final answer is $5$ .

$5$

Step 5

Solve the original function $f (x) = x^{3} + 4 x^{2} - 11 x + 11$ at $x = - \frac{11}{3}$ .

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Step 5.1

Replace the variable $x$ with $- \frac{11}{3}$ in the expression.

$f (- \frac{11}{3}) = {(- \frac{11}{3})}^{3} + 4 {(- \frac{11}{3})}^{2} - 11 (- \frac{11}{3}) + 11$

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.1.1

Apply the product rule to $- \frac{11}{3}$ .

$f (- \frac{11}{3}) = {(- 1)}^{3} {(\frac{11}{3})}^{3} + 4 {(- \frac{11}{3})}^{2} - 11 (- \frac{11}{3}) + 11$

Step 5.2.1.1.2

Apply the product rule to $\frac{11}{3}$ .

$f (- \frac{11}{3}) = {(- 1)}^{3} (\frac{11^{3}}{3^{3}}) + 4 {(- \frac{11}{3})}^{2} - 11 (- \frac{11}{3}) + 11$

Step 5.2.1.2

Raise $- 1$ to the power of $3$ .

$f (- \frac{11}{3}) = - \frac{11^{3}}{3^{3}} + 4 {(- \frac{11}{3})}^{2} - 11 (- \frac{11}{3}) + 11$

Step 5.2.1.3

Raise $11$ to the power of $3$ .

$f (- \frac{11}{3}) = - \frac{1331}{3^{3}} + 4 {(- \frac{11}{3})}^{2} - 11 (- \frac{11}{3}) + 11$

Step 5.2.1.4

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

$f (- \frac{11}{3}) = - \frac{1331}{27} + 4 {(- \frac{11}{3})}^{2} - 11 (- \frac{11}{3}) + 11$

Step 5.2.1.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.5.1

Apply the product rule to $- \frac{11}{3}$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + 4 ({(- 1)}^{2} {(\frac{11}{3})}^{2}) - 11 (- \frac{11}{3}) + 11$

Step 5.2.1.5.2

Apply the product rule to $\frac{11}{3}$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + 4 ({(- 1)}^{2} (\frac{11^{2}}{3^{2}})) - 11 (- \frac{11}{3}) + 11$

Step 5.2.1.6

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

$f (- \frac{11}{3}) = - \frac{1331}{27} + 4 (1 (\frac{11^{2}}{3^{2}})) - 11 (- \frac{11}{3}) + 11$

Step 5.2.1.7

Multiply $\frac{11^{2}}{3^{2}}$ by $1$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + 4 (\frac{11^{2}}{3^{2}}) - 11 (- \frac{11}{3}) + 11$

Step 5.2.1.8

Raise $11$ to the power of $2$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + 4 (\frac{121}{3^{2}}) - 11 (- \frac{11}{3}) + 11$

Step 5.2.1.9

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

$f (- \frac{11}{3}) = - \frac{1331}{27} + 4 (\frac{121}{9}) - 11 (- \frac{11}{3}) + 11$

Step 5.2.1.10

Multiply $4 (\frac{121}{9})$ .

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Step 5.2.1.10.1

Combine $4$ and $\frac{121}{9}$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{4 \cdot 121}{9} - 11 (- \frac{11}{3}) + 11$

Step 5.2.1.10.2

Multiply $4$ by $121$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484}{9} - 11 (- \frac{11}{3}) + 11$

Step 5.2.1.11

Multiply $- 11 (- \frac{11}{3})$ .

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Step 5.2.1.11.1

Multiply $- 1$ by $- 11$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484}{9} + 11 (\frac{11}{3}) + 11$

Step 5.2.1.11.2

Combine $11$ and $\frac{11}{3}$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484}{9} + \frac{11 \cdot 11}{3} + 11$

Step 5.2.1.11.3

Multiply $11$ by $11$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484}{9} + \frac{121}{3} + 11$

Step 5.2.2

Find the common denominator.

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Step 5.2.2.1

Multiply $\frac{484}{9}$ by $\frac{3}{3}$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484}{9} \cdot \frac{3}{3} + \frac{121}{3} + 11$

Step 5.2.2.2

Multiply $\frac{484}{9}$ by $\frac{3}{3}$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484 \cdot 3}{9 \cdot 3} + \frac{121}{3} + 11$

Step 5.2.2.3

Multiply $\frac{121}{3}$ by $\frac{9}{9}$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484 \cdot 3}{9 \cdot 3} + \frac{121}{3} \cdot \frac{9}{9} + 11$

Step 5.2.2.4

Multiply $\frac{121}{3}$ by $\frac{9}{9}$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484 \cdot 3}{9 \cdot 3} + \frac{121 \cdot 9}{3 \cdot 9} + 11$

Step 5.2.2.5

Write $11$ as a fraction with denominator $1$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484 \cdot 3}{9 \cdot 3} + \frac{121 \cdot 9}{3 \cdot 9} + \frac{11}{1}$

Step 5.2.2.6

Multiply $\frac{11}{1}$ by $\frac{27}{27}$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484 \cdot 3}{9 \cdot 3} + \frac{121 \cdot 9}{3 \cdot 9} + \frac{11}{1} \cdot \frac{27}{27}$

Step 5.2.2.7

Multiply $\frac{11}{1}$ by $\frac{27}{27}$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484 \cdot 3}{9 \cdot 3} + \frac{121 \cdot 9}{3 \cdot 9} + \frac{11 \cdot 27}{27}$

Step 5.2.2.8

Reorder the factors of $9 \cdot 3$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484 \cdot 3}{3 \cdot 9} + \frac{121 \cdot 9}{3 \cdot 9} + \frac{11 \cdot 27}{27}$

Step 5.2.2.9

Multiply $3$ by $9$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484 \cdot 3}{27} + \frac{121 \cdot 9}{3 \cdot 9} + \frac{11 \cdot 27}{27}$

Step 5.2.2.10

Multiply $3$ by $9$ .

$f (- \frac{11}{3}) = - \frac{1331}{27} + \frac{484 \cdot 3}{27} + \frac{121 \cdot 9}{27} + \frac{11 \cdot 27}{27}$

Step 5.2.3

Combine the numerators over the common denominator.

$f (- \frac{11}{3}) = \frac{- 1331 + 484 \cdot 3 + 121 \cdot 9 + 11 \cdot 27}{27}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Multiply $484$ by $3$ .

$f (- \frac{11}{3}) = \frac{- 1331 + 1452 + 121 \cdot 9 + 11 \cdot 27}{27}$

Step 5.2.4.2

Multiply $121$ by $9$ .

$f (- \frac{11}{3}) = \frac{- 1331 + 1452 + 1089 + 11 \cdot 27}{27}$

Step 5.2.4.3

Multiply $11$ by $27$ .

$f (- \frac{11}{3}) = \frac{- 1331 + 1452 + 1089 + 297}{27}$

Step 5.2.5

Simplify by adding numbers.

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Step 5.2.5.1

Add $- 1331$ and $1452$ .

$f (- \frac{11}{3}) = \frac{121 + 1089 + 297}{27}$

Step 5.2.5.2

Add $121$ and $1089$ .

$f (- \frac{11}{3}) = \frac{1210 + 297}{27}$

Step 5.2.5.3

Add $1210$ and $297$ .

$f (- \frac{11}{3}) = \frac{1507}{27}$

Step 5.2.6

The final answer is $\frac{1507}{27}$ .

$\frac{1507}{27}$

Step 6

The horizontal tangent lines on function $f (x) = x^{3} + 4 x^{2} - 11 x + 11$ are $y = 5, y = \frac{1507}{27}$ .

$y = 5, y = \frac{1507}{27}$

Step 7