Calculus Examples

$g (x) = 2 x^{2} - x - 1$ , $[3, 5]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

By the Sum Rule, the derivative of $2 x^{2} - x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 1]$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 1]$

Step 1.1.1.2.2

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

$2 (2 x) + \frac{d}{d x} [- x] + \frac{d}{d x} [- 1]$

Step 1.1.1.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [- x] + \frac{d}{d x} [- 1]$

Step 1.1.1.3

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

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Step 1.1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$4 x - \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.1.1.3.2

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

$4 x - 1 \cdot 1 + \frac{d}{d x} [- 1]$

Step 1.1.1.3.3

Multiply $- 1$ by $1$ .

$4 x - 1 + \frac{d}{d x} [- 1]$

Step 1.1.1.4

Differentiate using the Constant Rule.

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Step 1.1.1.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$4 x - 1 + 0$

Step 1.1.1.4.2

Add $4 x - 1$ and $0$ .

$f' (x) = 4 x - 1$

Step 1.1.2

The first derivative of $g (x)$ with respect to $x$ is $4 x - 1$ .

$4 x - 1$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $4 x - 1 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$4 x - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$4 x = 1$

Step 1.2.3

Divide each term in $4 x = 1$ by $4$ and simplify.

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Step 1.2.3.1

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $4$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 1.4

Evaluate $2 x^{2} - x - 1$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \frac{1}{4}$ .

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Step 1.4.1.1

Substitute $\frac{1}{4}$ for $x$ .

$2 {(\frac{1}{4})}^{2} - (\frac{1}{4}) - 1$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

Apply the product rule to $\frac{1}{4}$ .

$2 \frac{1^{2}}{4^{2}} - (\frac{1}{4}) - 1$

Step 1.4.1.2.1.2

One to any power is one.

$2 \frac{1}{4^{2}} - (\frac{1}{4}) - 1$

Step 1.4.1.2.1.3

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

$2 (\frac{1}{16}) - (\frac{1}{4}) - 1$

Step 1.4.1.2.1.4

Cancel the common factor of $2$ .

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Step 1.4.1.2.1.4.1

Factor $2$ out of $16$ .

$2 \frac{1}{2 (8)} - (\frac{1}{4}) - 1$

Step 1.4.1.2.1.4.2

Cancel the common factor.

$2 \frac{1}{2 \cdot 8} - (\frac{1}{4}) - 1$

Step 1.4.1.2.1.4.3

Rewrite the expression.

$\frac{1}{8} - (\frac{1}{4}) - 1$

$\frac{1}{8} - \frac{1}{4} - 1$

Step 1.4.1.2.2

Find the common denominator.

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Step 1.4.1.2.2.1

Multiply $\frac{1}{4}$ by $\frac{2}{2}$ .

$\frac{1}{8} - (\frac{1}{4} \cdot \frac{2}{2}) - 1$

Step 1.4.1.2.2.2

Multiply $\frac{1}{4}$ by $\frac{2}{2}$ .

$\frac{1}{8} - \frac{2}{4 \cdot 2} - 1$

Step 1.4.1.2.2.3

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

$\frac{1}{8} - \frac{2}{4 \cdot 2} + \frac{- 1}{1}$

Step 1.4.1.2.2.4

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

$\frac{1}{8} - \frac{2}{4 \cdot 2} + \frac{- 1}{1} \cdot \frac{8}{8}$

Step 1.4.1.2.2.5

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

$\frac{1}{8} - \frac{2}{4 \cdot 2} + \frac{- 1 \cdot 8}{8}$

Step 1.4.1.2.2.6

Reorder the factors of $4 \cdot 2$ .

$\frac{1}{8} - \frac{2}{2 \cdot 4} + \frac{- 1 \cdot 8}{8}$

Step 1.4.1.2.2.7

Multiply $2$ by $4$ .

$\frac{1}{8} - \frac{2}{8} + \frac{- 1 \cdot 8}{8}$

Step 1.4.1.2.3

Combine the numerators over the common denominator.

$\frac{1 - 2 - 1 \cdot 8}{8}$

Step 1.4.1.2.4

Simplify the expression.

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Step 1.4.1.2.4.1

Multiply $- 1$ by $8$ .

$\frac{1 - 2 - 8}{8}$

Step 1.4.1.2.4.2

Subtract $2$ from $1$ .

$\frac{- 1 - 8}{8}$

Step 1.4.1.2.4.3

Subtract $8$ from $- 1$ .

$\frac{- 9}{8}$

Step 1.4.1.2.4.4

Move the negative in front of the fraction.

$- \frac{9}{8}$

Step 1.4.2

List all of the points.

$(\frac{1}{4}, - \frac{9}{8})$

Step 2

Exclude the points that are not on the interval.

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = 3$ .

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Step 3.1.1

Substitute $3$ for $x$ .

$2 {(3)}^{2} - (3) - 1$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

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

$2 \cdot 9 - (3) - 1$

Step 3.1.2.1.2

Multiply $2$ by $9$ .

$18 - (3) - 1$

Step 3.1.2.1.3

Multiply $- 1$ by $3$ .

$18 - 3 - 1$

Step 3.1.2.2

Simplify by subtracting numbers.

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Step 3.1.2.2.1

Subtract $3$ from $18$ .

$15 - 1$

Step 3.1.2.2.2

Subtract $1$ from $15$ .

$14$

Step 3.2

Evaluate at $x = 5$ .

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Step 3.2.1

Substitute $5$ for $x$ .

$2 {(5)}^{2} - (5) - 1$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

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

$2 \cdot 25 - (5) - 1$

Step 3.2.2.1.2

Multiply $2$ by $25$ .

$50 - (5) - 1$

Step 3.2.2.1.3

Multiply $- 1$ by $5$ .

$50 - 5 - 1$

Step 3.2.2.2

Simplify by subtracting numbers.

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Step 3.2.2.2.1

Subtract $5$ from $50$ .

$45 - 1$

Step 3.2.2.2.2

Subtract $1$ from $45$ .

$44$

Step 3.3

List all of the points.

$(3, 14), (5, 44)$

Step 4

Compare the $g (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $g (x)$ value and the minimum will occur at the lowest $g (x)$ value.

Absolute Maximum: $(5, 44)$

Absolute Minimum: $(3, 14)$

Step 5