Calculus Examples

$f (x) = \frac{x + 1}{x^{2} + 3}$ , $- 1 \leq x \leq 2$

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

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x + 1$ and $g (x) = x^{2} + 3$ .

$\frac{(x^{2} + 3) \frac{d}{d x} [x + 1] - (x + 1) \frac{d}{d x} [x^{2} + 3]}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

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

$\frac{(x^{2} + 3) (\frac{d}{d x} [x] + \frac{d}{d x} [1]) - (x + 1) \frac{d}{d x} [x^{2} + 3]}{{(x^{2} + 3)}^{2}}$

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 $n = 1$ .

$\frac{(x^{2} + 3) (1 + \frac{d}{d x} [1]) - (x + 1) \frac{d}{d x} [x^{2} + 3]}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.2.3

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

$\frac{(x^{2} + 3) (1 + 0) - (x + 1) \frac{d}{d x} [x^{2} + 3]}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.2.4

Simplify the expression.

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Step 1.1.1.2.4.1

Add $1$ and $0$ .

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

Step 1.1.1.2.4.2

Multiply $x^{2} + 3$ by $1$ .

$\frac{x^{2} + 3 - (x + 1) \frac{d}{d x} [x^{2} + 3]}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.2.5

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

$\frac{x^{2} + 3 - (x + 1) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3])}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.2.6

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

$\frac{x^{2} + 3 - (x + 1) (2 x + \frac{d}{d x} [3])}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.2.7

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{x^{2} + 3 - (x + 1) (2 x + 0)}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.2.8

Simplify the expression.

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Step 1.1.1.2.8.1

Add $2 x$ and $0$ .

$\frac{x^{2} + 3 - (x + 1) (2 x)}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.2.8.2

Multiply $2$ by $- 1$ .

$\frac{x^{2} + 3 - 2 (x + 1) x}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3

Simplify.

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Step 1.1.1.3.1

Apply the distributive property.

$\frac{x^{2} + 3 + (- 2 x - 2 \cdot 1) x}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.2

Apply the distributive property.

$\frac{x^{2} + 3 - 2 x \cdot x - 2 \cdot 1 x}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.3

Simplify the numerator.

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Step 1.1.1.3.3.1

Simplify each term.

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Step 1.1.1.3.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.1.3.3.1.1.1

Move $x$ .

$\frac{x^{2} + 3 - 2 (x \cdot x) - 2 \cdot 1 x}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.3.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} + 3 - 2 x^{2} - 2 \cdot 1 x}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.3.1.2

Multiply $- 2$ by $1$ .

$\frac{x^{2} + 3 - 2 x^{2} - 2 x}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.3.2

Subtract $2 x^{2}$ from $x^{2}$ .

$\frac{- x^{2} + 3 - 2 x}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.4

Reorder terms.

$\frac{- x^{2} - 2 x + 3}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.5

Factor by grouping.

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Step 1.1.1.3.5.1

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 $a \cdot c = - 1 \cdot 3 = - 3$ and whose sum is $b = - 2$ .

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Step 1.1.1.3.5.1.1

Factor $- 2$ out of $- 2 x$ .

$\frac{- x^{2} - 2 (x) + 3}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.5.1.2

Rewrite $- 2$ as $1$ plus $- 3$

$\frac{- x^{2} + (1 - 3) x + 3}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.5.1.3

Apply the distributive property.

$\frac{- x^{2} + 1 x - 3 x + 3}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.5.1.4

Multiply $x$ by $1$ .

$\frac{- x^{2} + x - 3 x + 3}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.5.2

Factor out the greatest common factor from each group.

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Step 1.1.1.3.5.2.1

Group the first two terms and the last two terms.

$\frac{(- x^{2} + x) - 3 x + 3}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.5.2.2

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

$\frac{x (- x + 1) + 3 (- x + 1)}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.5.3

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

$\frac{(- x + 1) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.6

Factor $- 1$ out of $- x$ .

$\frac{(- (x) + 1) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.7

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{(- (x) - 1 (- 1)) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.8

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

$\frac{- (x - 1) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.9

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$\frac{- 1 (x - 1) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.1.1.3.10

Move the negative in front of the fraction.

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

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{(x - 1) (x + 3)}{{(x^{2} + 3)}^{2}}$ .

$- \frac{(x - 1) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $- \frac{(x - 1) (x + 3)}{{(x^{2} + 3)}^{2}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$- \frac{(x - 1) (x + 3)}{{(x^{2} + 3)}^{2}} = 0$

Step 1.2.2

Set the numerator equal to zero.

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

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

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

$x - 1 = 0$

$x + 3 = 0$

Step 1.2.3.2

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

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Step 1.2.3.2.1

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

$x - 1 = 0$

Step 1.2.3.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.3.3

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

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Step 1.2.3.3.1

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

$x + 3 = 0$

Step 1.2.3.3.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.3.4

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

$x = 1, - 3$

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 $\frac{x + 1}{x^{2} + 3}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 1$ .

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Step 1.4.1.1

Substitute $1$ for $x$ .

$\frac{(1) + 1}{{(1)}^{2} + 3}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Add $1$ and $1$ .

$\frac{2}{1^{2} + 3}$

Step 1.4.1.2.2

Simplify the denominator.

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Step 1.4.1.2.2.1

One to any power is one.

$\frac{2}{1 + 3}$

Step 1.4.1.2.2.2

Add $1$ and $3$ .

$\frac{2}{4}$

Step 1.4.1.2.3

Cancel the common factor of $2$ and $4$ .

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Step 1.4.1.2.3.1

Factor $2$ out of $2$ .

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

Step 1.4.1.2.3.2

Cancel the common factors.

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Step 1.4.1.2.3.2.1

Factor $2$ out of $4$ .

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

Step 1.4.1.2.3.2.2

Cancel the common factor.

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

Step 1.4.1.2.3.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 1.4.2

Evaluate at $x = - 3$ .

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Step 1.4.2.1

Substitute $- 3$ for $x$ .

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

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Add $- 3$ and $1$ .

$\frac{- 2}{{(- 3)}^{2} + 3}$

Step 1.4.2.2.2

Simplify the denominator.

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Step 1.4.2.2.2.1

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

$\frac{- 2}{9 + 3}$

Step 1.4.2.2.2.2

Add $9$ and $3$ .

$\frac{- 2}{12}$

Step 1.4.2.2.3

Reduce the expression by cancelling the common factors.

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Step 1.4.2.2.3.1

Cancel the common factor of $- 2$ and $12$ .

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Step 1.4.2.2.3.1.1

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{12}$

Step 1.4.2.2.3.1.2

Cancel the common factors.

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Step 1.4.2.2.3.1.2.1

Factor $2$ out of $12$ .

$\frac{2 \cdot - 1}{2 \cdot 6}$

Step 1.4.2.2.3.1.2.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 \cdot 6}$

Step 1.4.2.2.3.1.2.3

Rewrite the expression.

$\frac{- 1}{6}$

Step 1.4.2.2.3.2

Move the negative in front of the fraction.

$- \frac{1}{6}$

Step 1.4.3

List all of the points.

$(1, \frac{1}{2}), (- 3, - \frac{1}{6})$

Step 2

Exclude the points that are not on the interval.

$(1, \frac{1}{2})$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = - 1$ .

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Step 3.1.1

Substitute $- 1$ for $x$ .

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

Step 3.1.2

Simplify.

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Step 3.1.2.1

Add $- 1$ and $1$ .

$\frac{0}{{(- 1)}^{2} + 3}$

Step 3.1.2.2

Simplify the denominator.

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Step 3.1.2.2.1

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

$\frac{0}{1 + 3}$

Step 3.1.2.2.2

Add $1$ and $3$ .

$\frac{0}{4}$

Step 3.1.2.3

Divide $0$ by $4$ .

$0$

Step 3.2

Evaluate at $x = 2$ .

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Step 3.2.1

Substitute $2$ for $x$ .

$\frac{(2) + 1}{{(2)}^{2} + 3}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Add $2$ and $1$ .

$\frac{3}{2^{2} + 3}$

Step 3.2.2.2

Simplify the denominator.

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Step 3.2.2.2.1

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

$\frac{3}{4 + 3}$

Step 3.2.2.2.2

Add $4$ and $3$ .

$\frac{3}{7}$

Step 3.3

List all of the points.

$(- 1, 0), (2, \frac{3}{7})$

Step 4

Compare the $f (x)$ 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 $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(1, \frac{1}{2})$

Absolute Minimum: $(- 1, 0)$

Step 5