Calculus Examples

$f (x) = 4 \sqrt{4 - x^{2}}$ , $[0, 2]$

Step 1

If $f$ is continuous on the interval $[a, b]$ and differentiable on $(a, b)$ , then at least one real number $c$ exists in the interval $(a, b)$ such that $f' (c) = \frac{f (b) - f a}{b - a}$ . The mean value theorem expresses the relationship between the slope of the tangent to the curve at $x = c$ and the slope of the line through the points $(a, f (a))$ and $(b, f (b))$ .

If $f (x)$ is continuous on $[a, b]$

and if $f (x)$ differentiable on $(a, b)$ ,

then there exists at least one point, $c$ in $[a, b]$ : $f' (c) = \frac{f (b) - f a}{b - a}$ .

Step 2

Check if $f (x) = 4 \sqrt{4 - x^{2}}$ is continuous.

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Step 2.1

To find whether the function is continuous on $[0, 2]$ or not, find the domain of $f (x) = 4 \sqrt{4 - x^{2}}$ .

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Step 2.1.1

Set the radicand in $\sqrt{4 - x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$4 - x^{2} \geq 0$

Step 2.1.2

Solve for $x$ .

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Step 2.1.2.1

Subtract $4$ from both sides of the inequality.

$- x^{2} \geq - 4$

Step 2.1.2.2

Divide each term in $- x^{2} \geq - 4$ by $- 1$ and simplify.

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Step 2.1.2.2.1

Divide each term in $- x^{2} \geq - 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 4}{- 1}$

Step 2.1.2.2.2

Simplify the left side.

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Step 2.1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 4}{- 1}$

Step 2.1.2.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} \leq \frac{- 4}{- 1}$

Step 2.1.2.2.3

Simplify the right side.

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Step 2.1.2.2.3.1

Divide $- 4$ by $- 1$ .

$x^{2} \leq 4$

Step 2.1.2.3

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{4}$

Step 2.1.2.4

Simplify the equation.

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Step 2.1.2.4.1

Simplify the left side.

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Step 2.1.2.4.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{4}$

Step 2.1.2.4.2

Simplify the right side.

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Step 2.1.2.4.2.1

Simplify $\sqrt{4}$ .

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Step 2.1.2.4.2.1.1

Rewrite $4$ as $2^{2}$ .

$| x | \leq \sqrt{2^{2}}$

Step 2.1.2.4.2.1.2

Pull terms out from under the radical.

$| x | \leq | 2 |$

Step 2.1.2.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| x | \leq 2$

Step 2.1.2.5

Write $| x | \leq 2$ as a piecewise.

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Step 2.1.2.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 2.1.2.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 2$

Step 2.1.2.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 2.1.2.5.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 2$

Step 2.1.2.5.5

Write as a piecewise.

${\begin{cases} x \leq 2 & x \geq 0 \\ - x \leq 2 & x < 0 \end{cases}$

Step 2.1.2.6

Find the intersection of $x \leq 2$ and $x \geq 0$ .

$0 \leq x \leq 2$

Step 2.1.2.7

Solve $- x \leq 2$ when $x < 0$ .

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Step 2.1.2.7.1

Divide each term in $- x \leq 2$ by $- 1$ and simplify.

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Step 2.1.2.7.1.1

Divide each term in $- x \leq 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{2}{- 1}$

Step 2.1.2.7.1.2

Simplify the left side.

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Step 2.1.2.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{2}{- 1}$

Step 2.1.2.7.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{2}{- 1}$

Step 2.1.2.7.1.3

Simplify the right side.

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Step 2.1.2.7.1.3.1

Divide $2$ by $- 1$ .

$x \geq - 2$

Step 2.1.2.7.2

Find the intersection of $x \geq - 2$ and $x < 0$ .

$- 2 \leq x < 0$

Step 2.1.2.8

Find the union of the solutions.

$- 2 \leq x \leq 2$

Step 2.1.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 2, 2]$

Set-Builder Notation:

${x | - 2 \leq x \leq 2}$

Interval Notation:

$[- 2, 2]$

Set-Builder Notation:

${x | - 2 \leq x \leq 2}$

Step 2.2

$f (x)$ is continuous on $[0, 2]$ .

The function is continuous.

Step 3

Find the derivative.

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Step 3.1

Find the first derivative.

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Step 3.1.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x^{2}}$ as ${(4 - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [4 {(4 - x^{2})}^{\frac{1}{2}}]$

Step 3.1.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 {(4 - x^{2})}^{\frac{1}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [{(4 - x^{2})}^{\frac{1}{2}}]$ .

$4 \frac{d}{d x} [{(4 - x^{2})}^{\frac{1}{2}}]$

Step 3.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 - x^{2}$ .

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Step 3.1.2.1

To apply the Chain Rule, set $u$ as $4 - x^{2}$ .

$4 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [4 - x^{2}])$

Step 3.1.2.2

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

$4 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [4 - x^{2}])$

Step 3.1.2.3

Replace all occurrences of $u$ with $4 - x^{2}$ .

$4 (\frac{1}{2} {(4 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [4 - x^{2}])$

Step 3.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 (\frac{1}{2} {(4 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [4 - x^{2}])$

Step 3.1.4

Combine $- 1$ and $\frac{2}{2}$ .

$4 (\frac{1}{2} {(4 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [4 - x^{2}])$

Step 3.1.5

Combine the numerators over the common denominator.

$4 (\frac{1}{2} {(4 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [4 - x^{2}])$

Step 3.1.6

Simplify the numerator.

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Step 3.1.6.1

Multiply $- 1$ by $2$ .

$4 (\frac{1}{2} {(4 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [4 - x^{2}])$

Step 3.1.6.2

Subtract $2$ from $1$ .

$4 (\frac{1}{2} {(4 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [4 - x^{2}])$

Step 3.1.7

Move the negative in front of the fraction.

$4 (\frac{1}{2} {(4 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [4 - x^{2}])$

Step 3.1.8

Combine $\frac{1}{2}$ and ${(4 - x^{2})}^{- \frac{1}{2}}$ .

$4 (\frac{{(4 - x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [4 - x^{2}])$

Step 3.1.9

Move ${(4 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 (\frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [4 - x^{2}])$

Step 3.1.10

Combine $\frac{1}{2 {(4 - x^{2})}^{\frac{1}{2}}}$ and $4$ .

$\frac{4}{2 {(4 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [4 - x^{2}]$

Step 3.1.11

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2 {(4 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [4 - x^{2}]$

Step 3.1.12

Cancel the common factors.

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Step 3.1.12.1

Factor $2$ out of $2 {(4 - x^{2})}^{\frac{1}{2}}$ .

$\frac{2 \cdot 2}{2 ({(4 - x^{2})}^{\frac{1}{2}})} \frac{d}{d x} [4 - x^{2}]$

Step 3.1.12.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 {(4 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [4 - x^{2}]$

Step 3.1.12.3

Rewrite the expression.

$\frac{2}{{(4 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [4 - x^{2}]$

Step 3.1.13

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

$\frac{2}{{(4 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}])$

Step 3.1.14

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{2}{{(4 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}])$

Step 3.1.15

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$\frac{2}{{(4 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- x^{2}]$

Step 3.1.16

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$\frac{2}{{(4 - x^{2})}^{\frac{1}{2}}} (- \frac{d}{d x} [x^{2}])$

Step 3.1.17

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

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

Step 3.1.18

Combine fractions.

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Step 3.1.18.1

Multiply $2$ by $- 1$ .

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

Step 3.1.18.2

Combine $- 2$ and $\frac{2}{{(4 - x^{2})}^{\frac{1}{2}}}$ .

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

Step 3.1.18.3

Multiply $- 2$ by $2$ .

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

Step 3.1.18.4

Combine $\frac{- 4}{{(4 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

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

Step 3.1.18.5

Move the negative in front of the fraction.

$f' (x) = - \frac{4 x}{{(4 - x^{2})}^{\frac{1}{2}}}$

Step 3.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{4 x}{{(4 - x^{2})}^{\frac{1}{2}}}$ .

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

Step 4

Find if the derivative is continuous on $(0, 2)$ .

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Step 4.1

To find whether the function is continuous on $(0, 2)$ or not, find the domain of $f' (x) = - \frac{4 x}{{(4 - x^{2})}^{\frac{1}{2}}}$ .

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Step 4.1.1

Convert expressions with fractional exponents to radicals.

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Step 4.1.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 4.1.1.2

Anything raised to $1$ is the base itself.

$- \frac{4 x}{\sqrt{4 - x^{2}}}$

Step 4.1.2

Set the radicand in $\sqrt{4 - x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$4 - x^{2} \geq 0$

Step 4.1.3

Solve for $x$ .

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Step 4.1.3.1

Subtract $4$ from both sides of the inequality.

$- x^{2} \geq - 4$

Step 4.1.3.2

Divide each term in $- x^{2} \geq - 4$ by $- 1$ and simplify.

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Step 4.1.3.2.1

Divide each term in $- x^{2} \geq - 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 4}{- 1}$

Step 4.1.3.2.2

Simplify the left side.

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Step 4.1.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 4}{- 1}$

Step 4.1.3.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} \leq \frac{- 4}{- 1}$

Step 4.1.3.2.3

Simplify the right side.

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Step 4.1.3.2.3.1

Divide $- 4$ by $- 1$ .

$x^{2} \leq 4$

Step 4.1.3.3

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{4}$

Step 4.1.3.4

Simplify the equation.

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Step 4.1.3.4.1

Simplify the left side.

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Step 4.1.3.4.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{4}$

Step 4.1.3.4.2

Simplify the right side.

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Step 4.1.3.4.2.1

Simplify $\sqrt{4}$ .

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Step 4.1.3.4.2.1.1

Rewrite $4$ as $2^{2}$ .

$| x | \leq \sqrt{2^{2}}$

Step 4.1.3.4.2.1.2

Pull terms out from under the radical.

$| x | \leq | 2 |$

Step 4.1.3.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| x | \leq 2$

Step 4.1.3.5

Write $| x | \leq 2$ as a piecewise.

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Step 4.1.3.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 4.1.3.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 2$

Step 4.1.3.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 4.1.3.5.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 2$

Step 4.1.3.5.5

Write as a piecewise.

${\begin{cases} x \leq 2 & x \geq 0 \\ - x \leq 2 & x < 0 \end{cases}$

Step 4.1.3.6

Find the intersection of $x \leq 2$ and $x \geq 0$ .

$0 \leq x \leq 2$

Step 4.1.3.7

Solve $- x \leq 2$ when $x < 0$ .

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Step 4.1.3.7.1

Divide each term in $- x \leq 2$ by $- 1$ and simplify.

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Step 4.1.3.7.1.1

Divide each term in $- x \leq 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{2}{- 1}$

Step 4.1.3.7.1.2

Simplify the left side.

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Step 4.1.3.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{2}{- 1}$

Step 4.1.3.7.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{2}{- 1}$

Step 4.1.3.7.1.3

Simplify the right side.

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Step 4.1.3.7.1.3.1

Divide $2$ by $- 1$ .

$x \geq - 2$

Step 4.1.3.7.2

Find the intersection of $x \geq - 2$ and $x < 0$ .

$- 2 \leq x < 0$

Step 4.1.3.8

Find the union of the solutions.

$- 2 \leq x \leq 2$

Step 4.1.4

Set the denominator in $\frac{4 x}{\sqrt{4 - x^{2}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{4 - x^{2}} = 0$

Step 4.1.5

Solve for $x$ .

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Step 4.1.5.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{4 - x^{2}}}^{2} = 0^{2}$

Step 4.1.5.2

Simplify each side of the equation.

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Step 4.1.5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x^{2}}$ as ${(4 - x^{2})}^{\frac{1}{2}}$ .

${({(4 - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.1.5.2.2

Simplify the left side.

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Step 4.1.5.2.2.1

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

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Step 4.1.5.2.2.1.1

Multiply the exponents in ${({(4 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 4.1.5.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(4 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.1.5.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.1.5.2.2.1.1.2.1

Cancel the common factor.

${(4 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.1.5.2.2.1.1.2.2

Rewrite the expression.

${(4 - x^{2})}^{1} = 0^{2}$

Step 4.1.5.2.2.1.2

Simplify.

$4 - x^{2} = 0^{2}$

Step 4.1.5.2.3

Simplify the right side.

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Step 4.1.5.2.3.1

Raising $0$ to any positive power yields $0$ .

$4 - x^{2} = 0$

Step 4.1.5.3

Solve for $x$ .

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Step 4.1.5.3.1

Subtract $4$ from both sides of the equation.

$- x^{2} = - 4$

Step 4.1.5.3.2

Divide each term in $- x^{2} = - 4$ by $- 1$ and simplify.

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Step 4.1.5.3.2.1

Divide each term in $- x^{2} = - 4$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 4}{- 1}$

Step 4.1.5.3.2.2

Simplify the left side.

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Step 4.1.5.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 4}{- 1}$

Step 4.1.5.3.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 4}{- 1}$

Step 4.1.5.3.2.3

Simplify the right side.

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Step 4.1.5.3.2.3.1

Divide $- 4$ by $- 1$ .

$x^{2} = 4$

Step 4.1.5.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Step 4.1.5.3.4

Simplify $\pm \sqrt{4}$ .

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Step 4.1.5.3.4.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Step 4.1.5.3.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 4.1.5.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1.5.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 4.1.5.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 4.1.5.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 4.1.6

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- 2, 2)$

Set-Builder Notation:

${x | - 2 < x < 2}$

Interval Notation:

$(- 2, 2)$

Set-Builder Notation:

${x | - 2 < x < 2}$

Step 4.2

$f' (x)$ is continuous on $(0, 2)$ .

The function is continuous.

Step 5

The function is differentiable on $(0, 2)$ because the derivative is continuous on $(0, 2)$ .

The function is differentiable.

Step 6

$f (x)$ satisfies the two conditions for the mean value theorem. It is continuous on $[0, 2]$ and differentiable on $(0, 2)$ .

$f (x)$ is continuous on $[0, 2]$ and differentiable on $(0, 2)$ .

Step 7

Evaluate $f (a)$ from the interval $[0, 2]$ .

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Step 7.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 4 \sqrt{4 - {(0)}^{2}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 4 \sqrt{4 - 0}$

Step 7.2.2

Multiply $- 1$ by $0$ .

$f (0) = 4 \sqrt{4 + 0}$

Step 7.2.3

Add $4$ and $0$ .

$f (0) = 4 \sqrt{4}$

Step 7.2.4

Rewrite $4$ as $2^{2}$ .

$f (0) = 4 \sqrt{2^{2}}$

Step 7.2.5

Pull terms out from under the radical, assuming positive real numbers.

$f (0) = 4 \cdot 2$

Step 7.2.6

Multiply $4$ by $2$ .

$f (0) = 8$

Step 7.2.7

The final answer is $8$ .

$8$

Step 8

Evaluate $f (b)$ from the interval $[0, 2]$ .

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Step 8.1

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

$f (2) = 4 \sqrt{4 - {(2)}^{2}}$

Step 8.2

Simplify the result.

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Step 8.2.1

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

$f (2) = 4 \sqrt{4 - 1 \cdot 4}$

Step 8.2.2

Multiply $- 1$ by $4$ .

$f (2) = 4 \sqrt{4 - 4}$

Step 8.2.3

Subtract $4$ from $4$ .

$f (2) = 4 \sqrt{0}$

Step 8.2.4

Rewrite $0$ as $0^{2}$ .

$f (2) = 4 \sqrt{0^{2}}$

Step 8.2.5

Pull terms out from under the radical, assuming positive real numbers.

$f (2) = 4 \cdot 0$

Step 8.2.6

Multiply $4$ by $0$ .

$f (2) = 0$

Step 8.2.7

The final answer is $0$ .

$0$

Step 9

Solve $- \frac{4 x}{{(4 - x^{2})}^{\frac{1}{2}}} = \frac{- (f (b) + f (a))}{- (b + a)}$ for $x$ . $- \frac{4 x}{{(4 - x^{2})}^{\frac{1}{2}}} = \frac{- (f (2) + f (0))}{- (2 + 0)}$ .

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Step 9.1

Simplify $\frac{(0) - (8)}{(2) - (0)}$ .

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Step 9.1.1

Simplify the numerator.

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Step 9.1.1.1

Multiply $- 1$ by $8$ .

$- \frac{4 x}{{(4 - x^{2})}^{\frac{1}{2}}} = \frac{0 - 8}{2 - (0)}$

Step 9.1.1.2

Subtract $8$ from $0$ .

$- \frac{4 x}{{(4 - x^{2})}^{\frac{1}{2}}} = \frac{- 8}{2 - (0)}$

Step 9.1.2

Simplify the denominator.

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Step 9.1.2.1

Multiply $- 1$ by $0$ .

$- \frac{4 x}{{(4 - x^{2})}^{\frac{1}{2}}} = \frac{- 8}{2 + 0}$

Step 9.1.2.2

Add $2$ and $0$ .

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

Step 9.1.3

Divide $- 8$ by $2$ .

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

Step 9.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = \sqrt{2}$

Step 10

There is a tangent line found at $x = \sqrt{2}$ parallel to the line that passes through the end points $a = 0$ and $b = 2$ .

There is a tangent line at $x = \sqrt{2}$ parallel to the line that passes through the end points $a = 0$ and $b = 2$

Step 11