Calculus Examples

$x^{2} + y^{2} = 26 y$

Step 1

Solve the equation as $y$ in terms of $x$ .

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Step 1.1

Subtract $26 y$ from both sides of the equation.

$x^{2} + y^{2} - 26 y = 0$

Step 1.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.3

Substitute the values $a = 1$ , $b = - 26$ , and $c = x^{2}$ into the quadratic formula and solve for $y$ .

$\frac{26 \pm \sqrt{{(- 26)}^{2} - 4 \cdot (1 \cdot x^{2})}}{2 \cdot 1}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Rewrite $4 \cdot 1 \cdot x^{2}$ as ${(2 \cdot 1 x)}^{2}$ .

$y = \frac{26 \pm \sqrt{{(- 26)}^{2} - {(2 \cdot (1 x))}^{2}}}{2 \cdot 1}$

Step 1.4.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 26$ and $b = 2 \cdot 1 x$ .

$y = \frac{26 \pm \sqrt{(- 26 + 2 \cdot (1 x)) (- 26 - (2 \cdot (1 x)))}}{2 \cdot 1}$

Step 1.4.1.3

Simplify.

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Step 1.4.1.3.1

Multiply $2$ by $1$ .

$y = \frac{26 \pm \sqrt{(- 26 + 2 x) (- 26 - (2 \cdot (1 x)))}}{2 \cdot 1}$

Step 1.4.1.3.2

Factor $2$ out of $- 26 + 2 x$ .

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Step 1.4.1.3.2.1

Factor $2$ out of $- 26$ .

$y = \frac{26 \pm \sqrt{(2 \cdot - 13 + 2 x) (- 26 - (2 \cdot (1 x)))}}{2 \cdot 1}$

Step 1.4.1.3.2.2

Factor $2$ out of $2 \cdot - 13 + 2 x$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (- 26 - (2 \cdot (1 x)))}}{2 \cdot 1}$

Step 1.4.1.3.3

Combine exponents.

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Step 1.4.1.3.3.1

Multiply $2$ by $1$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (- 26 - (2 x))}}{2 \cdot 1}$

Step 1.4.1.3.3.2

Multiply $2$ by $- 1$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (- 26 - 2 x)}}{2 \cdot 1}$

Step 1.4.1.4

Factor $2$ out of $- 26 - 2 x$ .

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Step 1.4.1.4.1

Factor $2$ out of $- 26$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (2 (- 13) - 2 x)}}{2 \cdot 1}$

Step 1.4.1.4.2

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

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (2 (- 13) + 2 (- x))}}{2 \cdot 1}$

Step 1.4.1.4.3

Factor $2$ out of $2 (- 13) + 2 (- x)$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (2 (- 13 - x))}}{2 \cdot 1}$

$y = \frac{26 \pm \sqrt{2 (- 13 + x) \cdot (2 (- 13 - x))}}{2 \cdot 1}$

Step 1.4.1.5

Multiply $2$ by $2$ .

$y = \frac{26 \pm \sqrt{4 (- 13 + x) (- 13 - x)}}{2 \cdot 1}$

Step 1.4.1.6

Rewrite $4 (- 13 + x) (- 13 - x)$ as $2^{2} ((- 13 + x) (- 13 - x))$ .

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Step 1.4.1.6.1

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

$y = \frac{26 \pm \sqrt{2^{2} (- 13 + x) (- 13 - x)}}{2 \cdot 1}$

Step 1.4.1.6.2

Add parentheses.

$y = \frac{26 \pm \sqrt{2^{2} ((- 13 + x) (- 13 - x))}}{2 \cdot 1}$

Step 1.4.1.7

Pull terms out from under the radical.

$y = \frac{26 \pm 2 \sqrt{(- 13 + x) (- 13 - x)}}{2 \cdot 1}$

Step 1.4.2

Multiply $2$ by $1$ .

$y = \frac{26 \pm 2 \sqrt{(- 13 + x) (- 13 - x)}}{2}$

Step 1.4.3

Simplify $\frac{26 \pm 2 \sqrt{(- 13 + x) (- 13 - x)}}{2}$ .

$y = 13 \pm \sqrt{(- 13 + x) (- 13 - x)}$

Step 1.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Rewrite $4 \cdot 1 \cdot x^{2}$ as ${(2 \cdot 1 x)}^{2}$ .

$y = \frac{26 \pm \sqrt{{(- 26)}^{2} - {(2 \cdot (1 x))}^{2}}}{2 \cdot 1}$

Step 1.5.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 26$ and $b = 2 \cdot 1 x$ .

$y = \frac{26 \pm \sqrt{(- 26 + 2 \cdot (1 x)) (- 26 - (2 \cdot (1 x)))}}{2 \cdot 1}$

Step 1.5.1.3

Simplify.

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Step 1.5.1.3.1

Multiply $2$ by $1$ .

$y = \frac{26 \pm \sqrt{(- 26 + 2 x) (- 26 - (2 \cdot (1 x)))}}{2 \cdot 1}$

Step 1.5.1.3.2

Factor $2$ out of $- 26 + 2 x$ .

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Step 1.5.1.3.2.1

Factor $2$ out of $- 26$ .

$y = \frac{26 \pm \sqrt{(2 \cdot - 13 + 2 x) (- 26 - (2 \cdot (1 x)))}}{2 \cdot 1}$

Step 1.5.1.3.2.2

Factor $2$ out of $2 \cdot - 13 + 2 x$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (- 26 - (2 \cdot (1 x)))}}{2 \cdot 1}$

Step 1.5.1.3.3

Combine exponents.

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Step 1.5.1.3.3.1

Multiply $2$ by $1$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (- 26 - (2 x))}}{2 \cdot 1}$

Step 1.5.1.3.3.2

Multiply $2$ by $- 1$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (- 26 - 2 x)}}{2 \cdot 1}$

Step 1.5.1.4

Factor $2$ out of $- 26 - 2 x$ .

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Step 1.5.1.4.1

Factor $2$ out of $- 26$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (2 (- 13) - 2 x)}}{2 \cdot 1}$

Step 1.5.1.4.2

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

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (2 (- 13) + 2 (- x))}}{2 \cdot 1}$

Step 1.5.1.4.3

Factor $2$ out of $2 (- 13) + 2 (- x)$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (2 (- 13 - x))}}{2 \cdot 1}$

$y = \frac{26 \pm \sqrt{2 (- 13 + x) \cdot (2 (- 13 - x))}}{2 \cdot 1}$

Step 1.5.1.5

Multiply $2$ by $2$ .

$y = \frac{26 \pm \sqrt{4 (- 13 + x) (- 13 - x)}}{2 \cdot 1}$

Step 1.5.1.6

Rewrite $4 (- 13 + x) (- 13 - x)$ as $2^{2} ((- 13 + x) (- 13 - x))$ .

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Step 1.5.1.6.1

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

$y = \frac{26 \pm \sqrt{2^{2} (- 13 + x) (- 13 - x)}}{2 \cdot 1}$

Step 1.5.1.6.2

Add parentheses.

$y = \frac{26 \pm \sqrt{2^{2} ((- 13 + x) (- 13 - x))}}{2 \cdot 1}$

Step 1.5.1.7

Pull terms out from under the radical.

$y = \frac{26 \pm 2 \sqrt{(- 13 + x) (- 13 - x)}}{2 \cdot 1}$

Step 1.5.2

Multiply $2$ by $1$ .

$y = \frac{26 \pm 2 \sqrt{(- 13 + x) (- 13 - x)}}{2}$

Step 1.5.3

Simplify $\frac{26 \pm 2 \sqrt{(- 13 + x) (- 13 - x)}}{2}$ .

$y = 13 \pm \sqrt{(- 13 + x) (- 13 - x)}$

Step 1.5.4

Change the $\pm$ to $+$ .

$y = 13 + \sqrt{(- 13 + x) (- 13 - x)}$

Step 1.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Rewrite $4 \cdot 1 \cdot x^{2}$ as ${(2 \cdot 1 x)}^{2}$ .

$y = \frac{26 \pm \sqrt{{(- 26)}^{2} - {(2 \cdot (1 x))}^{2}}}{2 \cdot 1}$

Step 1.6.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 26$ and $b = 2 \cdot 1 x$ .

$y = \frac{26 \pm \sqrt{(- 26 + 2 \cdot (1 x)) (- 26 - (2 \cdot (1 x)))}}{2 \cdot 1}$

Step 1.6.1.3

Simplify.

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Step 1.6.1.3.1

Multiply $2$ by $1$ .

$y = \frac{26 \pm \sqrt{(- 26 + 2 x) (- 26 - (2 \cdot (1 x)))}}{2 \cdot 1}$

Step 1.6.1.3.2

Factor $2$ out of $- 26 + 2 x$ .

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Step 1.6.1.3.2.1

Factor $2$ out of $- 26$ .

$y = \frac{26 \pm \sqrt{(2 \cdot - 13 + 2 x) (- 26 - (2 \cdot (1 x)))}}{2 \cdot 1}$

Step 1.6.1.3.2.2

Factor $2$ out of $2 \cdot - 13 + 2 x$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (- 26 - (2 \cdot (1 x)))}}{2 \cdot 1}$

Step 1.6.1.3.3

Combine exponents.

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Step 1.6.1.3.3.1

Multiply $2$ by $1$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (- 26 - (2 x))}}{2 \cdot 1}$

Step 1.6.1.3.3.2

Multiply $2$ by $- 1$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (- 26 - 2 x)}}{2 \cdot 1}$

Step 1.6.1.4

Factor $2$ out of $- 26 - 2 x$ .

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Step 1.6.1.4.1

Factor $2$ out of $- 26$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (2 (- 13) - 2 x)}}{2 \cdot 1}$

Step 1.6.1.4.2

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

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (2 (- 13) + 2 (- x))}}{2 \cdot 1}$

Step 1.6.1.4.3

Factor $2$ out of $2 (- 13) + 2 (- x)$ .

$y = \frac{26 \pm \sqrt{2 (- 13 + x) (2 (- 13 - x))}}{2 \cdot 1}$

$y = \frac{26 \pm \sqrt{2 (- 13 + x) \cdot (2 (- 13 - x))}}{2 \cdot 1}$

Step 1.6.1.5

Multiply $2$ by $2$ .

$y = \frac{26 \pm \sqrt{4 (- 13 + x) (- 13 - x)}}{2 \cdot 1}$

Step 1.6.1.6

Rewrite $4 (- 13 + x) (- 13 - x)$ as $2^{2} ((- 13 + x) (- 13 - x))$ .

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Step 1.6.1.6.1

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

$y = \frac{26 \pm \sqrt{2^{2} (- 13 + x) (- 13 - x)}}{2 \cdot 1}$

Step 1.6.1.6.2

Add parentheses.

$y = \frac{26 \pm \sqrt{2^{2} ((- 13 + x) (- 13 - x))}}{2 \cdot 1}$

Step 1.6.1.7

Pull terms out from under the radical.

$y = \frac{26 \pm 2 \sqrt{(- 13 + x) (- 13 - x)}}{2 \cdot 1}$

Step 1.6.2

Multiply $2$ by $1$ .

$y = \frac{26 \pm 2 \sqrt{(- 13 + x) (- 13 - x)}}{2}$

Step 1.6.3

Simplify $\frac{26 \pm 2 \sqrt{(- 13 + x) (- 13 - x)}}{2}$ .

$y = 13 \pm \sqrt{(- 13 + x) (- 13 - x)}$

Step 1.6.4

Change the $\pm$ to $-$ .

$y = 13 - \sqrt{(- 13 + x) (- 13 - x)}$

Step 1.7

The final answer is the combination of both solutions.

$y = 13 + \sqrt{(- 13 + x) (- 13 - x)}$

$y = 13 - \sqrt{(- 13 + x) (- 13 - x)}$

$y = 13 + \sqrt{(- 13 + x) (- 13 - x)}$

$y = 13 - \sqrt{(- 13 + x) (- 13 - x)}$

Step 2

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

$y = 13 + \sqrt{(- 13 + x) (- 13 - x)} \to f (x) = 13 + \sqrt{(- 13 + x) (- 13 - x)}$

$y = 13 - \sqrt{(- 13 + x) (- 13 - x)} \to f (x) = 13 - \sqrt{(- 13 + x) (- 13 - x)}$

Step 3

Because the $y$ variable in the equation $x^{2} + y^{2} = 26 y$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

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Step 3.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} + y^{2}) = \frac{d}{d x} (26 y)$

Step 3.2

Differentiate the left side of the equation.

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Step 3.2.1

Differentiate.

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.2

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

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Step 3.2.2.1

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

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Step 3.2.2.1.1

To apply the Chain Rule, set $u$ as $y$ .

$2 x + \frac{d}{d u} [u^{2}] \frac{d}{d x} [y]$

Step 3.2.2.1.2

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

$2 x + 2 u \frac{d}{d x} [y]$

Step 3.2.2.1.3

Replace all occurrences of $u$ with $y$ .

$2 x + 2 y \frac{d}{d x} [y]$

Step 3.2.2.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$2 x + 2 y y'$

Step 3.2.3

Reorder terms.

$2 y y' + 2 x$

Step 3.3

Differentiate the right side of the equation.

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Step 3.3.1

Since $26$ is constant with respect to $x$ , the derivative of $26 y$ with respect to $x$ is $26 \frac{d}{d x} [y]$ .

$26 \frac{d}{d x} [y]$

Step 3.3.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$26 y'$

Step 3.4

Reform the equation by setting the left side equal to the right side.

$2 y y' + 2 x = 26 y'$

Step 3.5

Solve for $y'$ .

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Step 3.5.1

Subtract $26 y'$ from both sides of the equation.

$2 y y' + 2 x - 26 y' = 0$

Step 3.5.2

Subtract $2 x$ from both sides of the equation.

$2 y y' - 26 y' = - 2 x$

Step 3.5.3

Factor $2 y'$ out of $2 y y' - 26 y'$ .

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Step 3.5.3.1

Factor $2 y'$ out of $2 y y'$ .

$2 y' y - 26 y' = - 2 x$

Step 3.5.3.2

Factor $2 y'$ out of $- 26 y'$ .

$2 y' y + 2 y' \cdot - 13 = - 2 x$

Step 3.5.3.3

Factor $2 y'$ out of $2 y' y + 2 y' \cdot - 13$ .

$2 y' (y - 13) = - 2 x$

Step 3.5.4

Divide each term in $2 y' (y - 13) = - 2 x$ by $2 (y - 13)$ and simplify.

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Step 3.5.4.1

Divide each term in $2 y' (y - 13) = - 2 x$ by $2 (y - 13)$ .

$\frac{2 y' (y - 13)}{2 (y - 13)} = \frac{- 2 x}{2 (y - 13)}$

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Cancel the common factor of $2$ .

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Step 3.5.4.2.1.1

Cancel the common factor.

$\frac{2 y' (y - 13)}{2 (y - 13)} = \frac{- 2 x}{2 (y - 13)}$

Step 3.5.4.2.1.2

Rewrite the expression.

$\frac{y' (y - 13)}{y - 13} = \frac{- 2 x}{2 (y - 13)}$

Step 3.5.4.2.2

Cancel the common factor of $y - 13$ .

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Step 3.5.4.2.2.1

Cancel the common factor.

$\frac{y' (y - 13)}{y - 13} = \frac{- 2 x}{2 (y - 13)}$

Step 3.5.4.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 x}{2 (y - 13)}$

Step 3.5.4.3

Simplify the right side.

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Step 3.5.4.3.1

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

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Step 3.5.4.3.1.1

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

$y' = \frac{2 (- x)}{2 (y - 13)}$

Step 3.5.4.3.1.2

Cancel the common factors.

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Step 3.5.4.3.1.2.1

Cancel the common factor.

$y' = \frac{2 (- x)}{2 (y - 13)}$

Step 3.5.4.3.1.2.2

Rewrite the expression.

$y' = \frac{- x}{y - 13}$

Step 3.5.4.3.2

Move the negative in front of the fraction.

$y' = - \frac{x}{y - 13}$

Step 3.6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{x}{y - 13}$

Step 4

Set the numerator equal to zero.

$x = 0$

Step 5

Solve the function $f (x) = 13 + \sqrt{(- 13 + x) (- 13 - x)}$ at $x = 0$ .

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Step 5.1

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

$f (0) = 13 + \sqrt{(- 13 + 0) (- 13 - (0))}$

Step 5.2

Simplify the result.

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Step 5.2.1

Remove parentheses.

$f (0) = 13 + \sqrt{(- 13 + 0) (- 13 - (0))}$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Add $- 13$ and $0$ .

$f (0) = 13 + \sqrt{- 13 (- 13 - (0))}$

Step 5.2.2.2

Multiply $- 1$ by $0$ .

$f (0) = 13 + \sqrt{- 13 (- 13 + 0)}$

Step 5.2.2.3

Add $- 13$ and $0$ .

$f (0) = 13 + \sqrt{- 13 \cdot - 13}$

Step 5.2.2.4

Multiply $- 13$ by $- 13$ .

$f (0) = 13 + \sqrt{169}$

Step 5.2.2.5

Rewrite $169$ as $13^{2}$ .

$f (0) = 13 + \sqrt{13^{2}}$

Step 5.2.2.6

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

$f (0) = 13 + 13$

Step 5.2.3

Add $13$ and $13$ .

$f (0) = 26$

Step 5.2.4

The final answer is $26$ .

$26$

Step 6

Solve the function $f (x) = 13 - \sqrt{(- 13 + x) (- 13 - x)}$ at $x = 0$ .

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Step 6.1

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

$f (0) = 13 - \sqrt{(- 13 + 0) (- 13 - (0))}$

Step 6.2

Simplify the result.

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Step 6.2.1

Remove parentheses.

$f (0) = 13 - \sqrt{(- 13 + 0) (- 13 - (0))}$

Step 6.2.2

Simplify each term.

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Step 6.2.2.1

Add $- 13$ and $0$ .

$f (0) = 13 - \sqrt{- 13 (- 13 - (0))}$

Step 6.2.2.2

Multiply $- 1$ by $0$ .

$f (0) = 13 - \sqrt{- 13 (- 13 + 0)}$

Step 6.2.2.3

Add $- 13$ and $0$ .

$f (0) = 13 - \sqrt{- 13 \cdot - 13}$

Step 6.2.2.4

Multiply $- 13$ by $- 13$ .

$f (0) = 13 - \sqrt{169}$

Step 6.2.2.5

Rewrite $169$ as $13^{2}$ .

$f (0) = 13 - \sqrt{13^{2}}$

Step 6.2.2.6

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

$f (0) = 13 - 1 \cdot 13$

Step 6.2.2.7

Multiply $- 1$ by $13$ .

$f (0) = 13 - 13$

Step 6.2.3

Subtract $13$ from $13$ .

$f (0) = 0$

Step 6.2.4

The final answer is $0$ .

$0$

Step 7

The horizontal tangent lines are $y = 26, y = 0$

$y = 26, y = 0$

Step 8