Calculus Examples

$f (x) = \sqrt{x} + \frac{1}{\sqrt{x}}$ ; $(4, \frac{5}{2})$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\sqrt{x}$ from both sides of the equation.

$f (x) - \sqrt{x} = \frac{1}{\sqrt{x}}$

Step 1.2

Subtract $\frac{1}{\sqrt{x}}$ from both sides of the equation.

$f (x) - \sqrt{x} - \frac{1}{\sqrt{x}} = 0$

Step 2

Simplify each term.

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Step 2.1

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$f (x) - \sqrt{x} - (\frac{1}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}) = 0$

Step 2.2

Combine and simplify the denominator.

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Step 2.2.1

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$f (x) - \sqrt{x} - \frac{\sqrt{x}}{\sqrt{x} \sqrt{x}} = 0$

Step 2.2.2

Raise $\sqrt{x}$ to the power of $1$ .

$f (x) - \sqrt{x} - \frac{\sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}} = 0$

Step 2.2.3

Raise $\sqrt{x}$ to the power of $1$ .

$f (x) - \sqrt{x} - \frac{\sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}} = 0$

Step 2.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (x) - \sqrt{x} - \frac{\sqrt{x}}{{\sqrt{x}}^{1 + 1}} = 0$

Step 2.2.5

Add $1$ and $1$ .

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

Step 2.2.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 2.2.6.1

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

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

Step 2.2.6.2

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

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

Step 2.2.6.3

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

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

Step 2.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.6.4.1

Cancel the common factor.

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

Step 2.2.6.4.2

Rewrite the expression.

$f (x) - \sqrt{x} - \frac{\sqrt{x}}{x^{1}} = 0$

Step 2.2.6.5

Simplify.

$f (x) - \sqrt{x} - \frac{\sqrt{x}}{x} = 0$

Step 3

Find the value of $m$ using the formula for the equation of a line.

$y = m x + b$

Step 4

Substitute the value of $b$ into the equation.

$y = m x + f (x) - \sqrt{x} - \frac{\sqrt{x}}{x}$

Step 5

Substitute the value of $x$ into the equation.

$y = m (4) + f (4) - \sqrt{4} - \frac{\sqrt{4}}{4}$

Step 6

Substitute the value of $y$ into the equation.

$\frac{5}{2} = m (4) + f (4) - \sqrt{4} - \frac{\sqrt{4}}{4}$

Step 7

Find the value of $m$ .

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Step 7.1

Rewrite the equation as $m (4) + f (4) - \sqrt{4} - \frac{\sqrt{4}}{4} = \frac{5}{2}$ .

$m (4) + f (4) - \sqrt{4} - \frac{\sqrt{4}}{4} = \frac{5}{2}$

Step 7.2

Simplify $m (4) + f (4) - \sqrt{4} - \frac{\sqrt{4}}{4}$ .

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Move $4$ to the left of $m$ .

$4 m + f (4) - \sqrt{4} - \frac{\sqrt{4}}{4} = \frac{5}{2}$

Step 7.2.1.2

Move $4$ to the left of $f$ .

$4 m + 4 f - \sqrt{4} - \frac{\sqrt{4}}{4} = \frac{5}{2}$

Step 7.2.1.3

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

$4 m + 4 f - \sqrt{2^{2}} - \frac{\sqrt{4}}{4} = \frac{5}{2}$

Step 7.2.1.4

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

$4 m + 4 f - 1 \cdot 2 - \frac{\sqrt{4}}{4} = \frac{5}{2}$

Step 7.2.1.5

Multiply $- 1$ by $2$ .

$4 m + 4 f - 2 - \frac{\sqrt{4}}{4} = \frac{5}{2}$

Step 7.2.1.6

Simplify the numerator.

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Step 7.2.1.6.1

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

$4 m + 4 f - 2 - \frac{\sqrt{2^{2}}}{4} = \frac{5}{2}$

Step 7.2.1.6.2

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

$4 m + 4 f - 2 - \frac{2}{4} = \frac{5}{2}$

Step 7.2.1.7

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

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Step 7.2.1.7.1

Factor $2$ out of $2$ .

$4 m + 4 f - 2 - \frac{2 (1)}{4} = \frac{5}{2}$

Step 7.2.1.7.2

Cancel the common factors.

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Step 7.2.1.7.2.1

Factor $2$ out of $4$ .

$4 m + 4 f - 2 - \frac{2 \cdot 1}{2 \cdot 2} = \frac{5}{2}$

Step 7.2.1.7.2.2

Cancel the common factor.

$4 m + 4 f - 2 - \frac{2 \cdot 1}{2 \cdot 2} = \frac{5}{2}$

Step 7.2.1.7.2.3

Rewrite the expression.

$4 m + 4 f - 2 - \frac{1}{2} = \frac{5}{2}$

Step 7.2.2

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

$4 m + 4 f - 2 \cdot \frac{2}{2} - \frac{1}{2} = \frac{5}{2}$

Step 7.2.3

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

$4 m + 4 f + \frac{- 2 \cdot 2}{2} - \frac{1}{2} = \frac{5}{2}$

Step 7.2.4

Combine the numerators over the common denominator.

$4 m + 4 f + \frac{- 2 \cdot 2 - 1}{2} = \frac{5}{2}$

Step 7.2.5

Simplify the numerator.

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Step 7.2.5.1

Multiply $- 2$ by $2$ .

$4 m + 4 f + \frac{- 4 - 1}{2} = \frac{5}{2}$

Step 7.2.5.2

Subtract $1$ from $- 4$ .

$4 m + 4 f + \frac{- 5}{2} = \frac{5}{2}$

Step 7.2.6

Move the negative in front of the fraction.

$4 m + 4 f - \frac{5}{2} = \frac{5}{2}$

Step 7.3

Move all terms not containing $m$ to the right side of the equation.

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Step 7.3.1

Subtract $4 f$ from both sides of the equation.

$4 m - \frac{5}{2} = \frac{5}{2} - 4 f$

Step 7.3.2

Add $\frac{5}{2}$ to both sides of the equation.

$4 m = \frac{5}{2} - 4 f + \frac{5}{2}$

Step 7.3.3

Combine the numerators over the common denominator.

$4 m = - 4 f + \frac{5 + 5}{2}$

Step 7.3.4

Add $5$ and $5$ .

$4 m = - 4 f + \frac{10}{2}$

Step 7.3.5

Divide $10$ by $2$ .

$4 m = - 4 f + 5$

Step 7.4

Divide each term in $4 m = - 4 f + 5$ by $4$ and simplify.

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Step 7.4.1

Divide each term in $4 m = - 4 f + 5$ by $4$ .

$\frac{4 m}{4} = \frac{- 4 f}{4} + \frac{5}{4}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of $4$ .

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Step 7.4.2.1.1

Cancel the common factor.

$\frac{4 m}{4} = \frac{- 4 f}{4} + \frac{5}{4}$

Step 7.4.2.1.2

Divide $m$ by $1$ .

$m = \frac{- 4 f}{4} + \frac{5}{4}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

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

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Step 7.4.3.1.1

Factor $4$ out of $- 4 f$ .

$m = \frac{4 (- f)}{4} + \frac{5}{4}$

Step 7.4.3.1.2

Cancel the common factors.

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Step 7.4.3.1.2.1

Factor $4$ out of $4$ .

$m = \frac{4 (- f)}{4 (1)} + \frac{5}{4}$

Step 7.4.3.1.2.2

Cancel the common factor.

$m = \frac{4 (- f)}{4 \cdot 1} + \frac{5}{4}$

Step 7.4.3.1.2.3

Rewrite the expression.

$m = \frac{- f}{1} + \frac{5}{4}$

Step 7.4.3.1.2.4

Divide $- f$ by $1$ .

$m = - f + \frac{5}{4}$

Step 8

Now that the values of $m$ (slope) and $b$ (y-intercept) are known, substitute them into $y = m x + b$ to find the equation of the line.

$y = (- f + \frac{5}{4}) x + f (x) - \sqrt{x} - \frac{\sqrt{x}}{x}$

Step 9