Calculus Examples

$k \sqrt{x} - \ln (x)$

Step 1

Solve for $y$ .

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Step 1.1

Add $\ln (x)$ to both sides of the equation.

$y \sqrt{x} = \ln (x)$

Step 1.2

Divide each term in $y \sqrt{x} = \ln (x)$ by $\sqrt{x}$ and simplify.

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Step 1.2.1

Divide each term in $y \sqrt{x} = \ln (x)$ by $\sqrt{x}$ .

$\frac{y \sqrt{x}}{\sqrt{x}} = \frac{\ln (x)}{\sqrt{x}}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $\sqrt{x}$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{y \sqrt{x}}{\sqrt{x}} = \frac{\ln (x)}{\sqrt{x}}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (x)}{\sqrt{x}}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Multiply $\frac{\ln (x)}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$y = \frac{\ln (x)}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 1.2.3.2

Combine and simplify the denominator.

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Step 1.2.3.2.1

Multiply $\frac{\ln (x)}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$y = \frac{\ln (x) \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 1.2.3.2.2

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

$y = \frac{\ln (x) \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

Step 1.2.3.2.3

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

$y = \frac{\ln (x) \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

Step 1.2.3.2.4

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

$y = \frac{\ln (x) \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 1.2.3.2.5

Add $1$ and $1$ .

$y = \frac{\ln (x) \sqrt{x}}{{\sqrt{x}}^{2}}$

Step 1.2.3.2.6

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

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Step 1.2.3.2.6.1

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

$y = \frac{\ln (x) \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

Step 1.2.3.2.6.2

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

$y = \frac{\ln (x) \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Step 1.2.3.2.6.3

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

$y = \frac{\ln (x) \sqrt{x}}{x^{\frac{2}{2}}}$

Step 1.2.3.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.3.2.6.4.1

Cancel the common factor.

$y = \frac{\ln (x) \sqrt{x}}{x^{\frac{2}{2}}}$

Step 1.2.3.2.6.4.2

Rewrite the expression.

$y = \frac{\ln (x) \sqrt{x}}{x^{1}}$

Step 1.2.3.2.6.5

Simplify.

$y = \frac{\ln (x) \sqrt{x}}{x}$

Step 2

Find the domain for $k \sqrt{x} - \ln (x) = 0$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 2.1

Set the argument in $\ln (x)$ greater than $0$ to find where the expression is defined.

$x > 0$

Step 2.2

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

$x \geq 0$

Step 2.3

Set the denominator in $\frac{\ln (x) \sqrt{x}}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 2.4

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Step 3

To find the radical expression end point, substitute the $x$ value $0$ , which is the least value in the domain, into $f (x) = \frac{\ln (x) \sqrt{x}}{x}$ .

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Step 3.1

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

$f (0) = \frac{\ln (0) \sqrt{0}}{0}$

Step 3.2

The expression contains a division by $0$ . The expression is undefined.

$f (0) = Undefined$

Undefined

Step 4

The radical expression end point is $(0, Undefined)$ .

$(0, Undefined)$

Step 5

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

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Step 5.1

Substitute the $x$ value $1$ into $f (x) = \frac{\ln (x) \sqrt{x}}{x}$ . In this case, the point is $(1, 0)$ .

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Step 5.1.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{\ln (1) \sqrt{1}}{1}$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Divide $\ln (1) \sqrt{1}$ by $1$ .

$f (1) = \ln (1) \sqrt{1}$

Step 5.1.2.2

The natural logarithm of $1$ is $0$ .

$f (1) = 0 \sqrt{1}$

Step 5.1.2.3

Any root of $1$ is $1$ .

$f (1) = 0 \cdot 1$

Step 5.1.2.4

Multiply $0$ by $1$ .

$f (1) = 0$

Step 5.1.2.5

The final answer is $0$ .

$y = 0$

Step 5.2

Substitute the $x$ value $2$ into $f (x) = \frac{\ln (x) \sqrt{x}}{x}$ . In this case, the point is $(2, \frac{\ln (2) \sqrt{2}}{2})$ .

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Step 5.2.1

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

$f (2) = \frac{\ln (2) \sqrt{2}}{2}$

Step 5.2.2

The final answer is $\frac{\ln (2) \sqrt{2}}{2}$ .

$y = \frac{\ln (2) \sqrt{2}}{2}$

Step 5.3

The square root can be graphed using the points around the vertex $(0, Undefined), (1, 0), (2, 0.49)$

$\begin{array}{c} x & y \\ 0 & Undefined \\ 1 & 0 \\ 2 & 0.49 \end{array}$

Step 6