Calculus Examples

$\ln (x + \sqrt{x^{2} - 1})$

Step 1

Find the $y$ value at $x = 1$ .

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Step 1.1

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

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

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Add $1$ and $1$ .

$f (1) = \ln (1 + \sqrt{2 (1 - 1)})$

Step 1.2.1.2

Subtract $1$ from $1$ .

$f (1) = \ln (1 + \sqrt{2 \cdot 0})$

Step 1.2.1.3

Multiply $2$ by $0$ .

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

Step 1.2.1.4

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

$f (1) = \ln (1 + \sqrt{0^{2}})$

Step 1.2.1.5

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

$f (1) = \ln (1 + 0)$

Step 1.2.2

Add $1$ and $0$ .

$f (1) = \ln (1)$

Step 1.2.3

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

$f (1) = 0$

Step 1.2.4

The final answer is $0$ .

$0$

Step 1.3

The $y$ value at $x = 1$ is $0$ .

$y = 0$

Step 2

Find the $y$ value at $x = 2$ .

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Step 2.1

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

$f (2) = \ln ((2) + \sqrt{((2) + 1) ((2) - 1)})$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Add $2$ and $1$ .

$f (2) = \ln (2 + \sqrt{3 (2 - 1)})$

Step 2.2.1.2

Subtract $1$ from $2$ .

$f (2) = \ln (2 + \sqrt{3 \cdot 1})$

Step 2.2.1.3

Multiply $3$ by $1$ .

$f (2) = \ln (2 + \sqrt{3})$

Step 2.2.2

The final answer is $\ln (2 + \sqrt{3})$ .

$\ln (2 + \sqrt{3})$

Step 2.3

The $y$ value at $x = 2$ is $\ln (2 + \sqrt{3})$ .

$y = \ln (2 + \sqrt{3})$

Step 3

Find the $y$ value at $x = 3$ .

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Step 3.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \ln ((3) + \sqrt{((3) + 1) ((3) - 1)})$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Add $3$ and $1$ .

$f (3) = \ln (3 + \sqrt{4 (3 - 1)})$

Step 3.2.1.2

Subtract $1$ from $3$ .

$f (3) = \ln (3 + \sqrt{4 \cdot 2})$

Step 3.2.1.3

Multiply $4$ by $2$ .

$f (3) = \ln (3 + \sqrt{8})$

Step 3.2.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.2.1.4.1

Factor $4$ out of $8$ .

$f (3) = \ln (3 + \sqrt{4 (2)})$

Step 3.2.1.4.2

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

$f (3) = \ln (3 + \sqrt{2^{2} \cdot 2})$

Step 3.2.1.5

Pull terms out from under the radical.

$f (3) = \ln (3 + 2 \sqrt{2})$

Step 3.2.2

The final answer is $\ln (3 + 2 \sqrt{2})$ .

$\ln (3 + 2 \sqrt{2})$

Step 3.3

The $y$ value at $x = 3$ is $\ln (3 + 2 \sqrt{2})$ .

$y = \ln (3 + 2 \sqrt{2})$

Step 4

List the points to graph.

$(1, 0), (2, 1.31695789), (3, 1.76274717)$

Step 5

Select a few points to graph.

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 1.317 \\ 3 & 1.763 \end{array}$

Step 6