Finite Math Examples

\frac{e^{2 x} - 1}{2 x}

$\frac{e^{2 x} - 1}{2 x}$

Step 1

Write

\frac{e^{2 x} - 1}{2 x}

$\frac{e^{2 x} - 1}{2 x}$ as an equation.

y = \frac{e^{2 x} - 1}{2 x}

$y = \frac{e^{2 x} - 1}{2 x}$

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in

0

$0$ for

y

$y$ and solve for

x

$x$ .

0 = \frac{e^{2 x} - 1}{2 x}

$0 = \frac{e^{2 x} - 1}{2 x}$

Step 2.2

Solve the equation.

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Step 2.2.1

Set the numerator equal to zero.

e^{2 x} - 1 = 0

$e^{2 x} - 1 = 0$

Step 2.2.2

Solve the equation for

x

$x$ .

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Step 2.2.2.1

Add

1

$1$ to both sides of the equation.

e^{2 x} = 1

$e^{2 x} = 1$

Step 2.2.2.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

ln (e^{2 x}) = ln (1)

$\ln (e^{2 x}) = \ln (1)$

Step 2.2.2.3

Expand the left side.

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Step 2.2.2.3.1

Expand

ln (e^{2 x})

$\ln (e^{2 x})$ by moving

2 x

$2 x$ outside the logarithm.

2 x ln (e) = ln (1)

$2 x \ln (e) = \ln (1)$

Step 2.2.2.3.2

The natural logarithm of

e

$e$ is

1

$1$ .

2 x \cdot 1 = ln (1)

$2 x \cdot 1 = \ln (1)$

Step 2.2.2.3.3

Multiply

2

$2$ by

1

$1$ .

2 x = ln (1)

$2 x = \ln (1)$

2 x = ln (1)

$2 x = \ln (1)$

Step 2.2.2.4

Simplify the right side.

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Step 2.2.2.4.1

The natural logarithm of

1

$1$ is

0

$0$ .

2 x = 0

$2 x = 0$

2 x = 0

$2 x = 0$

Step 2.2.2.5

Divide each term in

2 x = 0

$2 x = 0$ by

2

$2$ and simplify.

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Step 2.2.2.5.1

Divide each term in

2 x = 0

$2 x = 0$ by

2

$2$ .

\frac{2 x}{2} = \frac{0}{2}

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.2.2.5.2

Simplify the left side.

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Step 2.2.2.5.2.1

Cancel the common factor of

2

$2$ .

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Step 2.2.2.5.2.1.1

Cancel the common factor.

\frac{2 x}{2} = \frac{0}{2}

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.2.2.5.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{0}{2}

$x = \frac{0}{2}$

x = \frac{0}{2}

$x = \frac{0}{2}$

x = \frac{0}{2}

$x = \frac{0}{2}$

Step 2.2.2.5.3

Simplify the right side.

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Step 2.2.2.5.3.1

Divide

0

$0$ by

2

$2$ .

x = 0

$x = 0$

x = 0

$x = 0$

x = 0

$x = 0$

x = 0

$x = 0$

Step 2.2.3

Exclude the solutions that do not make

0 = \frac{e^{2 x} - 1}{2 x}

$0 = \frac{e^{2 x} - 1}{2 x}$ true.

No solution

$No solution$

No solution

$No solution$

Step 2.3

To find the x-intercept(s), substitute in

0

$0$ for

y

$y$ and solve for

x

$x$ .

x-intercept(s):

None

$None$

x-intercept(s):

None

$None$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in

0

$0$ for

x

$x$ and solve for

y

$y$ .

y = \frac{e^{2 (0)} - 1}{2 (0)}

$y = \frac{e^{2 (0)} - 1}{2 (0)}$

Step 3.2

The equation has an undefined fraction.

Undefined

Step 3.3

To find the y-intercept(s), substitute in

0

$0$ for

x

$x$ and solve for

y

$y$ .

y-intercept(s):

None

$None$

y-intercept(s):

None

$None$

Step 4

List the intersections.

x-intercept(s):

None

$None$

y-intercept(s):

None

$None$

Step 5

⎡ ⎢ ⎣ \begin{matrix} x^{2} & \frac{1}{2} \sqrt{π} & \int x d x \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$