Calculus Examples

$y = e^{x}$ , $y = e^{- 2 x}$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

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

Step 1.2

Solve $e^{x} = e^{- 2 x}$ for $x$ .

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Step 1.2.1

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x = - 2 x$

Step 1.2.2

Solve for $x$ .

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Step 1.2.2.1

Move all terms containing $x$ to the left side of the equation.

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Step 1.2.2.1.1

Add $2 x$ to both sides of the equation.

$x + 2 x = 0$

Step 1.2.2.1.2

Add $x$ and $2 x$ .

$3 x = 0$

Step 1.2.2.2

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 1.2.2.2.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 1.2.2.2.2

Simplify the left side.

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Step 1.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 1.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 1.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.3.1

Divide $0$ by $3$ .

$x = 0$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = e^{- 2 \cdot 0}$

Step 1.3.2

Simplify $e^{- 2 \cdot 0}$ .

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Step 1.3.2.1

Multiply $- 2$ by $0$ .

$y = e^{0}$

Step 1.3.2.2

Anything raised to $0$ is $1$ .

$y = 1$

Step 1.4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 1)$

Step 2

The area between the given curves is unbounded.

Unbounded area

Step 3