Calculus Examples

$f (x) = x^{2} e^{- x}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{2} e^{- x} = 0$ .

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

Step 1.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$e^{- x} = 0$

Step 1.2.3

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 1.2.3.2

Solve $x^{2} = 0$ for $x$ .

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Step 1.2.3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 1.2.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.3.2.2.1

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

$x = \pm \sqrt{0^{2}}$

Step 1.2.3.2.2.2

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

$x = \pm 0$

Step 1.2.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.4

Set $e^{- x}$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $e^{- x}$ equal to $0$ .

$e^{- x} = 0$

Step 1.2.4.2

Solve $e^{- x} = 0$ for $x$ .

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Step 1.2.4.2.1

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

$\ln (e^{- x}) = \ln (0)$

Step 1.2.4.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 1.2.4.2.3

There is no solution for $e^{- x} = 0$

No solution

Step 1.2.5

The final solution is all the values that make $x^{2} e^{- x} = 0$ true.

$x = 0$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Simplify ${(0)}^{2} e^{- (0)}$ .

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Step 2.2.3.1

Raising $0$ to any positive power yields $0$ .

$y = 0 e^{- (0)}$

Step 2.2.3.2

Multiply $- 1$ by $0$ .

$y = 0 e^{0}$

Step 2.2.3.3

Anything raised to $0$ is $1$ .

$y = 0 \cdot 1$

Step 2.2.3.4

Multiply $0$ by $1$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(0, 0)$

y-intercept(s): $(0, 0)$

Step 4