Calculus Examples

$z = e^{- (x^{2} + y^{2})}$

Step 1

Rewrite the equation as $e^{- (x^{2} + y^{2})} = z$ .

$e^{- (x^{2} + y^{2})} = z$

Step 2

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

$\ln (e^{- (x^{2} + y^{2})}) = \ln (z)$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{- (x^{2} + y^{2})})$ by moving $- (x^{2} + y^{2})$ outside the logarithm.

$- (x^{2} + y^{2}) \ln (e) = \ln (z)$

Step 3.2

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

$- (x^{2} + y^{2}) \cdot 1 = \ln (z)$

Step 3.3

Apply the distributive property.

$(- x^{2} - y^{2}) \cdot 1 = \ln (z)$

Step 3.4

Multiply $- x^{2} - y^{2}$ by $1$ .

$- x^{2} - y^{2} = \ln (z)$

Step 4

Add $x^{2}$ to both sides of the equation.

$- y^{2} = \ln (z) + x^{2}$

Step 5

Divide each term in $- y^{2} = \ln (z) + x^{2}$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- y^{2} = \ln (z) + x^{2}$ by $- 1$ .

$\frac{- y^{2}}{- 1} = \frac{\ln (z)}{- 1} + \frac{x^{2}}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} = \frac{\ln (z)}{- 1} + \frac{x^{2}}{- 1}$

Step 5.2.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{\ln (z)}{- 1} + \frac{x^{2}}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Move the negative one from the denominator of $\frac{\ln (z)}{- 1}$ .

$y^{2} = - 1 \cdot \ln (z) + \frac{x^{2}}{- 1}$

Step 5.3.1.2

Rewrite $- 1 \cdot \ln (z)$ as $- \ln (z)$ .

$y^{2} = - \ln (z) + \frac{x^{2}}{- 1}$

Step 5.3.1.3

Move the negative one from the denominator of $\frac{x^{2}}{- 1}$ .

$y^{2} = - \ln (z) - 1 \cdot x^{2}$

Step 5.3.1.4

Rewrite $- 1 \cdot x^{2}$ as $- x^{2}$ .

$y^{2} = - \ln (z) - x^{2}$

Step 6

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

$y = \pm \sqrt{- \ln (z) - x^{2}}$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 7.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

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

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

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

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

Step 8

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

$z > 0$

Step 9

Set the radicand in $\sqrt{- \ln (z) - x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$- \ln (z) - x^{2} \geq 0$

Step 10

Add $x^{2}$ to both sides of the inequality.

$- \ln (z) \geq x^{2}$

Step 11

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

$[N o (Max i m u m), \infty)$

Set-Builder Notation:

${z | z \geq N o (Max i m u m)}$