Calculus Examples

$x = e^{t} - 3$ , $y = e^{2 t}$

Step 1

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

$x = e^{t} - 3$

Step 2

Rewrite the equation as $e^{t} - 3 = x$ .

$e^{t} - 3 = x$

Step 3

Add $3$ to both sides of the equation.

$e^{t} = x + 3$

Step 4

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

$\ln (e^{t}) = \ln (x + 3)$

Step 5

Expand the left side.

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Step 5.1

Expand $\ln (e^{t})$ by moving $t$ outside the logarithm.

$t \ln (e) = \ln (x + 3)$

Step 5.2

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

$t \cdot 1 = \ln (x + 3)$

Step 5.3

Multiply $t$ by $1$ .

$t = \ln (x + 3)$

Step 6

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

$y = e^{2 (\ln (x + 3))}$

Step 7

Simplify $e^{2 (\ln (x + 3))}$ .

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Step 7.1

Simplify $2 (\ln (x + 3))$ by moving $2$ inside the logarithm.

$y = e^{\ln ({(x + 3)}^{2})}$

Step 7.2

Exponentiation and log are inverse functions.

$y = {(x + 3)}^{2}$

Step 7.3

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$y = (x + 3) (x + 3)$

Step 7.4

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 7.4.1

Apply the distributive property.

$y = x (x + 3) + 3 (x + 3)$

Step 7.4.2

Apply the distributive property.

$y = x \cdot x + x \cdot 3 + 3 (x + 3)$

Step 7.4.3

Apply the distributive property.

$y = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Step 7.5

Simplify and combine like terms.

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Step 7.5.1

Simplify each term.

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Step 7.5.1.1

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Step 7.5.1.2

Move $3$ to the left of $x$ .

$y = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Step 7.5.1.3

Multiply $3$ by $3$ .

$y = x^{2} + 3 x + 3 x + 9$

Step 7.5.2

Add $3 x$ and $3 x$ .

$y = x^{2} + 6 x + 9$