Question 1

Given the parametric equations $$x = e$$^{4t}$$ + 1$$ and $$y = e$$^{4t}$$, which of the following is the correct Cartesian equation relating x$ and y$?

Accepted Answer

$$x = y + 1$$

Question 2

If $$x = e$$^{4t}$$ + 1$$ and $$y = e$$^{4t}$$, what is the value of t$ when x = 5$?

Accepted Answer

$$t = \frac{1}{4} \ln(4)$$

Question 3

For the parametric equations $$x = e$$^{4t}$$ + 1$$ and $$y = e$$^{4t}$$, what is the (x$$, y)$$ pair when t = 0$?

Accepted Answer

$$(2, 1)$$

Question 4

Which of the following best describes the graph of the Cartesian equation $$x = y + 1$$?

Accepted Answer

A straight line with slope 1 and $$y-$$intercept $$-1$$

Question 5

Suppose a population grows according to $$y = e$$^{4t}$$, where t$ is measured in years. If the initial population is 1, what will the population be after 2 years?

Accepted Answer

$$e^{8}$$

Question 6

Given $$x = e$$^{4t}$$ + 1$$, solve for $$t in $$terms of $$x.$$

Accepted Answer

$$t = \frac{1}{4} \ln(x - 1)$$

Question 7

If $$y = e$$^{4t}$$, what is $$\frac{dy}{dt}$$?

Accepted Answer

$$4e^{4t}$$

Question 8

For the parametric equations $$x = e$$^{4t}$$ + 1$$ and $$y = e$$^{4t}$$, what is $$\frac{dx}{dt}$$?

Accepted Answer

$$4e^{4t}$$

Question 9

Given $$x = e$$^{4t}$$ + 1$$ and $$y = e$$^{4t}$$, what is $$\frac{dx}{dy}$$?

Accepted Answer

$$1$$

Question 10

If $$x = e$$^{4t}$$ + 1$$ and $$y = e$$^{4t}$$, which of the following is the correct expression for t$ in terms of y$?

Accepted Answer

$$t = \frac{1}{4} \ln(y)$$

Question 11

A scientist models the concentration of a substance over time as $$y = e$$^{4t}$$. If the concentration doubles every k$ years, what is the value of k$?

Accepted Answer

$$k = \frac{\ln(2)}{4}$$

Question 12

If $$x = e$$^{4t}$$ + 1$$ and $$y = e$$^{4t}$$, what is the value of x$ when y = 7$?

Accepted Answer

$$8$$

Question 13

Which of the following statements is true about the parametric curve defined by $$x = e$$^{4t}$$ + 1$$, $$y = e$$^{4t}$$ for t$$ \in \mathbb{R}$$?

Accepted Answer

The curve lies entirely in the region where $$x \u003e 1$$ and $$y \u003e 0$$

Question 14

If $$x = e$$^{4t}$$ + 1$$ and $$y = e$$^{4t}$$, what is the inverse function expressing t$ in terms of x$?

Accepted Answer

$$t = \frac{1}{4} \ln(x - 1)$$

Question 15

A medical researcher uses the model $$y = e$$^{4t}$$ to describe the growth of bacteria. If the bacteria count is 100 at t = 0$, what will the count be at t = 0.5$?

Accepted Answer

$$100 e$$^{2}$$

Parametric Equations and Cartesian Conversion

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