Question 1

Given the parametric equations $$x = f(t) = t^2$ + 4t$$ and $$y = g(t) = t^2$, which value of t$ corresponds to a point where the tangent line is vertical?

Accepted Answer

$$t = -2$$

Question 2

For the parametric curve $$x = t^2$ + 4t$$, $$y = t^2$, what is the equation of the tangent line at t = -2$?

Accepted Answer

$$x = -4$$

Question 3

For the parametric equations $$x = t^2$ + 4t$$, $$y = t^2$, at which value of t$ is the tangent line horizontal?

Accepted Answer

$$t = 0$$

Question 4

Find the slope of the tangent line to the curve $$x = t^2$ + 4t$$, $$y = t^2$ at t = 1$.

Accepted Answer

$$m = \frac{1}{6}$$

Question 5

Given $$x = t^2$ + 4t$$, $$y = t^2$, what is the equation of the tangent line at t = 0$?

Accepted Answer

$$y = 0$$

Question 6

For the parametric curve $$x = 2t + 1$$, $$y = 2t - t^2$, what is the area enclosed by the curve and the x$-axis from t = 0$ to t = 2$?

Accepted Answer

$$\frac{8}{3}$$

Question 7

Which of the following expressions gives $$\frac{dy}{dx}$$ for the parametric equations $$x = t^2$ + 4t$$, $$y = t^2$?

Accepted Answer

$$\frac{dy}{dx} = \frac{2t}{2t + 4}$$

Question 8

For the parametric curve $$x = 2t + 1$$, $$y = 2t - t^2$, at what value of t$ does the curve intersect the x$-axis?

Accepted Answer

$$t = 2$$

Question 9

Given $$x = t^2$ + 4t$$, $$y = t^2$, what is the point on the curve at t = -2$?

Accepted Answer

$$(-4, 4)$$

Question 10

For the parametric curve $$x = 2t + 1$$, $$y = 2t - t^2$, what is the slope of the tangent line at t = 1$?

Accepted Answer

$$0$$

Question 11

Which of the following is the correct formula for the area enclosed by a parametric curve $$x = f(t)$$, $$y = g(t)$$ from $$t = a to$$ $$t = b$$?

Accepted Answer

$$A = \int_a^b y \frac{dx}{dt} dt$$

Question 12

For the parametric equations $$x = t^2$ + 4t$$, $$y = t^2$, what is the value of $$\frac{dy}{dx}$$ at t = 0$?

Accepted Answer

$$0$$

Question 13

Given $$x = 2t + 1$$, $$y = 2t - t^2$, what is the equation of the tangent line at t = 0$?

Accepted Answer

$$y = 2x - 2$$

Question 14

For the parametric curve $$x = t^2$ + 4t$$, $$y = t^2$, which of the following best describes the behavior of the tangent line at t = -2$?

Accepted Answer

The tangent line is vertical

Question 15

For the parametric curve $$x = 2t + 1$$, $$y = 2t - t^2$, what is the x$-coordinate when y = 0$?

Accepted Answer

$$x = 5$$

Question 16

Given $$x = t^2$ + 4t$$, $$y = t^2$, what is the equation of the tangent line at t = 1$?

Accepted Answer

$$y - 1 = \frac{1}{6}(x - 5)$$

Question 17

For the parametric curve $$x = 2t + 1$$, $$y = 2t - t^2$, what is the maximum value of y$ for t$ in $$[0,2]$$?

Accepted Answer

$$y = 1$$

Question 18

For the parametric equations $$x = t^2$ + 4t$$, $$y = t^2$, what is the x$-coordinate when the tangent line is horizontal?

Accepted Answer

$$x = 0$$

Question 19

Given $$x = 2t + 1$$, $$y = 2t - t^2$, what is the value of $$\frac{dy}{dx}$$ at t = 2$?

Accepted Answer

$$-1$$

Question 20

For the parametric curve $$x = t^2$ + 4t$$, $$y = t^2$, what is the y$-coordinate when the tangent line is vertical?

Accepted Answer

$$y = 4$$

Study Notes: Tangent Lines and Enclosed Area for Parametric Curves

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