Question 1

Given the parametric equations $$x = t^2$ - 1$$, $$y = t^3$ - t$$, which of the following expressions gives the slope of the tangent line at any point $$t$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{3t^2$ - 1}{2t}$$

Question 2

For the parametric curve $$x = t^2$ - 1$$, $$y = t^3$ - t$$, at which value of $$t is $$the tangent line horizontal?

Accepted Answer

$$t = 0 or$$ $$t = 1 or$$ $$t = -1 or$$ $$t = \sqrt{\frac{1}{3}} or$$ $$t = -\sqrt{\frac{1}{3}}$$

Question 3

For the same curve, at which value(s) of $$t is $$the tangent line vertical?

Accepted Answer

$$t = 0$$

Question 4

What is the equation of the tangent line to the curve $$x = t^2$ - 1$$, $$y = t^3$ - t at$$ $$t = -2$$?

Accepted Answer

$$y + 6 = -7(x + 3)$$

Question 5

Which integral represents the area enclosed by the parametric curve $$x = t^2$ - 1$$, $$y = t^3$ - t$$ and the $$x-$$axis for $$t in$$ $$[-1, 1]$$?

Accepted Answer

$$\int_{-1}^{1} (t^3$ - t) \cdot 2t \, dt$$

Question 6

Suppose a particle moves along the curve $$x = t^2$ - 1$$, $$y = t^3$ - t. $$What is the speed of the particle at $$t = 1$$?

Accepted Answer

$$\sqrt{4 + 4} = 2\sqrt{2}$$

Question 7

If $$x = t^2$ - 1$$ and $$y = t^3$ - t$$, what is $$\frac{d^2$y}{dx^2$} at$$ $$t = 1$$?

Accepted Answer

$$\frac{3}{2}$$

Question 8

Which of the following best describes the geometric meaning of $$\frac{dy}{dx}$$ for a parametric curve $$x = f(t)$$, $$y = g(t)$$?

Accepted Answer

It is the slope of the tangent line to the curve at the point corresponding to $$t.$$

Question 9

For the parametric curve $$x = t^2$ - 1$$, $$y = t^3$ - t$$, which of the following is true about the symmetry of the curve?

Accepted Answer

The curve is symmetric about the origin.

Question 10

A chemical reaction's rate is modeled by the parametric equations $$x = t^2$ - 1$$, $$y = t^3$ - t$$, where $$x is $$time and $$y is $$concentration. At what time is the concentration changing most rapidly?

Accepted Answer

$$t = 0$$

Question 11

Given $$x = t^2$ - 1$$, $$y = t^3$ - t$$, what is the $$y-$$coordinate when the tangent line is vertical?

Accepted Answer

$$y = 0$$

Question 12

If $$x = t^2$ - 1$$, $$y = t^3$ - t$$, what is the area enclosed by the curve and the $$x-$$axis for $$t in$$ $$[-1, 1]$$?

Accepted Answer

$$0$$

Question 13

Which of the following is the correct formula for the arc length of the parametric curve $$x = t^2$ - 1$$, $$y = t^3$ - t$$ from $$t = a to$$ $$t = b$$?

Accepted Answer

$$\int_a^b \sqrt{(2t)^2$ + (3t^2$ - 1)^2$} \, dt$$

Question 14

Suppose $$x = t^2$ - 1$$, $$y = t^3$ - t$$ models the path of a robot. At $$t = 0$$, what is the direction of motion (as a vector)?

Accepted Answer

$$(0, -1)$$

Study Notes: Parametric Equations, Tangent Lines, and Area Enclosed by Curves

Guida di studio - Domande di pratica

Study guide - Flashcard