Question 1

Given the parametric equations $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, find the equation of the tangent line at $$t = -2.$$

Accepted Answer

$$y = 12x + 34$$

Question 2

For the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, at which value of $$t is $$the tangent line vertical?

Accepted Answer

$$t = 0$$

Question 3

For the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, at which value of $$t is $$the tangent line horizontal?

Accepted Answer

$$t = 0$$

Question 4

Given $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, what is the slope of the tangent line at $$t = 1$$?

Accepted Answer

$$m = 1.5$$

Question 5

Which of the following integrals gives the area enclosed by the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$ and the $$x-$$axis for $$t in$$ $$[a, b]$$?

Accepted Answer

$$\int_a^b y \frac{dx}{dt} \, dt$$

Question 6

Suppose a particle moves along the curve $$x = t^2$ - 4$$, $$y = t^3$ - 2. At$$ $$t = 2$$, what is the velocity vector?

Accepted Answer

$$(4, 12)$$

Question 7

For the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, what is the equation of the tangent line at $$t = 1$$?

Accepted Answer

$$y = 1.5x + 0.5$$

Question 8

If $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, what is $$\frac{dy}{dx} in $$terms of $$t$$?

Accepted Answer

$$\frac{3t^2$}{2t}$$

Question 9

For the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, which value of $$t$$ gives a point where the tangent is horizontal?

Accepted Answer

$$t = 0$$

Question 10

Given $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, what is the area enclosed by the curve and the $$x-$$axis for $$t in$$ $$[0, 2]$$?

Accepted Answer

$$\$$int_0$$^2 (t^3$ - 2) \cdot 2t \, dt$$

Question 11

A curve is defined by $$x = t^2$ - 4$$, $$y = t^3$ - 2. $$What is the $$y-$$coordinate when the tangent line is vertical?

Accepted Answer

$$y = -2$$

Question 12

For the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, what is the $$x-$$coordinate when the tangent line is horizontal?

Accepted Answer

$$x = -4$$

Question 13

Given $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, what is the equation of the tangent line at $$t = 0$$?

Accepted Answer

$$y = -2$$

Question 14

For the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, what is the value of $$\frac{dy}{dx} at$$ $$t = -2$$?

Accepted Answer

$$6$$

Question 15

Suppose $$x = t^2$ - 4$$, $$y = t^3$ - 2. $$What is the $$x-$$coordinate when $$t = 1$$?

Accepted Answer

$$x = -3$$

Question 16

Suppose $$x = t^2$ - 4$$, $$y = t^3$ - 2. $$What is the $$y-$$coordinate when $$t = 1$$?

Accepted Answer

$$y = -1$$

Question 17

Given $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, what is the equation of the tangent line at $$t = -2$$?

Accepted Answer

$$y = 6x + 34$$

Question 18

For the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, what is the $$y-$$intercept of the tangent line at $$t = 1$$?

Accepted Answer

$$0.5$$

Question 19

Suppose $$x = t^2$ - 4$$, $$y = t^3$ - 2. $$What is the value of $$\frac{dx}{dt} at$$ $$t = 2$$?

Accepted Answer

$$4$$

Question 20

Suppose $$x = t^2$ - 4$$, $$y = t^3$ - 2. $$What is the value of $$\frac{dy}{dt} at$$ $$t = 2$$?

Accepted Answer

$$12$$

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

Guida di studio - Domande di pratica

Study guide - Flashcard