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

$$y = 12x + 14$$

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 $$rac{dy}{dx} in $$terms of $$t$$?

Accepted Answer

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

Question 5

Find the area enclosed by the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$ and the $$x-$$axis for $$t in$$ $$[a, b]. $$Which integral correctly represents this area?

Accepted Answer

$$\int_a^b (t^3$ - 2) \frac{dx}{dt} \ dt$$

Question 6

Suppose a particle moves along the curve $$x = t^2$ - 4$$, $$y = t^3$ - 2. At$$ $$t = 1$$, what is the slope of the tangent line to the curve?

Accepted Answer

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

Question 7

For the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, which of the following is the correct formula for the tangent line at $$t = a$$?

Accepted Answer

$$y = y(a) + \frac{dy}{dx}|_{t=a} (x - x(a))$$

Question 8

If $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, what is the $$x-$$coordinate of the point where the tangent is horizontal?

Accepted Answer

$$x = -4$$

Question 9

For the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, what is the $$y-$$coordinate of the point where the tangent is vertical?

Accepted Answer

$$y = -2$$

Question 10

A medical device traces the path $$x = t^2$ - 4$$, $$y = t^3$ - 2 as it $$moves. At what $$t$$ does the device cross the $$x-$$axis?

Accepted Answer

$$t = 0$$

Question 11

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

Accepted Answer

$$-6$$

Question 12

Which of the following best describes the geometric meaning of $$\frac{dy}{dx}$$ for a parametric curve?

Accepted Answer

It is the rate of change of $$y$$ with respect to $$x$$ along the curve.

Question 13

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

Accepted Answer

$$y = -2$$

Question 14

For the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, which of the following is true about the tangent line at $$t = 0$$?

Accepted Answer

The tangent line is horizontal.

Question 15

A chemical reaction follows the path $$x = t^2$ - 4$$, $$y = t^3$ - 2 in a $$reactor. At $$t = 2$$, what is the instantaneous rate of change of $$y$$ with respect to $$x$$?

Accepted Answer

$$6$$

Question 16

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

Accepted Answer

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

Question 17

For the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$, what is the value of $$y$$ when $$x = 0$$?

Accepted Answer

$$y = -2$$

Question 18

Suppose you are asked to graph the tangent lines at points where the parametric curve $$x = t^2$ - 4$$, $$y = t^3$ - 2$$ has vertical and horizontal tangents. Which points should you label?

Accepted Answer

$$(x, y) = (-4, -2)$$ and $$(x, y) = (0, 6)$$

Question 19

A robot arm traces the curve $$x = t^2$ - 4$$, $$y = t^3$ - 2. At$$ $$t = -2$$, what is the equation of the tangent line?

Accepted Answer

$$y = -6x + 14$$

Parametric Equations, Tangent Lines Homework

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