Q: For the parametric curve $$x = f(t)$$, $$y = g(t)$$, at what values of $$t is $$the tangent line horizontal?

When $$\frac{dy}{dt} = 0$$ and $$\frac{dx}{dt} \neq 0$$

Q: If $$x = f(t)$$ and $$y = g(t)$$, at what values of $$t is $$the tangent line vertical?

When $$\frac{dx}{dt} = 0$$ and $$\frac{dy}{dt} \neq 0$$

Q: If a curve is described by $$x = f(t)$$, $$y = g(t)$$, and $$\frac{dx}{dt} = 0 at$$ $$t = t_0$, what does this imply about the tangent line at t_0$?

The tangent line is vertical (provided $$\frac{dy}{dt} \neq 0)$$

Question 1

Suppose a curve is defined parametrically by $$x = f(t)$$ and $$y = g(t)$$, where $$f$$ and $$g$$ are differentiable functions. What is the formula for the slope of the tangent line at a point on the curve?

Accepted Answer

$$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$

Question 2

Given the parametric equations $$x = r(	heta - \sin	heta)$$ and $$y = r(1 - \cos	heta)$$ for a cycloid, which integral gives the area under one arch of the cycloid?

Accepted Answer

$$\int_{0}^{2\pi} y \, dx$$

Question 3

For the parametric curve $$x = f(t)$$, $$y = g(t)$$, at what values of $$t is $$the tangent line horizontal?

Accepted Answer

When $$\frac{dy}{dt} = 0$$ and $$\frac{dx}{dt} 
eq 0$$

Question 4

Which formula gives the length $$L of a $$curve described by $$x = f(t)$$, $$y = g(t)$$, $$a \leq t \leq b$$?

Accepted Answer

$$L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \$$right)^2$$ + \left( \frac{dy}{dt} \$$right)^2$$} \, dt$$

Question 5

A curve is given by $$x = \cos t$$, $$y = \sin t$$, $$0 \leq t \leq 2\pi. $$What is the total length of the curve?

Accepted Answer

$$2\pi$$

Question 6

For the parametric equations $$x = r \cos t$$, $$y = r \sin t$$, $$0 \leq t \leq \pi$$, what is the surface area generated by rotating the curve about the $$x-$$axis?

Accepted Answer

$$4\pi r^2$

Question 7

If $$x = f(t)$$ and $$y = g(t)$$, at what values of $$t is $$the tangent line vertical?

Accepted Answer

When $$\frac{dx}{dt} = 0$$ and $$\frac{dy}{dt} 
eq 0$$

Question 8

Given $$x = r(	heta - \sin	heta)$$ and $$y = r(1 - \cos	heta)$$, what is $$\frac{dx}{d	heta}$$?

Accepted Answer

$$r(1 - \cos	heta)$$

Question 9

Which of the following is the correct formula for the surface area $$S$$ generated by rotating a parametric curve $$x = f(t)$$, $$y = g(t)$$ about the $$x-$$axis?

Accepted Answer

$$S = \int_{a}^{b} 2\pi y \sqrt{\left( \frac{dx}{dt} \$$right)^2$$ + \left( \frac{dy}{dt} \$$right)^2$$} \, dt$$

Question 10

A parametric curve is given by $$x = t^2$, y$$ = $$t^3$$, $$0 \leq t \leq 2. $$What is the value of $$\frac{dy}{dx} at$$ $$t = 1$$?

Accepted Answer

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

Question 11

Suppose a particle moves along a path given by $$x = t^2$, y$$ = $$t^3. At$$ $$t = 0$$, what is the direction of the tangent line?

Accepted Answer

Vertical

Question 12

Which theorem allows you to compute the length of a curve described by parametric equations $$x = f(t)$$, $$y = g(t)$$, where $$f$$ and $$g$$ are continuous and differentiable?

Accepted Answer

Theorem 5

Question 13

For the parametric equations $$x = r \cos t$$, $$y = r \sin t$$, $$0 \leq t \leq \pi$$, what real-world object does the surface generated by rotating this curve about the $$x-$$axis represent?

Accepted Answer

A sphere

Question 14

Given $$x = t^2$, y$$ = $$t^3$$, $$0 \leq t \leq 2$$, what is the length of the curve from $$t = 0 to$$ $$t = 2$$?

Accepted Answer

$$\int_{0}^{2} \sqrt{4t^2$ + 9t^4$} \, dt$$

Question 15

If a curve is described by $$x = f(t)$$, $$y = g(t)$$, and $$\frac{dx}{dt} = 0 at$$ $$t = t_0$, what does this imply about the tangent line at t_0$?

Accepted Answer

The tangent line is vertical (provided $$\frac{dy}{dt} 
eq 0)$$

Question 16

For the parametric curve $$x = r(	heta - \sin	heta)$$, $$y = r(1 - \cos	heta)$$, what is the value of $$\frac{dy}{d	heta}$$?

Accepted Answer

$$r \sin	heta$$

Question 17

Which of the following integrals gives the surface area generated by rotating the parametric curve $$x = f(t)$$, $$y = g(t)$$ about the $$y-$$axis?

Accepted Answer

$$S = \int_{a}^{b} 2\pi x \sqrt{\left( \frac{dx}{dt} \$$right)^2$$ + \left( \frac{dy}{dt} \$$right)^2$$} \, dt$$

Question 18

A medical device traces a path given by $$x = 2t$$, $$y = t^2$ for 0$$ \leq t \leq 3$$. What is the total distance the device travels?

Accepted Answer

$$\int_{0}^{3} \sqrt{4 + 4t^2$} \, dt$$

Section 1.2 - Calculus with Parametric Curves

Guida di studio - Domande di pratica

Study guide - Flashcard