Q 1. True/false - Calculus

Question

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: Every function $y = f \left(x\right)$ can be written in terms of parametric equations.

(b) True or False: Given parametric equations $x = x\left(t\right)$ and $y = y\left(t\right)$ the parameter can be eliminated to obtain the form y = f (x).

(c) True or False: Every parametric curve passes the vertical line test.

(d) True or False: Every curve in the plane has a unique expression in terms of parametric equations.

(e) True or False: If the functions x = f (t) and y = g(t) are differentiable for every t ∈ R, then the parametric curve defined by x and y is differentiable for every value of t.

(f) True or False: A curve parametrized by x = x(t), y = y(t) has a horizontal tangent line at (x(t0), y(t0)) if y (t) = 0.

(g) True or False: A curve parametrized by x = x(t), y = y(t) has a horizontal tangent line at (x(t0), y(t0)) if x (t) = 0 and y (t) = 0.

(h) True or False: The cycloid curve associated with a circle of radius r is made up of a series of semicircles of radius r.

Step-by-Step Solution

Answer

(a) True

(b) False

(c) False

(d) False

(e) False

(f) False

(g) True

(h) False 

Part (a) Step 1: Explanation

Parametric equations can be used to express any function $y=f\left(x\right)$

The statement is true.

Example: consider a function $y=x^2$

For any parameter $t$ the function is expressed as $y=t^2$

That is $x=t,y=t^2$

As a result, the answer is true.

Part (b) Step 1: Explanation

The parameter can be deleted to obtain the form $y=f \left(x\right)$ for a given parameter.

It is not always possible to remove a parameter and write it in the form $y=f \left(x\right)$

Example: $x=t+e^t,y=\sin t+\frac{1}{1+t^2}$ from these parametric equations it is not possible to remove the parameter $t$

Therefore, the answer is False.

Part (c) Step 1: Explanation

"The vertical line test passes every parametric curve." 

The vertical line test does not pass for every parametric curve. A curve is a function if it passes the vertical line test. A parametric curve does not have to be a function. Therefore, the answer is False. 

Part (d) Step 1: Explanation

"In terms of parametric equations, every curve in the plane has a unique expression." 

There are several parametric representations for every curve in a plane.

We can represent this equation by using the parameter $x=t$ then $y\left(t\right)=t^2-4$

It can also be represented by letting $x=t^3$ then $y\left(t\right)=t^6-4$

Therefore, the answer is False.

Part (e) Step 1: Explanation

The parametric curve produced by the functions $x=f\left(t\right), y=g\left(t\right)$ and $t\in \mathrm{ℝ}$ is not differentiable if they are differentiable for every $t\in \mathrm{ℝ}$

Therefore, the answer is False.

Part (f) Step 1: Explanation

 A curve is parameterized by $x=x\left(t\right), y=y\left(t\right)$ has a horizontal tangent line a $\left(x\left(t_0\right),y\left(t_0\right)\right)$ if $y^{\text{'}}\left(t\right)=0$ is not correct.

A curve will have a horizontal tangent line if $x^{\text{'}}\left(t\right)=0$

Therefore, the answer is False.

Part (g) Step 1: Explanation

 A curve is parameterized by $x=x\left(t\right), y=y\left(t\right)$ has a horizontal tangent line $\left(x\left(t_0\right),y\left(t_0\right)\right)$ if $x^{\text{'}}\left(t\right)\ne 0$ then $y^{\text{'}}\left(t\right)=0$ is correct.

Therefore, the answer is True.

Part (h) Step 1: Explanation

A cycloid is a curve that is related to a circle of radius $r$ and is composed of a succession of circles that are not semicircles of radius $r$ Without slipping, a circle rolls down a horizontal line. A cycloid is a curve traced out by a point on the circle as it rolls along the line. Therefore, the answer is False.