Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

e. There are two points on the curve x=−4 cos t, y=sin t, for 0≤t≤2π, at which there is a vertical tangent line.

Accepted Answer

First, recall that a vertical tangent line occurs where the derivative $$ \frac{dy}{dx}  is $$undefined or infinite. This happens when $$ \frac{dx}{dt} = 0 $$ but $$ \frac{dy}{dt} 
eq 0 . $$Given the parametric equations $$ x = -4 \cos t $$ and $$ y = \sin t $$, compute the derivatives with respect to $$ t $$: $$ \frac{dx}{dt} = -4 (-\sin t) = 4 \sin t $$ $$ \frac{dy}{dt} = \cos t $$ Set $$ \frac{dx}{dt} = 0  to $$find potential points of vertical tangents: $$ 4 \sin t = 0 \implies \sin t = 0 $$ Solve for $$ t  in $$the interval $$ 0 \leq t \leq 2\pi $$: $$ t = 0, \pi, 2\pi . $$For each $$ t $$, check if $$ \frac{dy}{dt} 
eq 0  to $$confirm vertical tangents: $$ \frac{dy}{dt} = \cos t $$ Evaluate $$ \cos t  at $$these points: - At $$ t=0 $$, $$ \cos 0 = 1 
eq 0  - At$$ $$ t=\pi $$, $$ \cos \pi = -1 
eq 0  - At$$ $$ t=2\pi $$, $$ \cos 2\pi = 1 
eq 0 $$ Since $$ \frac{dy}{dt} 
eq 0  at $$all these points, vertical tangents occur at $$ t=0, \pi, 2\pi . $$However, note that $$ t=0 $$ and $$ t=2\pi $$ correspond to the same point on the curve because the parametric equations are periodic with period $$ 2\pi . $$Therefore, there are exactly two distinct points on the curve where vertical tangent lines occur: one at $$ t=0  (or$$ $$ 2\pi ) $$and one at $$ t=\pi .$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

e. There are two points on the curve x=−4 cos t, y=sin t, for 0≤t≤2π, at which there is a vertical tangent line.

Risposta video verificata per un problema simile:

Concetti chiave

Parametric Curves and Their Derivatives

Vertical Tangent Lines in Parametric Form

Analyzing the Interval for Parameter t

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. e. There are two points on the curve x=−4 cos t, y=sin t, for 0≤t≤2π, at which there is a vertical tangent line.

Risposta video verificata per un problema simile:

Concetti chiave

Parametric Curves and Their Derivatives

Vertical Tangent Lines in Parametric Form

Analyzing the Interval for Parameter t

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

e. There are two points on the curve x=−4 cos t, y=sin t, for 0≤t≤2π, at which there is a vertical tangent line.