Determine whether the functions y 1 and y 2 are linearly dependent on the interval (0,1). y 1equals2 cosine squared t minus 1, y 2equals6 cosine 2 t Select the correct choice below and, if necessary, fill in the answer box within your choice. A. Since y 1equals(nothing )y 2 on (0,1), the functions are linearly independent on (0,1). (Simplify your answer.) B. Since y 1equals(nothing )y 2 on (0,1), the functions are linearly dependent on (0,1). (Simplify your answer.) C. Since y 1 is not a constant multiple of y 2 on (0,1), the functions are linearly dependent on (0,1). D. Since y 1 is not a constant multiple of y 2 on (0,1), the functions are linearly independent on (0,1).

D. Since y₁ is not a constant multiple of y₂ on (0,1), the functions are linearly independent on (0,1).

Step-by-step explanation:

y₁=2Cos²t-1

y₂=6Cos2t

We want to determine if y₁ and y₂ are linearly independent in the interval (0,1).

To do this, we show that there does not exist any c₁ and c₂ in (0,1) that makes the expression:

f(t)=c₁(2Cos²t-1)+c₂6Cos2t=0.

If c₁ and c₂=0

f(t)=0

Let c₁=1 and c₂=-⅓

f(t)=c₁(2Cos²t-1)+c₂6Cos2t

=2Cos²t-1-⅓*6(cos²t-sin²t)

=2Cos²t-1-2cos²t+2sin²t

=2sin²t-1

Since f(t)≠0, y₁ is not a constant multiple of y₂ and the functions are linearly independent on (0,1).

the distance from any point on the parabola to the focus is the exact same as the distance from that point to the directrix.

if you could provide answer choices that might allow me to answer the question easier if my above answer isn't an option! (: