Differentiation and Trigonometric Functions

The video tutorial explains a particle motion problem where the position function is given as X(T) = sin^2(T). The goal is to find the acceleration when the velocity is 1/2. The process involves deriving the velocity from the position function using the chain rule, solving for the time when velocity equals 1/2 using trigonometric identities, and then calculating the acceleration by deriving the velocity function again and evaluating it at the specific time.

What is the position function given in the particle motion problem?

• X(T) = cos^2(T)

• X(T) = sec^2(T)

• X(T) = sin^2(T)

• X(T) = tan^2(T)

What is the first step in finding the velocity of the particle?

• Multiply the position function by a constant

• Solve the position function

• Differentiate the position function

• Integrate the position function

Which mathematical rule is used to differentiate the position function?

• Quotient rule

• Power rule

• Product rule

• Chain rule

What is the expression for velocity V(T) after differentiating the position function?

• V(T) = sin(T) + cos(T)

• V(T) = 2 sin(T) cos(T)

• V(T) = 2 cos(T) sin(T)

• V(T) = sin^2(T) + cos^2(T)

What formula is used to simplify 2 sin(T) cos(T)?

• Half-angle formula

• Product-to-sum formula

• Sum-to-product formula

• Double angle formula

At what angle is sin(2T) first equal to 1/2 for T ≥ 0?

• π/4

• π/3

• π/6

• π/2

What is the value of T when the velocity is first 1/2?

• π/8

• π/10

• π/12

• π/14

What is the next step after finding the time when velocity is 1/2?

• Find the position

• Find the acceleration

• Find the speed

• Find the displacement

What is the expression for acceleration A(T) after differentiating the velocity function?

• A(T) = 2 sin(2T)

• A(T) = 2 cos(2T)

• A(T) = sin(2T)

• A(T) = cos(2T)

What is the derivative of 2T with respect to T?

• 1

• 0

• T

• 2