A projectile is fired vertically upward into the air; its position (in feet) above the ground after t seconds is given by the function s (t). For the following functions, use limits to determine the instantaneous velocity of the projectile at t = a seconds for the given value of a.
s(t) = -16t² + 128t + 192; a = 2

Verified step by step guidance

1. Step 1: Understand that the instantaneous velocity of the projectile at time t = a is given by the derivative of the position function s(t) evaluated at t = a. This is because the derivative of a position function with respect to time gives the velocity function.
2. Step 2: The position function is s(t) = -16t^2 + 128t + 192. To find the instantaneous velocity, we need to find the derivative of s(t) with respect to t, which is denoted as s'(t).
3. Step 3: Differentiate the function s(t) = -16t^2 + 128t + 192. Use the power rule for differentiation: if f(t) = at^n, then f'(t) = n*at^(n-1). Apply this rule to each term in s(t).
4. Step 4: After differentiating, you will obtain s'(t) = -32t + 128. This is the velocity function, which gives the velocity of the projectile at any time t.
5. Step 5: Evaluate the velocity function s'(t) at t = a, where a = 2. Substitute t = 2 into s'(t) to find the instantaneous velocity at that specific time.