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.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of instantaneous velocity, limits help us find the derivative of a function at a specific point, which is the slope of the tangent line to the curve at that point. This concept is crucial for understanding how functions behave near particular values.

Derivatives

The derivative of a function measures how the function's output changes as its input changes, essentially providing the rate of change at any given point. For the position function s(t) of the projectile, the derivative s'(t) gives the instantaneous velocity. This concept is key to solving problems involving motion, as it allows us to determine how fast an object is moving at any moment.

Projectile Motion

Projectile motion describes the motion of an object that is launched into the air and is subject to gravitational acceleration. The position function s(t) = -16t² + 128t + 192 models the height of the projectile over time, where the coefficients represent the effects of gravity and initial velocity. Understanding this concept is essential for analyzing the behavior of the projectile and applying calculus to find its instantaneous velocity.