Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
Determine the velocity and acceleration of the object at t = 1. 
f(t) = 2t³ - 21t² + 60t; 0 ≤ t ≤ 6

Verified step by step guidance

1. Step 1: Understand that the velocity of an object is the derivative of its position function with respect to time. Therefore, to find the velocity function v(t), differentiate the position function f(t) = 2t^3 - 21t^2 + 60t with respect to t.
2. Step 2: Apply the power rule of differentiation to find the derivative of each term in f(t). The power rule states that d/dt [t^n] = n*t^(n-1).
3. Step 3: Differentiate the first term: d/dt [2t^3] = 3*2*t^(3-1) = 6t^2.
4. Step 4: Differentiate the second term: d/dt [-21t^2] = 2*(-21)*t^(2-1) = -42t.
5. Step 5: Differentiate the third term: d/dt [60t] = 60, since the derivative of t with respect to t is 1. Combine these results to get the velocity function v(t) = 6t^2 - 42t + 60.

Key Concepts

Position Function

The position function, denoted as s = f(t), describes the location of an object at any given time t. In this case, the function f(t) = 2t³ - 21t² + 60t provides a mathematical representation of the object's position in feet as a function of time in seconds. Understanding this function is crucial for determining the object's velocity and acceleration.

Velocity

Velocity is the rate of change of position with respect to time, mathematically represented as the first derivative of the position function, v(t) = f'(t). For the given function, calculating the derivative will yield the velocity at any time t, including t = 1. This concept is essential for understanding how fast the object is moving at a specific moment.

Acceleration

Acceleration is the rate of change of velocity with respect to time, represented as the second derivative of the position function, a(t) = f''(t). By finding the second derivative of the position function, we can determine the object's acceleration at any time, including t = 1. This concept helps in understanding how the object's velocity is changing over time.