**Question:** **True or False:** If \( f(t) \) is the distance of a particle from a fixed point after \( t \) units of time, then \( f'(t) \) is the instantaneous velocity of the particle after \( t \) units of time. **Explanation:** This statement involves understanding the relationship between a function and its derivative in calculus. Here, \( f(t) \) denotes the distance function, where \( t \) represents time. The derivative \( f'(t) \) gives the rate of change of the function \( f(t) \) at any specific moment \( t \). In this context, \( f'(t) \) represents the instantaneous velocity of the particle, as velocity is defined as the rate of change of distance with respect to time. Thus, the statement is **True**.

8) A runner was training for a 21 mile long race. The distance D (in miles) in function of time t (in weeks) is given by: D = 2(t-1)² +15 a) b) c) How many miles have done the runner in the first week? What is the average rate of training between the first and the third week? dD Calculate the derivative of D with respect to time dt and explain the obtained answer.

The displacement of a particle moving back and forth along a straight line is given by the motion equation s(t) = 2t^2 + 5t – 1 meters, where t is time in seconds. The value of the average velocity during the time period [0, 1] is --- - s(t) = 2t² + 5t – 1 The slope of the tangent line to the graph of f(x) = (x + 1)² at x = 10 is