📐 Part 1: Differentiation in Kinematics

a) Find the velocity function v(t) by differentiating the displacement function.

b) Find the acceleration function a(t) by differentiating the velocity function.

c) What is the velocity at t = 2 seconds?

d) What is the acceleration at t = 3 seconds?

∫ Part 2: Integration in Kinematics

a) Find the displacement function s(t) if the initial position s(0) = 10 feet.

b) Calculate the total displacement from t = 1 to t = 4 seconds using definite integration.

a) Find the velocity function v(t).

b) Find the displacement function s(t).

📊 Part 3: Graph Interpretation

Sketch a velocity vs. time graph where v(t) starts at 0, increases linearly to 20 ft/s at t = 4s, remains constant until t = 6s, then decreases linearly back to 0 at t = 10s.

a) During which time intervals is the acceleration positive, negative, or zero?

b) Calculate the total displacement from t = 0 to t = 10 seconds by finding the area under the velocity curve.

c) What is the average velocity over the entire 10-second interval?

🚗 Part 4: Real-World Application

a) Find when the car is momentarily at rest (velocity = 0).

b) Determine the car's maximum acceleration during this time interval.

c) At what time(s) is the acceleration equal to zero?

d) Explain the physical meaning of your answer from part (c) in terms of the car's motion.

🎯 Part 5: Conceptual Understanding