INTRODUCTION

Solving motion problems is a fundamental application of integral calculus to real-world scenarios. The Fundamental Theorem of Calculus is a conceptually important part of integral calculus.

The Evaluation Theorem is the second part of the fundamental theorem of calculus: "If f is continuous over [a, b] and F is any antiderivative of f on [a, b], then abf(x)dx = F(b) - F(a)."

SCENARIO

You are tracking the velocity and position of a rocket-propelled object near the surface of Mars. The velocity is v(t) and the position is s(t), where t is measured in seconds, s in meters, and v in meters per second. It is known that the v(t) = ds/dt = 4.94 - 3.72t and s(0) = 5.

REQUIREMENTS

A. Explain why the condition "f is continuous over [a, b]" from the Evaluation Theorem is fulfilled by this scenario.

B. Explain why the condition "F is any antiderivative of f on [a, b]" from the Evaluation Theorem is fulfilled by this scenario.