Unit 2: Derivatives

Answer: C — The correct answer is C because the derivative of the profit function represents the rate of change of profit with respect to the number of units sold. This is a measure of how quickly the profit is changing as the number of units sold increases. Option A is incorrect because the derivative does not represent the total profit. Option B is incorrect because the cost of producing x units is not directly related to the derivative of the profit function. Option D is incorrect because the revenue generated by selling x units is not the same as the derivative of the profit function.

Answer: C — The derivative s'(t) represents the instantaneous velocity (rate of change of position) at time t. The value s'(2) = -3 means the instantaneous velocity at t = 2 is -3 m/s, indicating the object is moving in the negative direction at that instant. Option A confuses position with velocity—it uses the derivative value to describe location rather than motion. Option B is nearly correct but uses imprecise language ('rate of 3 meters per second in the negative direction' is awkward; the proper statement is '-3 m/s'). Option D incorrectly suggests the derivative predicts total distance traveled over an interval, which requires integration or distance calculations, not just the velocity at a point. Only option C correctly identifies that the derivative gives instantaneous velocity and properly interprets the negative sign as indicating direction of motion.

Answer: B — The correct answer is B because the derivative of the position function s(t) with respect to time represents the velocity of the particle. This is a fundamental concept in calculus, where the derivative of a function represents the rate of change of the function with respect to its variable. In this case, the derivative of s(t) represents the rate of change of the position with respect to time, which is the velocity. Options A, C, and D are incorrect because they do not accurately represent the physical interpretation of the derivative of s(t).

Answer: A — The correct answer is A because the derivative f'(x) represents the rate of change of the function f(x) with respect to x, which in this context is the instantaneous velocity of the object at time x. Option B is incorrect because the average velocity is represented by the difference quotient, not the derivative. Option C is incorrect because acceleration is represented by the second derivative, f''(x). Option D is incorrect because distance traveled is represented by the integral of the velocity function, not the derivative.

Answer: B — The correct answer is B. The derivative dT/dt represents the instantaneous rate of change of temperature with respect to time. At t = 5, the negative value -2.14 indicates the temperature is decreasing (cooling) at a rate of 2.14°F per minute at that specific moment. This is the foundational interpretation of a derivative in applied contexts. Why the other options are incorrect: A) Misinterprets the derivative as a future temperature value rather than a rate of change; confuses the sign of the derivative with an impossible absolute temperature. C) Confuses the derivative value with the actual temperature T(5); students making this error fail to distinguish between the function value and its rate of change. D) Confuses the instantaneous rate of change with total accumulated change; students selecting this option incorrectly believe derivatives measure cumulative quantities rather than instantaneous rates.