An object's position is f(t) = t³ − 3t². You find f'(2) = 0. A classmate concludes the object is at rest and not accelerating at t = 2. What is wrong with this reasoning?
ANothing — zero velocity implies zero acceleration at that instant
Bf'(2) = 0 means the object is at rest, but the second derivative f''(2) = 6(2) − 6 = 6 ≠ 0, so the object has nonzero acceleration even while momentarily stationary
CThe classmate is correct, but only for polynomial functions
Df'(2) does not mean the object is at rest — velocity is the second derivative, not the first
f''(t) = 6t − 6, so f''(2) = 6. The object has zero velocity at t = 2 but nonzero acceleration — it is momentarily stopped but being accelerated (like a ball thrown upward at its peak). Zero velocity and zero acceleration are independent conditions; neither implies the other. Each derivative order provides genuinely new information that the previous order cannot supply.
Question 2 Multiple Choice
What does the notation d²y/dx² represent?
AThe square of the first derivative: (dy/dx)²
BThe second derivative of y with respect to x — differentiating y twice with respect to x
CThe second power of x in the denominator divided by the square of y
DAn alternate notation for the differential dy multiplied by dx
d²y/dx² means differentiate y twice with respect to x. The '²' in the numerator counts how many times y has been differentiated; the '²' in the denominator counts how many times x appears in the differential. This is entirely distinct from (dy/dx)², which squares the first derivative. Confusing them leads to errors throughout concavity analysis, Taylor series, and differential equations.
Question 3 True / False
The expression f^(n)(x) denotes the nth power of f(x), i.e., [f(x)]^n.
TTrue
FFalse
Answer: False
f^(n)(x) denotes the nth derivative of f — the result of differentiating f exactly n times. The parentheses around the superscript exist precisely to distinguish it from the nth power [f(x)]^n. For example, f^(2)(x) = f''(x), the second derivative, while [f(x)]^2 is the square of f. This notational confusion is common and causes persistent errors when working with Taylor series and ODEs.
Question 4 True / False
Every higher-order derivative of e^x equals e^x, meaning e^x is unchanged by differentiation regardless of how many times it is differentiated.
TTrue
FFalse
Answer: True
This is the defining property of the exponential function with base e. d/dx[e^x] = e^x, d²/dx²[e^x] = e^x, and so on for all orders. No other elementary function shares this property, which is why e is the natural base for exponential functions in calculus and why e^x appears throughout differential equations, Taylor series, and growth models.
Question 5 Short Answer
What additional physical information does the second derivative (acceleration) provide that the first derivative (velocity) cannot, and what is the third derivative called?
Think about your answer, then reveal below.
Model answer: Velocity tells you how fast position is changing at each moment — your speed and direction. But two objects can have the same velocity while one is speeding up and the other is slowing down. Acceleration captures that rate of change of velocity — it is what you physically feel (being pushed back in your seat, not the speed itself). The third derivative is called jerk — the rate of change of acceleration. Engineers control jerk in elevators and vehicles to prevent the lurching sensation of abrupt acceleration changes.
Each derivative order captures something the previous one cannot. Position doesn't tell you if you're moving; velocity doesn't tell you if you're accelerating; acceleration doesn't tell you if the ride will feel smooth. The progression continues: position → velocity → acceleration → jerk, with each level describing the behavior of the level below it.