Contextual Applications of Differentiation — High School Calculus practice questions

A particle is speeding up when:

1. its acceleration is positive
2. its velocity and acceleration have the same sign
3. its velocity is positive
4. its velocity is zero

Answer: B — its velocity and acceleration have the same sign

B) Correct — Speed grows when acceleration pushes in the direction of motion. A) Half right: if velocity is negative, positive acceleration slows it down. C) Moving right says nothing about speeding up. D) That is a moment of rest.

A sphere's volume grows at 100 cm³/s. How fast is the radius growing when r = 5 cm?

1. 5/π cm/s
2. 100 cm/s
3. 1/π cm/s
4. π cm/s

Answer: C — 1/π cm/s

C) Correct — 100 = 4π(5)²·dr/dt = 100π·dr/dt, so dr/dt = 1/π. A) Uses 4πr instead of 4πr². B) Copies dV/dt as if it were dr/dt. D) Divides the wrong way: 4πr²/100.

A 13-ft ladder slides down a wall; its base moves out at 2 ft/s. How fast is the top sliding down when the base is 5 ft from the wall?

1. 5/6 ft/s
2. 2 ft/s
3. 24/5 ft/s
4. 5/12 ft/s

Answer: A — 5/6 ft/s

A) Correct — x² + y² = 169, so x·dx/dt + y·dy/dt = 0. With x = 5, y = 12: 5(2) + 12·dy/dt = 0, so dy/dt = −5/6 (sliding down at 5/6 ft/s). B) Assumes the top moves as fast as the base. C) Swaps x and y when solving. D) Forgets to multiply by dx/dt = 2.

Position is s(t) = t³ − 6t² + 9t. What is the velocity at t = 2?

1. 2
2. −3
3. 3
4. 0

Answer: B — −3

B) Correct — v(t) = 3t² − 12t + 9; v(2) = 12 − 24 + 9 = −3. A) Is the position s(2), not the velocity. C) Drops the negative sign. D) Is the acceleration a(2) = 6(2) − 12.

Evaluate lim_{x→∞} (ln x)/x.

1. 1
2. 0
3. ∞
4. 1/e

Answer: B — 0

B) Correct — L'Hôpital's Rule: lim (1/x)/1 = 0. x grows much faster than ln x. A) Assumes ln x and x grow at the same rate. C) Thinks ln x eventually outgrows x. D) Confuses this with the maximum value of (ln x)/x, which occurs at x = e.

V(t) is the number of gallons in a tank after t minutes, and V'(5) = −3. What does this mean?

1. After 5 minutes the tank holds −3 gallons
2. The tank lost 3 gallons during the first 5 minutes
3. The tank is filling at 3 gallons per minute
4. At t = 5, water is draining at 3 gallons per minute

Answer: D — At t = 5, water is draining at 3 gallons per minute

D) Correct — A negative rate means the amount is decreasing at that moment. A) Confuses the rate with the amount. B) Confuses an instantaneous rate with a total change. C) Ignores the negative sign.

What is L'Hôpital's Rule?

If a limit gives 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter exists.