Unit 4: Contextual Applications
Calculus AB · Unit 4 · Paper 3

Contextual Applications unit test

A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.

Paper 1Paper 2Paper 3
Each paper is built from this unit’s 40 terms and is the same for everyone, so a teacher can assign “Unit 4, Paper 3” and every student sits the identical test. Multiple choice is marked objectively; the written sections you mark yourself against the model answer and rubric.
Suggested time 36 min 33 points0/17 attempted
1

Units of a definite integral

2

Total distance

3

Increasing at a decreasing rate

4

Related rates: one equation, one unknown rate

5

Marginal cost as a derivative

6

Displacement vs total distance

7

Changing direction test

8

Related rates with a cone or sphere

9

Related rates procedure

10

Cone volume relation

11

Position from velocity

12

The rate is decreasing

Short answer 1. Define or explain: Linearization formula

3 pts

Short answer 2. Define or explain: Units of a derivative

3 pts

Short answer 3. Define or explain: Particle at rest vs changing direction

3 pts

Short answer 4. Define or explain: Rate versus change versus amount

3 pts

Free response

9 pts

An inverted right circular cone has a radius of 5 cm at its top and a height of 10 cm. Water drains out of the cone at a constant rate of 8 cubic centimeters per minute. Let h be the depth of the water in centimeters and r the radius of the water surface in centimeters. The volume of a cone is V = (1/3)πr²h.

Use similar triangles to express r in terms of h, and hence write V as a function of h alone.

Find the rate at which the water level is falling at the instant when the depth is 6 cm. Include units.

Find dr/dt at that same instant. Then explain why dh/dt is not constant even though dV/dt is.