Unit 2: Differentiation: Definition & Fundamental Properties
Calculus BC · Unit 2 · Paper 3

Differentiation: Definition & Fundamental Properties 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 22 terms and is the same for everyone, so a teacher can assign “Unit 2, 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

Matching a piecewise function differentiably

2

Cusp

3

Derivative of an inverse function

4

Alternate-form definition

5

Recognizing a derivative in disguise

6

Product rule

7

Higher-order derivatives in motion

8

Difference-quotient definition

9

When only the definition works

10

Which rule governs

11

Differentiability implies continuity

12

The smooth join is the tangent line

Short answer 1. Define or explain: Corner

3 pts

Short answer 2. Define or explain: Derivatives of inverse trig functions

3 pts

Short answer 3. Define or explain: Tangent versus normal line

3 pts

Short answer 4. Define or explain: Symmetric difference quotient

3 pts

Free response

9 pts

This course has no free-response prompt tagged to this unit, so one from elsewhere in the course is used. It is still worth writing — the skill transfers.

CALCULATOR PERMITTED. The polar curve C is defined by r(θ) = 3 + 2 sin θ for 0 ≤ θ ≤ 2π. The circle r = 4 is also graphed in the xy-plane. (Your calculator should be in radian mode.)

A. Find the area of the region enclosed by curve C. Show the setup for your calculations.

B. Find the rate of change of r with respect to θ at θ = π/6. Show the setup for your calculations.

C. Find the area of the region that lies inside curve C and outside the circle r = 4. Show the setup for your calculations.

D. A particle travels along curve C so that dθ/dt = 3 for all times t. Find the rate at which the particle’s distance from the origin is changing when the particle is at the point where θ = π/6.