Differentiation: Definition & Rules 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 1 Paper 2 Paper 3

Each paper is built from this unit’s 17 terms and is the same for everyone, so a teacher can assign “Unit 2, Paper 2” 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

Derivative of aˣ and log_a x

Using a tangent line to approximate

Recognizing a limit as a derivative

Differentiability implies continuity

Product rule with three factors

Derivative as a limit

Derivative from a table of values

Product rule

Where a function fails to be differentiable

Alternate form of the definition

Higher-order derivatives

Power rule

Short answer 1. Define or explain: Derivative of eˣ and ln x

3 pts

Short answer 2. Define or explain: Tangent vs normal line

3 pts

Short answer 3. Define or explain: Quotient rule

3 pts

Short answer 4. Define or explain: Derivatives of trig functions

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.

Water flows into an inverted right circular cone-shaped tank at a constant rate of 3 cubic meters per minute. The tank has height 10 m and top radius 5 m, so the water-surface radius r and the water depth h always satisfy r = h/2.

Show that the volume of water in the tank is V = (π/12)h³.

Find the rate at which the depth of the water is rising at the instant when h = 4 m. Include units.

Find the rate at which the radius of the water surface is increasing at that same instant. Include units.