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 1 Paper 2 Paper 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 1” 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

Differentiability implies continuity

Corner

Product rule

Alternate-form definition

Quotient rule order

Logarithmic differentiation

Derivatives of inverse trig functions

Vertical tangent as a failure

Symmetric difference quotient

Difference-quotient definition

Recognizing a derivative in disguise

The constant subtracted is f(a)

Short answer 1. Define or explain: When only the definition works

3 pts

Short answer 2. Define or explain: Matching a piecewise function differentiably

3 pts

Short answer 3. Define or explain: Derivative of a sum

3 pts

Short answer 4. Define or explain: Cusp

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.

NO CALCULATOR. Answer the following questions about improper integrals and arc length.

A. Determine whether ∫₁^∞ x^(−3/2) dx converges or diverges. If it converges, find its value.

B. Determine whether ∫₀¹ x^(−1/2) dx converges or diverges. If it converges, find its value.

C. Find the length of the curve y = (2/3)x^(3/2) from x = 0 to x = 3.

D. Determine whether ∫₁^∞ (1/x) dx converges or diverges. Justify your answer, and explain how the result compares with part A.