Composite & Implicit Differentiation 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 38 terms and is the same for everyone, so a teacher can assign “Unit 3, 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

Where to evaluate an inverse derivative

Higher-order derivatives of sine

Implicit second derivative sign

Chain rule with a logarithm

Related rates as implicit differentiation

Vertical tangent from an implicit derivative

Chain rule

Inverse derivative formula

Chain rule with an exponential

Derivative of a variable base with a variable exponent

Chain rule with a table

Vertical and horizontal tangents implicitly

Short answer 1. Define or explain: Why dy/dx involves y

3 pts

Short answer 2. Define or explain: Candidate points must lie on the curve

3 pts

Short answer 3. Define or explain: Sign of an implicit second derivative

3 pts

Short answer 4. Define or explain: Logarithmic differentiation

3 pts

Free response

9 pts

NO CALCULATOR. Consider the curve defined by the equation x² + 2xy + 4y² = 12.

A. Show that dy/dx = −(x + y)/(x + 4y).

B. Find the equation of the line tangent to the curve at the point (2, 1).

C. Find the coordinates of a point on the curve at which the tangent line is horizontal, or explain why no such point exists.

D. Determine whether the curve has a vertical tangent line at any point where x = −4y, and justify your answer.