Calculus AB · Unit 7 · Paper 1

Differential Equations 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 28 terms and is the same for everyone, so a teacher can assign “Unit 7, 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

Match a slope field by structure

Slope field

Equilibrium solution

k as a continuous rate

Resolve C before simplifying

Logistic behavior qualitatively

Sketching a slope field solution

Choose the branch from the initial condition

Separation of variables

Newton’s law of cooling

Zero slopes locate the factors

Two checks on a candidate solution

Short answer 1. Define or explain: Sketching a particular solution

3 pts

Short answer 2. Define or explain: Sign of the constant in exponential models

3 pts

Short answer 3. Define or explain: Exponentiating a sum gives a product

3 pts

Short answer 4. Define or explain: Verifying a proposed solution

3 pts

Free response

9 pts

Consider the differential equation dy/dx = xy², and let y = f(x) be the particular solution with f(0) = 1.

Write an equation for the line tangent to the graph of f at (0, 1), and use it to approximate f(0.5).

Use separation of variables to find the particular solution y = f(x).

State the largest open interval containing x = 0 on which the solution from part (b) is defined, and explain what limits it.