Unit 7: Differential Equations
Calculus AB · Unit 7 · Paper 3

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 1Paper 2Paper 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 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

Sketching a slope field solution

2

State the domain of a particular solution

3

Why the constant matters

4

Equilibrium solution

5

Solutions cannot cross an equilibrium

6

Zero slopes locate the factors

7

dy/dt = ky solution

8

Domain of a particular solution

9

Exponential growth and decay

10

Half-life from k

11

Two checks on a candidate solution

12

k as a continuous rate

Short answer 1. Define or explain: Half-life and doubling time

3 pts

Short answer 2. Define or explain: Match a slope field by structure

3 pts

Short answer 3. Define or explain: Reading equilibrium solutions

3 pts

Short answer 4. Define or explain: Exponentiating a sum gives a product

3 pts

Free response

9 pts

NO CALCULATOR. Consider the differential equation dy/dx = x(y − 3).

A. Find d²y/dx² in terms of x and y.

B. 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 x = 0, and use it to approximate f(0.2).

C. Find the particular solution y = f(x) to the differential equation with the initial condition f(0) = 1.

D. Determine whether the approximation in part B is an overestimate or an underestimate of f(0.2). Justify your answer.