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FRQ Precision Grading & Diagnostic for AP Exams

Question

(c) Use separation of variables to find y = H(t), the particular solution to the differential equation

Student Answer
Student submission sample grading
Scoring Overview
Estimated Grade 3 / 5

Step 1 Separation of Variables

Correctly separated the variables H and t.

Req: 1 pt for separation.

Ans:$ \frac{dH}{H-1} = \frac{1}{2}\cos\left(\frac{t}{2}\right)dt $

Step 2 Antiderivatives

Both antiderivatives evaluated correctly.

Req: 2 pts (1 per side).

Ans:$ \ln|H-1| = \sin\left(\frac{t}{2}\right) + C $

Step 3 Constant & IVP

Included +C but failed to substitute H(0)=4.

Req: 1 pt for finding C.

Ans:$ H(0) = 4,\ H > 1 \ldots H-1 = e^{\sin(t/2)+C} $

Step 4 Solve for H(t)

Stopped prematurely. Did not isolate H(t).

Req: 1 pt for final particular solution.

Ans:$ H-1 = e^{\sin(t/2)+C} $