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.
Solutions cannot cross
Finding concavity without the solution
Exponentiating a sum gives a product
Choosing the step size
Logistic differential equation
Fastest growth in a logistic model
AP does not require solving the logistic equation
Euler recomputes the slope
Verify a candidate solution twice
Logistic carrying capacity
State the domain of a particular solution
Match a slope field by structure
Short answer 1. Define or explain: Zero slopes locate the factors
3 ptsShort answer 2. Define or explain: Separation needs a product form
3 ptsShort answer 3. Define or explain: Euler's method
3 ptsShort answer 4. Define or explain: Fastest logistic growth at M/2
3 ptsFree response
9 ptsNO CALCULATOR. Let y = f(x) be the particular solution to the differential equation dy/dx = (3 − x)y² with initial condition f(1) = −1.
A. Find f″(1), the value of d²y/dx² at the point (1, −1). Show the work that leads to your answer.
B. Write the second-degree Taylor polynomial for f about x = 1.
C. The second-degree Taylor polynomial for f about x = 1 is used to approximate f(1.1). Given that |f‴(x)| ≤ 60 for all x in the interval 1 ≤ x ≤ 1.1, use the Lagrange error bound to show that this approximation differs from f(1.1) by at most 0.01.
D. Use Euler’s method, starting at x = 1 with two steps of equal size, to approximate f(1.4). Show the work that leads to your answer.