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.
Choose the branch from the initial condition
Choosing the step size
Exponentiating a sum gives a product
Logistic differential equation
Show the Euler table
Identifying the carrying capacity
State the domain of a particular solution
Logistic equilibria
Euler's method
Zero slopes locate the factors
Separation of variables with initial conditions
Solutions cannot cross
Short answer 1. Define or explain: Verify a candidate solution twice
3 ptsShort answer 2. Define or explain: Match a slope field by structure
3 ptsShort answer 3. Define or explain: Logistic solution behavior
3 ptsShort answer 4. Define or explain: Fastest growth in a logistic model
3 ptsFree response
9 ptsNO CALCULATOR. A pie is taken from a hot oven and put on a table. The internal temperature of the pie at time t minutes can be modeled by the function H that satisfies the differential equation dH/dt = −(1/15)(H − 20), where H(t) is measured in degrees Celsius and H(0) = 75. For t > 0, it is known that 20 < H(t) < 75. A proposed slope-field figure shows short segments at grid points for 0 < t < 30 and 25 < H < 75 in which EVERY segment has positive slope, with the segments steeper at larger values of H and the slopes constant along each horizontal row.
A. Explain why the figure described could not be a slope field for the differential equation dH/dt = −(1/15)(H − 20).
B. Find the slope of the line tangent to the graph of H at time t = 0. Show the work that leads to your answer.
C. It can be shown that d²H/dt² = (1/225)(H − 20). The line tangent to the graph of H at time t = 0 is used to approximate H(5). Is this approximation an overestimate or an underestimate for the actual value of H(5)? Give a reason for your answer.
D. Use separation of variables to find an expression for H(t), the particular solution to the given differential equation with initial condition H(0) = 75.