Calc 2, Spring 2024
Problem Set 4 due: 1 Apr 2024
You do not have to submit your work on this worksheet. You may work on your own paper or using tablet applications (such as Goodnotes, OneNote, etc).
1. The pair of equations
(a) Solve the equation
dy = 2xy−2e−x. dx
You may leave your answer in an implicit form (i.e. you do not need to solve for y).
(b) Determine the explicit solution which satisfies the condition that y(0) = 1.
(c) Find the (exact) limit lim y(x) for your solution. x→∞
dx = −500x + xy dt
dy = −600y + 3xy dt
models two coexisting species with populations x(t) and y(t). One equilibrium solution is the pair of constant functions x(t) = 0, y(t) = 0, representing both species being extinct. Find an equilibrium solution for this model where both species have positive population.
Calc 2, Spring 2024
Problem Set 4 due: 1 Apr 2024
3. In an engineering project, your colleague comes up with a model dy = x2 + y4. Based on dx
(a) Write the first five terms of the sequence given by ?(−1)n(2n + 1)2 ?∞
experiments, your team has determined that the function you are looking for (a.k.a. the solution y(x) to the model) cannot have a local minimum. Determine if the model has this property. Hint: use the First Derivative Test from Calculus I.
22n−1 + 1
n=1
(b) If it converges, make a guess of the value of its limit.
(c) Justify your answer to (b) formally. Hint: use the Squeeze/Sandwich theorem.
Calc 2, Spring 2024
Problem Set 4 due: 1 Apr 2024
5. Consider the sequence defined recursively by
an+1 = 3an + 4 a0 = 6
2an + 3
(a) Write the terms a1, a2, a3, a4, and a5 approximated to 8 decimal places.
(b) Using appropriate justification, find the exact value of the limit.
6. Find the exact value of the limits of the recursively defined sequences.
1? 2? (a)an+1=2 an+a witha0=1.
n
1? 3? (b)an+1=2 an+a witha0=1.
n
(c) More generally: for a constant K satisfying K ≥ 0, find the value of lim an where n→∞
1? K?
an+1=2 an+a witha0=1.
n
(d)an+1=2an+ 4 witha0=1 3 3a2n
Calc 2, Spring 2024
Problem Set 4 due: 1 Apr 2024
7. By first finding a formula for the partial sums, determine whether each of the following series converges or diverges. If it converges, find the exact value that it approaches.
(a) 5−2+0.8−0.32+...
(b) 1 + 1 + 1 + 1 + . . . 4 16 64
(c) 1+1+ 1 + 1 +... 3 8 15 24
8. Let x be a (constant) real number. Determine, with proper justification, if the sequence ? cos(n4 x) ?∞
n3 converges. If it does, find the value of the limit.
n=1
