Thursday, Oct 24, 2013
STUDENT NAME:
EXAM 2 – MATH 1330, FALL 2013
READ EACH PROBLEM CAREFULLY! SHOW ALL WORK! NO WORK, NO CREDIT!
Make sure you turn in ALL pages. No graphing calculators allowed!
• Problem 1 [10 pts]
(i) Find the following limits, if they exist. Justify your answers to receive full credit. If a limit does
not exist, write ”DNE” and explain why it does not exist.
x2 + x − 6
x→3
x−2
(a) lim
x2 + x − 6
x→2
x−2
(b) lim
3
x→0 x2
(c) lim
3
x→2 x − 2
(d) lim
• Problem 2 [10 pts] Consider the function graphed below:
(ii) Consider f (x) = 3x2 . Note that lim 3x2 = 12. How close must the input x be to 2 to guarantee
x→2
that the output is within 0.01 of the limit 12? Give your answer to at least three decimal places.
• Problem 3 [10 pts] A hedgehog rolls down a hill, and the distance that she rolls as a function of
time is given by
f (t) = t2 + 3t.
(a) Find a formula for the average speed (average rate of change of f ) between t = 2 and t = 2 + ∆t,
as a function of ∆t alone.
(b) Using the limit definition of the instantaneous rate of change, find the speed of the hedgehog at
time t = 2.
(c) Find the derivative of f using the rules for derivatives and recompute the speed in part (b).
• Problem 4 [10 pts] Compute the derivative of the functions
(a) f (x) =
(b) g(y) =
√
x(x − 3)
2y + 1
y2 − 1
(c) V (r) = 43 πr3 + 5πr2
• Problem 5 [10 pts] For each of the following discrete time dynamical systems, determine the equilibria (if any) and specify whether each equilibrium is stable or unstable. Explain! (You may use
cobwebbing to justify your work!)
(a) (Lung Model) ct+1 = (1 − q)ct + qγ, q = 0.3 and γ = 4.
(b) (Mutation Model) pt+1 =
(c) (AV node model) Vt+1 =
1.5pt
1.5pt +2.0(1−pt ) .


 cVt + u,


cVt ,
1
Vt > Vc
c
1
Vt ≤ V c
c
,
c = 0.7, Vc = 20 mV, u = 10 mV