Math 171 — Calculus I Spring 2016
Name:
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Weekly Write-Up on Sections 4.3-4.5
Problem 1.
From the suggested homework:
[4.4 # 60] Sketch the graph of a function f satisfying all of the given conditions.
(i) f 0 (x) < 0 for x < 0 and f 0 (x) > 0 for x > 0
(ii) f 00 (x) < 0 for |x| > 2 and f 00 (x) > 0 for |x| < 2
Problem 2.
From an old exam:
The lapse rate, r, is the negative of the rate of temperature change with altitude, thus
the lapse rate is given by:
dT
r=−
dz
where T = temperature, and z = altitude. If the lapse rate exceeds 7◦ C / km in a layer of
the atmosphere, it is favorable for tornado formation. Assume the temperature function
is continuous and differentiable in terms of z. Concurrent measurements indicate that
at an elevation of 6 km, the temp is −10◦ C and at 4 km the temp is 8◦ C.
(a) (Fill in the blank) You can conclude that the lapse rate is
intermediate elevation.
◦
C/km at some
(b) (Circle one) Which theorem allows you to answer (a)?
Intermediate Value Theorem (IVT) or the Mean Value Theorem (MVT)
(c) (Circle one) Are conditions favorable for a tornado? Yes or No
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Problem 3.
For in-class discussion:
A weight of mass m is attached to a spring suspended from a support. The weight is set
in motion by moving the support up and down according to the formula h = A cos(ωt),
where A and ω are positive constants and t is time. If frictional forces are negligible,
then the displacement s of the weight from its initial position at time t is given by
s=
where ω0 =
p
Aω 2
(cos(ωt) − cos(ω0 t)),
ω02 − ω 2
k/m for some constant k and with ω 6= ω0 .
Use L’Hopital’s Rule to determine lim s.
ω→ω0