Download Math 156 Applied Honors Calculus II Fall 2016 hw4 , due: Tuesday

Math 156
Applied Honors Calculus II
Fall 2016
hw4 , due: Tuesday, October 4
Please write neatly, explain your reasoning, show intermediate steps, and erase mistakes.
1. A cylindrical water tank is lying on its side as shown.
The tank is full of water. Find the work done in pumping
the water to the top of the outlet. Use ρ = 1000 kg/m3 for
the density of water.
1m
1.5 m
-
6m
2. An object of mass m is moving in a straight line subject to a force F (x), where x is the
object’s position. Let v(x) be the object’s velocity as a function of position.
a) Show that the work done in moving
the object from x0 to x1 is equal to the change in
R x1
the object’s kinetic energy, W = x0 F (x)dx = 21 mv12 − 21 mv02 , where v0 = v(x0 ), v1 = v(x1 ).
, where a is the object’s acceleration and t is time, then by the chain
(hint: F (x) = ma = m dv
dt
dv
dv dx
dv
rule, dt = dx dt = dx v)
b) How many foot-pounds of work does it take to pitch a baseball at 90 mi/hr? Assume the
baseball weighs 5 oz. (hint: 1 lb = 16 oz, 1 mile = 5280 ft, g = 32 ft/sec2 , 1 hr = 3600 sec; also
recall that weight = force = mass × acceleration)
∞
∞
√
3. Consider the following integrals. a) −∞
xe−x dx b) −∞
e−|x| dx c) 01 xdx
x
Sketch the functions and determine whether the integral converges or diverges; if it converges,
find the value.
R
2
R
R
x
4. Show that 1∞ √1+x
6 dx converges. (hint: first explain the result heuristically using asymptotics,
then prove it rigorously using the comparison test)
R
5. The speed of gas molecules at equilibrium is a random variable v, and the average speed is
v̄ =
√4
π
M
2RT
3/2R
∞ 3 −M v 2 /(2RT )
dv,
0 v e
where M is the molecular weight of the gas, R is the gas
constant, and T is the gas temperature. Show that v̄ =
6. The Gamma function Γ(x) is defined by Γ(x) =
q
8RT
.
πM
(hint: substitute x = M v 2 /2RT )
R ∞ x−1 −t
t e dt.
0
a) Evaluate Γ(1), Γ(2), Γ(3).
b) Show that Γ(n + 1) = nΓ(n), for n ≥ 1. Use this to compute Γ(4).
Note that from (b) it follows that Γ(n+1) = n! (“n factorial”), where n! = n·(n−1)·(n−2) · · · 2·1.
R
R
R √
7. Find the antiderivative. a) sin x cos x dx b) sin2 x cos x dx c) x x2 + a2 dx
8. Submit the problem from the Maple worksheet.
announcement
The 1st midterm exam is on Wednesday, October 5, 6:15-7:45pm, in 1202 SEB (School of
Education Building). If you have a conflict, please tell your instructor. The exam will cover: 1.1
sigma notation, 1.2 area, 1.3 definite integral, 1.4 FTC, 1.5 work, 1.6 improper integrals, 1.7
arclength, plus the homework and lecture notes. A review sheet will be distributed before the
exam. Calculators are not allowed on the exam. You may use one sheet of paper (one side) for
handwritten notes. We will supply the exam booklets.