Math 53 X11 Problem Set 2 Optimization, Integration and Its Applications Instructions: Suppose your student number is
20XY-ABCDE
with binary representation of BCD given by
(BCD)10 = (n1 n2 n3 n4 n5 n6 n7 n8 n9 n10 )2 .
For each i 2 {1, 2, 3, ..., 10}, if ni = 0, answer item i(a), else if ni = 1, answer item i(b) instead. e.g. if the middle three digits of your student number is 072 with binary representation
(072)10 = (0 0 0 1 0 0 1 0 0 0)2 ,
you need to answer 1(a), 2(a), 3(a), 4(b), 5(a), 6(a), 7(b), 8(a), 9(a), and 10(a).
Work Independently! Do not consult anyone except your instructor about these problems.
Questions:
I. Evaluate the following integrals (4 pts each) 1. Integrals of Powers
Z
p
6
(a)
x 1 dx
(b) Z z2 (z 2 1 + z + 1) 2 3. Definite Integral
(a)
dz (b) Z ⇡
2 ⇡
6
Z 4
1 2. Integrals of Trigonometric Functions
Z s
sin y
(a)
dy
cos5 y
Z
(b)
sec2 ✓ cos2 ✓ tan ✓ d✓ cos cos (⇡ sin ) d
p 1
p
3 dw
w ( w + 2) 4. Integrals involving Absolute Value
(a)
(b) Z
Z ⇡
2 1
du
2 cos u 0
2 3s2 2s 1 ds 0 II. Do as indicated.
5. Absolute Extrema (4 pts) (a) Find the absolute extrema of f (x) = 4x3 (b) Find the absolute extrema of f (x) = 4x 3 9x2 + 12x
9x 6. Optimization 2 12x 3 on the interval [0, 1].
3 on the interval [ 1, 0].
(5 pts) (a) An open box is to be made from a 10-cm by 10-cm piece of cardboard by cutting out squares of
equal size from the four corners and bending up the sides. What size should the squares be to obtain
a box with the largest volume?
(b) Find the least amount of material that can be used to construct a rectangular box with an open top
and square base if its volume is 32 in3 .
7. Average Value (2 pts) (a) Find b > 0 such that the average value of g(x) = 6x2 on [0, b] is equal to 16.
Z 3
(b) If g is continuous on [1, 3] and
g(x) dx = 4, show that there is a c 2 [1, 3] such that g(c) = 2.
1 1 8. Derivative of Integrals
(a) Let H(x) = x
2x sin 3 x
⇡
2
H 0 ⇡2 i. Find H
ii. Find Z (1 pt, 3 pts)
q
2+ t 2
⇡ (b) Let H(v) = dt. Z x3 +x
tan x 2 cos v
dv.
v+1 i. Find H (0)
ii. Find H 0 (0) 9. Rectilinear Motion (2 pts, 3 pts) (a) A ball is thrown vertically upwards with initial velocity of 32 ft/s from the top of a building. The
ball hit the ground after 3 seconds. (Assume acceleration due to gravity is equal to -32 ft/s2 )
i. When will the ball reach its maximum height?
ii. What is the height of the building?
(b) The acceleration, in m/sec2 , of a particle moving along a line at t seconds is given by a(t) = 12. If
at t = 1, the particle is moving at the speed of 6 m/sec and is one unit to the right of the origin.
i. Find the velocity of the particle when t = 2.
ii. Find the position of the particle when t = 0.
III. For the following regions R, SET UP the integral(s) needed to find the following: (3 pts each) 1. the area of region R
2. the perimeter of region R
3. volume of the solid of revolution when R is revolved about the given line:
(a) using the method of Washers
(b) using the method of Cylindrical Shells
10. Plane Regions
(a) R bounded by C1 : y = x2 , C2 : y = 6
the y-axis. x and (b) R bounded by C1 : y = 3 2x3 , C2 : y =
and the line x = 1. x+1
2 6
6 4
4 R
2 2 R
1 2
Axis of revolution: y = 1 Axis of revolution: x = Total: 40 points (Bonus: 8 points) 1
2
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