Name:
Summer Math 0090, 7/25/13
Midterm Exam 2
2 Hours, No Calculators
Please read all instructions carefully. Show your work when it is reasonable to do so,
since partial credit will be awarded for showing progress toward a solution. If you get
stuck, don’t be afraid to leave the problem and come back to it later. If you have any
questions, raise your hand and I will come to you.
Good luck.
P.S. Don’t Forget To Read Instructions Carefully for Each Problem
1
Problem 1. (14 pts) Consider the function f (x) = x + sin(x) with domain [0, 2π].
a) Find all critical points (if any) of f (x) and identify if each is a local max, a local min,
or neither. Explain your reasoning.
b) Identify all points of inflection (if any) and give the intervals where f (x) is concave up
and concave down.
c) Identify the absolute max and the absolute min of the function, if they exist.
d) Using all the information found in parts (a)-(c), sketch a graph of f (x) on the axes
below, labeling all important features.
y
3Π
2Π
Π
0
Π
2
Π
"Π
"2 Π
"3 Π
2
3Π
2
x
2Π
Problem 2. (14 pts) Evaluate the following anti-derivatives:
a)
!
b)
! $
2
6
− 2
x x
c)
!
√
−5
dt =
1 − t2
d)
!
(sin(x) − 3 cos(x)) dx =
"
#
8y 8 + 3 dy =
%
dx =
3
Problem 3. (14 pts) Compute the following limits or infinite limits, if they exist (that
is, if they are a number, ∞, or −∞). If they don’t exist, explain why not.
13x6 + 7x + 2
=
x→−∞ 7x6 + 5x5 + 4
a) lim
b) lim
x→0
sin x − x
=
x3
c) lim+ x2 ln x =
x→0
cos θ − 1
=
θ→0 eθ − θ − 1
d) lim
4
Problem 4. (14 pts) A rectangle can be inscribed under a line in the first quadrant as
in the figure below. Find the maximum possible area of such a rectangle.
12
5
5
Problem 5. (14 pts) A water filtration system consists of an inverted conical tank
suspended over a cylindrical tank; water drains out of the bottom of the cone and falls
into the cylinder below, as shown in the picture, at a rate of 2 m3 /s. The cone has a base
radius of 10 meters and a height of 5 meters; the cylinder has a radius of 3 meters.
a) How fast is the water in the cylindrical tank rising when the water in the cone is 4
meters deep?
b) How fast is the water in the conical tank falling at this time?
6
Problem 6. (14 pts) Suppose a particle has acceleration a(t) = 12t2 + 2 m/s2 . Also
suppose that at t = 1 s the particle has velocity v(1) = 12 m/s and when t = 2 s the
particle has position s(2) = 20 m. Give the formulas for v(t) and s(t).
7
Problem 7. (14 pts) Compute the derivatives of the following functions. Logarithmic
differentiation may be appropriate for some of them, at least in part.
a) f (x) =
ln(ln(x))
ln x
b) g(x) = arctan(x2 )
c) h(x) = cos(x)xx
(x − 2)5 (x − 3)2 (x + 4)17
√
√
d) s(x) =
3
x+2 x−5
8