Description
1. Compute the following limits, if they exist. Else, argue why the limit does not exist.
(a) lims→−1
1
s − 1
s
3 − 1
(b) limx→∞
e
2x + x
3 + ln x
3 e2x − x
3 + cos x
(c) limr→1
|r − 1|
r
2 − 1
(5+5+5)
2. The function f(x) is defined on the interval [0, 2] and is between 4 − x and x
2 + 2 for
all x in this interval. Does it have to be continuous at x = 1? Explain why or why
not. (5)
3. Show that the equation x
7 −3 x−1 = 0 has at least one solution in the interval [−1, 1].
(5)
4. (a) Show that
d
dx
arctan x =
1
1 + x
2
.
(b) Consider the function
f(x) = 2 arctan x − x .
Find its domain, horizontal and vertical asymptotes, local minima, local maxima,
and inflection points of f. Identify the regions where the graph of f is concave
upward or concave downward. Finally, sketch the graph of the function.
(5+10)
5. An airplane is flying towards a radar station at a constant height of 6 km above the
ground. The distance s between the airplane and the radar station is decreasing at a
rate of 400 km/h when s = 10 km. What is the horizontal speed of the plane? (10)
1
6. Compute the following definite or indefinite integrals.
(a) Z
x
−3
e
1/x dx
(b) Z
x + 1
x
2
(x
2 + 1) dx
(c) Z 2π
0
(cos2 φ − sin2 φ) dφ
(10+10+5)
7. Find the derivative of the function
F(x) = Z x
√
x
e
t
t
dt .
(5)
2