MATH 138
Question 1. [10 marks] Consider the functions f1, f2, . . . with domain [0, ∞) given by:
Assignment 4
fn(x) =
1 n
− 1 x + 1 + 1 nn
0
if x ∈ [0,n)
if x ∈ [n, n + 1) if x ∈ [n + 1, ∞)
Prove the following:
(a)
(b) (c)
The sequence {fn}∞n=1 converges to the function f : [0,∞) → R given by f(x) = 0 for all x ∈ [0,∞). i.e. for all x ∈ [0,∞), we have lim fn(x) = f(x).
n→∞
∞ ∞
Each integral fn(x)dx is convergent, and the integral f(x)dx is also convergent. 00
∞ ∞
lim fn (x) dx ̸= f (x) dx. That is, the limit of the integrals is NOT the same as the
n→∞ 0 0 integral of the limit.
Fun fact: the functions f1, f2, . . . given above actually converge to f in a much stronger sense called uni- form convergence (which you will learn in the third-year real analysis courses). And uniform convergence on bounded intervals [a, b] does imply convergence of the integrals on that interval. But as we saw above, this fails when we work with unbounded intervals.
MC Questions
2. Suppose there is an m such that the line y = mx divides the area under the curve y = x(1 − x) and above the x-axis into two regions of equal areas. In what interval does m lie:
(a) m∈[0,1] 5
(b) m∈(1,2] 55
(c) m∈(2,3] 55
(d) m∈(3,4] 55
(e) m∈(4,1] 5
0
(a) I diverges by comparison with (b) I diverges by comparison with (c) I diverges by comparison with (d) I diverges by comparison with
0
11
x + x2 dx.
3. Consider the integral I =
Which of the following can be concluded by the Comparison Theorem?
11
x dx
0 11
√x dx 11
2x dx 11
0
0
1 + x2 dx
(e) none of the above
4. The integral
∞ x2/3 + x5/3
1
x2 dx:
(a) diverges by comparison with x4/3 dx 1
∞1
∞1
(b) converges by comparison with
(c) diverges by comparison with
√3 x dx
5. Given the following descriptions of a region R along with expressions involving integrals over x and
y respectively, determine which pairs of integrals represent the area of the same region. (a) Ristheregionlyingbelowy=−x3 andabovey=−4xforx∈[0,2].
x4/3 dx ∞1
√3 x dx ∞1
1
(d) converges by comparison with (e) none of the above
1
1
2 8y√ −x3+4xdx, − +3ydy
4
(b) Ristheregionlyingbelowy=4x−x2 andabovey=2xforx∈[0,2].
00
2 4y (4x−x2)−2xdx, 2 −(2− 4−y)dy
00
(c) R is the region lying above the x-axis, below y = x on x ∈ [0,1], and below y = [1, 2].
1 2√ 1
xdx+ 2−xdx, 2−y2 −ydy
010
√
(d) R is the region in the first quadrant bounded by the x-axis, y = √x and y = 1x − 3. 22
2−x on
3√ 9√ 1 3 3 xdx+ x− 2x−2 dx,
2y+3−y2dy
(e) none of the above
6. Which (a) If (b) If
(c) If (d) If
of the following are true:
∞ 1
xp dx converges then xp dx diverges. 10
∞ t
f (x) dx diverges then lim f (x) dx = ±∞
a t→∞ a
a < b and ∞ f(x) dx converges then ∞ f(x) dx converges
∞ 1
xp dx diverges then xp dx converges. 10
030
ba
(e) none of the above
7. Which of the following are true:
(a) If lim xf (x) = A, where A > 0 is a constant, then x→∞
∞ a
(b) If lim f(x) exists and x→∞
∞
0 x→∞
(c) If lim xf(x) = 0, then there is an a ∈ R such that x→∞
∞
(d) x3 dx = 0
−∞
(e) none of the above
8. Assume n ∈ N (i.e. n = 1,2,3···)
∞ a
f(x)dx converges.
f (x) dx diverges. f(x)dx converges then lim f(x) = 0
Let A(n) represent the area in the first quadrant bound by y = x and y = xn. Which of the following are true:
(a) A(n)=A(1) n
(b) There is an n such that A(n) = 1 10
(c) lim A(n) = 1 n→∞
(d) There is an n such that A(n) = 5 11
(e) none of the above
9. Let A represent the area of the “triangular” region bounded by the functions y = ln(x), y = ln(1−x)
and
(a) (b) (c)
(d) (e)
y = b for b ≥ 0. Which of the following are true:
A=2eb −b−ln(2e).
It is possible to calculate A by evaluating a single integral.
To find A we must compute an improper integral
2
A = 2
b −1
e−t dt none of the above