Resit no. 1. (Test no. 1 & 2)
Exercise 1. Let
A = {x ∈ R : |x − 1| ≤ 2} , B = ©
x ∈ R : x2 − 5x + 4 < 0ª . Find A ∪ B, A ∩ B, A\B. Mark the result on the real line.
Exercise 2. Find the limits:
(a) limn→∞ 4n4 + 2n 3n4 + 2n2 + 1 (b) limn→∞
µ
1 + 1 3n
¶n
(c) lim
x→2−2x−21 Exercise 3. Let
f (x) = 5
4x4 − 5x3.
Find f00 and then the intervals where f is convex and the intervals where f is concave. Find the points of in ection of f.
Exercise 4.
(a) Find the integrals Z
3
3x + 5dx
(b) Find the area between the curves y = −x2 + 6x − 8 and y = x − 4.
Exercise 5. Given matrices A =
1 0 1 2 −1 3 2 −3 1
B =
2 0 5
nd: det A, A−1, AB.
Exercise 6. Solve the following Cramer's system:
x1 − 3x2 + 5x3 = −4 2x1 + 5x2 − x3 = 3
− x1 − x2 + 3x3 = −4 .
ód¹, January 23, 2009.
Resit no. 1. (Test no. 1 & 2)
Exercise 1. Let
A = {x ∈ R : |x − 2| ≤ 1} , B = ©
x ∈ R : x2 − 6x + 8 < 0ª . Find A ∪ B, A ∩ B, A\B. Mark the result on the real line.
Exercise 2. Find the limits:
(a) limn→∞ 3n5 + 2n + 1 2n5 − 3n2 (b) limn→∞
µ
1 + 1 4n
¶n
(c) lim
x→3−3x−31 Exercise 3. Let
f (x) = x4 − 4x3.
Find f00 and then the intervals where f is convex and the intervals where f is concave. Find the points of in ection of f.
Exercise 4.
(a) Find the integrals Z
2
2x + 3dx
(b) Find the area between the curves y = −x2− 6x − 8 and y = −x − 4.
Exercise 5. Given matrices A =
1 0 1 2 −1 3 2 −3 1
B =
2 0 5
nd: det A, A−1, AB.
Exercise 6. Solve the following Cramer's system:
− x1 + 2x2 − x3 = 2 3x1 − x2 + x3 = 1 3x1 + 8x2 − 3x3 = 12
.
ód¹, January 23, 2009.
