Mathematics for Engineering I

Brandenburg Technical University (BTU) Cottbus

Chair of Mathematics for Enineering Prof. Dr. R. Reemtsen, Dr. F. Kemm

Mathematics for Engineering I

Problem Sheet No. 2, October 29/30, 2007 www.math.tu-cottbus.de/˜kemm/lehre/erm

Class Problems

1. Compute the following sets:

[0,1]∪[−1,2] ;

n

\

k=1

[0, k], n= 1,2, . . . ;

n

[

k=1

[0, k], n = 1,2, . . . ; R\ (a, b]∪[c, d)

, a < c < b < d; R\ (a, b]∩[c, d)

, a < c < b < d . 2. Let G be the set of all students at BTU and

J :={x∈G|x comes from Germany}

K :={x∈G|x is a student of ERM}

L:={x∈G|x visits the library at least three times a week}

M :={x∈G|x speaks more than three languages fluently}

Describe the following sets:

a) J∩K ∩L∩M

b) K ∩C_G(C_G(J)∪L)

∪C_G(M) c) C_G(J ∩L)∪J

d) C_G(K∩M)∩(C_G(K)∪M)

3. Sketch the curves given by the following equations. Check also for each if there is a function f with y=f(x) or x=f(y)

a) y= 2 sin(x+^π₂), x∈R b) x²+y² = 3², −3≤x, y ≤3 c) y= (x−3)(x−1)x(x+ 2), x∈R d) f(x) = xⁿ, x∈R, n= 1, . . . ,5 4. Simplify as much as you can:

(a−b−c)²+ (a+b)²−(b+c)²

2

15 −₁₀¹ +¹₅

1

2 − ¹₃ − ¹₆ − ¹₉ − ₄₅² a− ^a2

a−^b_a²

b−₁₊^ba b

1

3ln(a²−ab+b²) + 1

3ln(a+b)

Homework

1. Simplify as much as you can:

a) ax+bx

ax+bx+ay+by b) a−3

a+ 2 + 4a+ 8 (a+ 2)²(a−3)

c) 1 b +1

a a+b

b −a+b a

d)

9y² 3a²x²

2 3ax³

9y² 3

e) √ x²−1

rx−1

x+ 1 f) ^2p+1√ b^4p+2

g) (x+ 1)⁶(x−2) + 2(x+ 1)⁵ h) log₁₀(√³

x²) + log₁₀(¹⁰_x) 2. Give all solutions of the following equations.

[cos(p

x²/5)]²+ [sin(p

x²/5)]² = 1 x⁶+ 9x³+ 8 = 0 e^x−7(x+ 3)²

(x−3)² = 0 q

25−√

3x−12 = 5 3. Let

J := (0,1), K := [3,7], L:= [−4,2), M := (−13,−5]. Compute the following sets:

a) J∩K ∩L∩M

b) K ∩C_R(C_R(J)∪L)

∪C_R(M) c) C_R(J∩L)∪J

d) C_R(K∩M)∩(C_R(K)∪M)

4. Sketch the curves given by the following equations and mark the relevant numbers.

Check also for each if there is a functionf with y=f(x) or x=f(y) a) ^(x−1)₂ ² +^(y+1)₄ ² = 1 b)y−(x−1)² = 1

c) y= 3 sin(0.5x) d)y=−tan(x+ 1) e) y= 0.5^x f) y=x(x−3)(x+ 1)