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. 1, October 22/23, 2007 www.math.tu-cottbus.de/˜kemm/lehre/erm

Class Problems

1. Prove the following relation by mathematical induction:

n

X

k=1

(2k−1) =n², n= 0,1,2, . . .

2. Given is the following mathematical relation:

n

X

k=1

k³ = n²(n+ 1)²

4 , n = 1,2, . . . a) Check this relation for n= 4.

b) Prove this relation bymathematical induction.

3. Simplify the following expressions:

a) lna+nln(a+b) +nln(a−b) b) e^x+y2 e^x−y2

c) e^−nlnn√x d) (√³

x⁵+√⁵

x³)²−(√³

x⁵−√⁵ x³)²

e)

√ a²−x²

a²−x² + √ ^x2

(a²−x²)³ f) (1 +a)²−(1−a)²

4. Find the zeros of the following equations for x∈R: a) 3x²−3x−18 = 0 b) 3 sin(^x₂) = 0

Homework

1. Prove the following relation by mathematical induction:

n

X

k=1

(2k) =n(n+ 1), n = 1,2, . . .

2. Prove the following relation by mathematical induction :

n

Y

k=1

1

2^k = 1

√

2ⁿ⁽ⁿ⁺¹⁾, n= 1,2, . . .

3. Simplify the following expressions:

a) ^e

x+y 2

e^x−y2 b) ^a3_b+a^+b3 c)

qe^(x+y)2

e^(x−y)2 d) ln(x²−y²)−ln(x−y)

4. Find the zeros of the following equations for x∈R: a) 2x²−20x=−50 b) 3 tan(3x) := ^{3 sin(3x)}_cos(3x) = 0