Faculty of Management Mathematics Exercises Contents

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Faculty of Management Mathematics Exercises

Sheet 1. Logic and Sets

Exercise 1.1. For given sets A and B nd A ∪ B, A ∩ B, A \ B. Mark the results on the real axis.

a) A =

x ∈ R : 3x³

x²− 1 −4x + 16 x + 1 = 0

B = {x ∈ R : |x − 1| + |x − 5| > 8}

b) A = (

x ∈ R : 3√

log x + 2 logr 1 x = 2

)

B =x ∈ R : log²(x − 1) − 2 log (x − 1) > 0 c) A = {x ∈ R : ||x + 1| + 2| = 2} B = x ∈ R :√

x + 1 −√

x − 1 = 1 d) A = {x ∈ R : |x − 3| + |x + 4| = 9} B =n

x ∈ R : ¹₂^1−x_|x|

≤ 1o e) A = {x ∈ R : cos 3x = cos x} B = {x ∈ R : cos²2x = 1}

f) A = {x ∈ R : 3 sin x = 2 cos²x} B = {x ∈ R : sin x − cos 2x = 0}

g) A =

x ∈ R : x < 1 x

B =

x ∈ R : 1 + x 1 − x > 1

h) A =

x ∈ R : x²+ 1

x > x² x + 1

B = {x ∈ R : |x + 2| > 3}

i) A = (

x ∈ R : (x + 3)²(x²+ x + 1) (4 − x) x ≥ 0

)

B = {x ∈ R : |x − 1| ≤ 5}

j) A = x ∈ R : logx−2(x − 1) > 1

B = {x ∈ R : |2x − 1| < |x + 3|}

k) A =n

x ∈ R : 7^3x−5^x+2 ≥√ 7o

B =n

x ∈ R : ¹₃^x+3_1−x

< 9o

l) A = {x ∈ R : 2^x+1+ 2^x−1 ≤ 20} B = {x ∈ R : 5^x+ 3 · 5^x−2 > 28}

Exercise 1.2. Decide whether the following propositions are true or false. Write their negations:

a) ∀x∈R x = 2x b) ∀x∈N

x²

x + 1 ≥ x + 2

x + 1 c) ∃x∈N

1

x + 1 ≥ 1 x + 2 d) ∀x∈N

3x + 1

2x + 1 ≥ 0 e) ∃x∈R

−2x²+ x − 4

−3x²− 2 ≤ 0 f) ∃x∈C

2x²− 4x + 2

−2x²− 3 ≤ 0 g) ∀x∈R

|x + 1|

x²+ 1 ≥ 0 h) ∃m∈N ∃_n∈N m²+ n² = 10 i) ∀x∈R ∀_y∈R⁺ y = x²− 4

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1 Logic and Sets

a) A = {x ∈ R : x = 1, 2, . . .} B = {y ∈ R : y = 0}

b) A = {y ∈ R : y ≤ 2} B = {x ∈ R : x > 2}

c) A = {x ∈ R : |x − 2| > 3} B = {y ∈ R : |y + 2| ≤ 3}

d) A =

x ∈ R : x²− 2x + 1 4x − x² ≥ 0

B = {y ∈ R : 0 < |y| < 3}

e) A = {y ∈ R : 1 < |y| < 5} B =

x ∈ R : 16 − x² x³ + 27 ≥ 0

f) A = {x ∈ C : log2(x²− 1) < 3} B =

y ∈ R : 2y − 1 y + 1 < 1

g) A = {x ∈ C : log2(x + 1) + log₂(x − 1) < 3} B = (−1, 2) h) A =n

t ∈ R : log¹

3 (− |1 − t| + 4) < −1o

B =

x ∈ R : x³− x²− 4x + 4

x − 1 ≤ 0

i) A = {x ∈ R : 4^x+1+ 2^2−2x < 17} B =

x ∈ R : 1

2^x− 1 < 1 − 2^x+1 1 − 2^x−1

Exercise 1.4. In the Cartesian coordinate system mark the given sets of points:

A = {(x, y) : 2x − y − 2 > 0} B = {(x, y) : x + 3y + 6 ≤ 0}

C = {(x, y) : x − 2y < 0} D = {(x, y) : x − y ≥ 4 ∧ 2x − y < 6}

E = {(x, y) : 2x + y ≥ 2 ∧ 4x + 2y ≤ 12} F = {(x, y) : x + 2y > 0 ∧ x < −2}

G = {(x, y) : |x| − 1 < y} H = {(x, y) : |y − 1| + x > 3}

I = {(x, y) : y ≤ 3 − |x − 2|} J =(x, y) : ¹₂x + |y − 2| ≤ 2 K = {(x, y) : |y − 1| − 2 < 3x} L = {(x, y) : |x − 1| < y}

M = {(x, y) : |x| − |y − 2| ≤ 2} N = {(x, y) : |2x + 4| − |y| = 4}

O = {(x, y) : |x| + |y| ≤ 4} P = {(x, y) : |y − 3| < 2}

Q = {(x, y) : |1 − x| ≥ 3} R = {(x, y) : |x − 3| ≥ 2}

S = {(x, y) : |y + 1| < 3} T = {(x, y) : x²− y² ≤ 0}

U = {(x, y) : x²+ y²− 2x − 4y ≤ 0} W = {(x, y) : |x| + x = y + |y|} . Exercise 1.5. Compute:

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Sheet 2. Properties of a Function of One Variable

Exercise 2.1. Given the function

f (x) = 1 − x 1 + x,

nd: f (0), f (−x), f (x + 1), f (x) + 1, f _x¹ , _{f (x)}¹ . Exercise 2.2. Given the function

f (x) =

( 2^x for |x| ≤ 2 x²− 1 for |x| > 2 ,

nd: f (−1), f (0), f (2), f (−8), f (8).

Exercise 2.3. Let f (x) = x³−xand g (x) = sin 2x. Find: f g ₁₂^π

, g (f (1)), g (f (2)), f (f (f (1))).

Exercise 2.4. Find: f (f (x)), g (g (x)), f (g (x)), g (f (x)), where f (x) = x² and g (x) = 2^x. Exercise 2.5. Find the domains of the following functions:

a) f (x) = x²

x + 1 b) f (x) = √⁴

1 − x² c) f (x) = 1

√x²− 4x d) f (x) = (x − 2)r 1 + x 1 − x e) f (x) =√

2 + x − x²+ 1

√x²− 3x f) f (x) = e^x2−x−2¹ g) f (x) = 1

log (1 − x) +√

x + 2 h) f (x) = log |x|

i) f (x) = log (sin x) j) f (x) = ln (e^x− e) k) f (x) = logx2 l) f (x) = arcsin 2x 1 + x m) f (x) = arccos 2x

1 + x² n) f (x) = 1 + x

π 6

2

− (arcsin x)² o) f (x) =√

3 − x + arcsin3 − 2x

5 p) f (x) = arcsin (1 − x) + log (log x) Exercise 2.6. Are the following functions identical:

a) f (x) = x²+ 1 and g (z) = z²+ 1 b) f (x) =√

x² and g (z) = z c) f (x) = |x| and g (z) =√

z² d) f (x) = ^x_x and g (z) = 1

e) f (x) = 1 and g (z) = sin²z + cos²z f) f (x) = 1 and g (z) = tan z · cot z ? Exercise 2.7. Let

A) f (x) = x³ B) f (x) = sin x C) f (x) = 1

x for x 6= 0

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2 Properties of a Function of One Variable

Exercise 2.8. Using the graphs nd the range of the following functions:

a) f (x) = x − ¹₂2

+ 3 b) f (x) = ln (2 − x) + 1 dla x < 2 c) f (x) = |1 − |x − 2|| + 3 d) f (x) = 1 − e^2−x

Exercise 2.9. Using the graphs describe properties of the given functions:

a) f (x) = |x| b) f (x) = |x| + 1 c) f (x) = |x − 2|

d) f (x) = − |x + 1| e) f (x) = 2 − |x + 1| f) f (x) = |4 − x²| g) f (x) = x²− 3x h) f (x) = (x − 1)²− 4 i) f (x) = 2^x+1 j) f (x) = 2^x− 2 k) f (x) = 3^x−2− 1 l) f (x) = 1 − ²₃x

m) f (x) = log3(x + 2) n) f (x) = log¹

2 (−x) + 1 o) f (x) = tan x − ^π₂ p) f (x) = −2 sin x q) f (x) = sin 2x r) f (x) = 2 + sin 2x s) f (x) =

( x + 1 for x < 0 1 − x² for x ≥ 0

Exercise 2.10. Decide which of the below functions are even or odd:

a) f (x) = (x − 1)² b) f (x) = x |x| c) f (x) = 2 + x² x² d) f (x) =√

1 + x² e) f (x) = 3^x− 3^−x f) f (x) = x log x² g) f (x) = 1 + cos 2x h) f (x) = sin²x i) f (x) = |sin x|

j) f (x) = sin x

x³ k) f (x) = sin x³

Exercise 2.11. Find the composition of the following functions: f ◦ f, f ◦ g, g ◦ f, g ◦ g, where:

a) f (x) = x², g (x) = 2^x b) f (x) = 2 + cos x, g (x) =√ x

Exercise 2.12. Write the following functions as compositions of other functions:

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Exercise 2.13. Find the inverse function to the given one. Draw the graphs of the both functions in one coordinate system:

a) f (x) = 3x + 5 b) f (x) = x² dla x ≤ 1 c) f (x) =√ 2x + 3 d) f (x) = 3^x+2− 1 e) f (x) = log2(x + 3) f) f (x) = 1 + log¹

2 x

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Faculty of Management Mathematics Exercises

Sheet 3. Sequences and Their Limits

Exercise 3.1. Give the rst ve terms of each sequence dened below a) an = 2 b) an= n⁽⁻¹⁾ⁿ c) an= (−1)ⁿ

n + 1 + (−1)ⁿ 2 d) an = (−1)ⁿ⁺¹· 3

n + 1 e) an= −n (2 + (−1)ⁿ) f) an = sin^nπ₂ g) an = (−1)ⁿ+ sin^nπ₂ h) an= 1 + n sin^nπ₂ i) an= 1 + n

n + 1cos^nπ₂

Exercise 3.2. Find the fth term of the sequence (an) , if the sum of its rst n terms is equal to 4n²− 3n.

Exercise 3.3. Find the limits:

a) lim

n→∞(n²+ 5n − 6) b) lim

n→∞(−2n⁷ + 3n² − 4) c) lim

n→∞

n²+ 3n n²− 1 d) lim

n→∞

6n³− 1

3n³+ 2n − 4 e) lim

n→∞

n²− 2

n f) lim

n→∞

−3n³+ 1 n²+ 4 g) lim

n→∞

n − 1

n²+ 2n − 1 h) lim

n→∞

n³+ 2n − 1

n⁴ + n i) lim

n→∞

(1 − 2n)³ (2n + 3)²(1 − 7n) j) lim

n→∞

2n + 3 n + 1

3

k) lim

n→∞

1 + 2 + 3 + . . . + n

(3n − 1)² l) lim

n→∞

2 + 4 + 6 + . . . + 2n (1 − 9n²) m) lim

n→∞

1 − 2n 2 +√

n n) lim

n→∞

2 +√ n

1 − 2n o) lim

n→∞

(3 −√ n)² 5 + 4n p) lim

n→∞

r9n²+ 4n

n²+ 3 q) lim

n→∞

√2n − 1 −√ n − 7

r) lim

n→∞ 3n −√

9n² + 1 s) lim

n→∞

√4n²+ 9n − 2 − 2n

t) lim

n→∞

√3

n³+ 5 − n

u) lim

n→∞eⁿ⁺¹ⁿ v) lim

n→∞2ⁿ¹ w) lim

n→∞

4ⁿ⁻¹− 5

2²ⁿ− 7 x) lim

n→∞

2ⁿ⁺¹− 3ⁿ⁺² 3ⁿ⁺² y) lim

n→∞

√n

2ⁿ+ 3ⁿ z) lim

n→∞

√n

4n²+ n + 5 aa) lim

n→∞

n

q 1 2

n

+ ²₃n

+ ³₅n

ab) lim

n→∞

sin n

n + 1 ac) lim

n→∞

n

n²+ 1sin (3n + 1) ad) lim

n→∞

√3

n²sin n n + 1 ae) lim

n→∞

n − 1 n + 2

n

af) lim

n→∞

1 + 2 n + 1

n+1

ag) lim

n→∞

n + 4 n

2n

ah) lim

n→∞

n²+ 9 n²

n²

ai) lim

n→∞

1

√n²+ 1 + 1

√n²+ 2 + . . . + 1

√n²+ n

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Sheet 4. Limits and Continuity of Functions

Exercise 4.1. Find the limits (if they exists):

a) lim

x→1

(x − 1)√ 2 − x

x²− 1 b) lim

x→¹₂

8x³− 1

6x²− 5x + 1 c) lim

x→1

1

1 − x − 3 1 − x³

d) lim

x→4

√1 + 2x − 3

√x − 2 e) lim

x→3

√x + 13 − 2√ x + 1

x²− 9 f) lim

x→0

sin 5x sin 3x g) lim

x→^π₂

cos x

π − 2x h) lim

x→0

1 − cos x

x² i) lim

x→∞

x²+ 1 x²− 2

x²

j) lim

x→∞

√1 + x + x²−√

1 − x + x²

k) lim

x→∞

√x + 3 −√ x + 1

l) lim

x→∞x sin¹_x m) lim

x→0x cot 3x, n) lim

x→∞

2x + 3 2x + 1

x+1

o) lim

x→∞

3x − 1 2x + 1

2x−5

p) lim

x→0

√cos x − 1

x² q) lim

x→^π₄

cos x − sin x

cos 2x r) lim

x→0

sin 5x − sin 3x sin x Exercise 4.2. Compute the one-sided limits of the given functions as x approaches x0:

a) f (x) = 1

x − 3, x₀ = 3 b) f (x) = 1

3 − x, x₀ = 3 c) f (x) = 1

(3 − x)², x₀ = 3 d) f (x) = x + 1

x − 1, x₀ = 1 e) f (x) = 1

x² − 4, x₀ = 2 f) f (x) = 2^x−1¹ , x₀ = 1 g) f (x) = 4^x2−4¹ , x₀ = 2 h) f (x) = e^4−x2¹ , x₀ = −2 i) f (x) = x

1 + e^x¹, x₀ = 0 Exercise 4.3. Finding the one-sided limits verify if the given limits exists:

a) lim

x→1

x + 1

x − 1 b) lim

x→0x [x] c) lim

x→1

|x − 1|³

x³− x² d) lim

x→1e^1−x2¹ Exercise 4.4. Decide, if f is a continuous function:

a) f (x) =

( 2^x+ 3 if x ≤ 0

(x − 2)² if x > 0 b) f (x) =

( x − 1 if x < 0 3^x if x ≥ 0

c) f (x) =

( e^1−x^x if x 6= 1

0 if x = 1 d) f (x) =

( _{sin x}

x if x 6= 0 0 if x = 0

e) f (x) =

( cos_x¹ if x 6= 0

0 if x = 0 f) f (x) =

( arctan¹_x if x 6= 0 0 if x = 0

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4 Limits and Continuity of Functions

Exercise 4.5. Find the values of a and b that make the following functions continuous:

a) f (x) =

( 2^x+ 8 if x ≤ 0

(x − a)² if x > 0 b) f (x) =

( cos^πx₂ if x ≤ 1 a |x − 1| if x > 1

c) f (x) =







−a

x if x ≤ −1

2x + 3 if −1 < x ≤ 1 b (x − 2)²+ 3 if x > 1

d) f (x) =







2 + e^x¹ if x < 0

sin ax

3x if x > 0 b if x = 0

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Sheet 5. Derivatives. L'Hospital's Rule

Exercise 5.1. Dierentiate the following functions:

1) f(x) = 3 2) f(x) = x⁴ + 3x²− ¹_x +√

x 3) f(x) = 2x³− x² 4) f(x) = 5x − 1

3 − 2x 5) f(x) = x²− 1

x² + 1 6) f(x) = 2

x³− 1 7) f(x) = x√

1 + x² 8) f(x) = (√

x + 1)( 1

√x − 1) 9) f(x) = x²e 10) f(x) =

x³+ 1 x²

e^x 11) f(x) = 10^x 12) f(x) = x 4^x 13) f(x) = 2√

x − 3 ln x + 1 14) f(x) = x ln x 15) f(x) = ln x 1 + x² 16) f(x) = log3x 17) f(x) = sin x + cos x 18) f(x) = x³sin x 19) f(x) =√

x cos x 20) f(x) = sin x

x⁴+ 4 21) f(x) = sin x − cos x sin x + cos x 22) f(x) = arcsin x + arccos x 23) f(x) = x arcsin x 24) f(x) = x + arctg x 25) f(x) =q

1−x

1+x 26) f(x) = ln(e^x+√

1 + e^x) 27) f(x) = e^(x²^−3x−4) 28) f(x) = cos1 −√

x 1 +√

x 29) f(x) = (2x³− 1)⁵ 30) f(x) = 1 + x² 1 + x

5

31) f(x) =

sin x 1 + cos x

3

32) f(x) = cos³4x 33) f(x) =

√4x²+ 2 3x⁴ 34) f(x) = (2x + 1) 2^2x+1 35) f(x) = (1 +√⁴

x) tg (√

x) 36) f(x) = sin 2x cos²x 37) f(x) = arcsin^x₂ 38) f(x) = arcsin√⁴

1 − 5x 39) f(x) = arctg 2x 1 − x² Exercise 5.2. For the given functions f nd f⁰, f⁰⁰, f⁰⁰⁰:

a) f(x) = x ln x b) f(x) = (x²+ x + 1) cos x c) f(x) = √ x²+ 1 Exercise 5.3. Verify if the given function fulls the condition:

a) y = e^xsin x, y⁰⁰− 2y⁰+ 2y = 0 b) y = ln²x − 2 ln x, y⁰⁰+ 1

xy⁰− 2 x² = 0

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5 Derivatives. L'Hospital's Rule

Exercise 5.4. Using L'Hospital's Rule nd the limits:

a) lim

x→1

x³− 1

x²− 1 b) lim

x→0⁺

x ln x c) lim

x→−∞x

e^x¹ − 1

d) lim

x→0

e^x− x − 1

x² e) lim

x→0

ln (1 + x)

x f) lim

x→e

ln x − 1 x − e g) lim

x→0

1 − cos x

x² h) lim

x→0

sin x

x i) lim

x→0

sin x x cos x j) lim

x→+∞

e^x

x k) lim

x→+∞

ln x

x l) lim

x→+∞

ln x√ x

m) lim

x→1⁺

x

x − 1 − 1 ln x

n) lim

x→0⁺x^{sin x} o) lim

x→^π₂⁻

(sin x)^{tg x}

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Sheet 6. Investigation of Functions

Exercise 6.1. Find the asymptotes of the graphs of the given functions:

a) f(x) = 1

1 − x² b) f(x) = x²

2x + 3 c) f(x) = x x²+ 1 d) f(x) = x³+ x²

x²− 4 e) f(x) = x − 3

√x²− 9 f) f(x) =√

1 + x²+ 2x g) f(x) =

√1 + x²

x h) f(x) = sin x

x i) f(x) = x²e^−x

Exercise 6.2. Find the local maximum and minimum values of the following functions:

a) f(x) = 2x³− 15x²+ 36x − 14 b) f(x) = x⁴+ 4x − 2 c) f(x) = x x²+ 4 d) f(x) = (1 − x)²

2x e) f(x) = x −√

x f) f(x) = e^x+ e^−x Exercise 6.3. Find the intervals of increase and decrease of the given functions:

a) f(x) = xe^−3x b) f(x) = x − ln(1 + x) c) f(x) = (x²− 3) e^−x

Exercise 6.4. Find the absolute maximum and absolute minimum of a function f on the given interval:

a) f(x) = x²− 2x + 3, x ∈ [−2, 5] b) f(x) = 2x³− 3x²− 36x − 8, x ∈ [−3, 6]

c) f(x) = x − 2√

x, x ∈ [0, 5] d) f(x) = x²ln x, x ∈ [1, e]

e) f(x) = 2 sin x + sin 2x, x ∈ 0,³₂π

Exercise 6.5. Find the points of inection and the intervals on which the given functions are convex and concave:

a) f(x) = x⁴− 12x³+ 48x² b) f(x) = x²− 5x + 6

x + 1 c) (x) = x + sin 2x d) f(x) = xe^−x e) f(x) = ln x

x f) f(x) = x⁴

12 − x³ 3 + x² Exercise 6.6. Investigate the function f and then sketch its graph:

a) f(x) = x³− 3x²+ 4 b) f(x) = (x − 1)²(x + 2) c) f(x) = x 1 − x² d) f(x) = x³

x − 1 e) f(x) = x√

1 − x² f) f(x) =√ x − x g) f(x) = ln x

x h) f(x) = e^−x² i) f(x) = e^x

x + 1

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Sheet 7. Applications to Economics

Theoretical background

Let C(x) be a theoretical cost that a company incurs in producing x units of a certain commodity.

The derivative C⁰(x) is called a marginal cost. Let n be a number of units of a certain commodity.

For large n we have that

C⁰(n) ≈ C(n + 1) − C(n)

thus the marginal cost of producing n units is approximately equal to the cost of producing one more unit (the (n + 1)−st unit).

The average cost function

c(x) = C(x) x

represents the cost per unit when x units are produced. We have the following property

If the average cost is a minimum, then marginal cost is the same as average cost.

Let p(x) be the price per unit that a company can charge if it sells x units. Then p is called the demand function (or the price function). If x units are sold and the price per unit is p(x), then the total revenue is

R(x) = x · p(x)

and r is called the revenue function (or sales function). The derivative R⁰ of the revenue function is called the marginal revenue function and is rate of change of revenue with respect to the number of units sold.

If x units are sold, then the total prot is

P (x) = R(x) − C(x)

and P is called the prot function. The marginal prot function is P⁰. We have that

If the prot is a maximum, then marginal revenue is the same as marginal cost.

Exercises

Exercise 7.1. For each of the given cost functions nd (a) the cost, average cost and marginal cost of producing 1000 units; (b) the production level that will minimize the average cost; and (c) the minimum average cost

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e) C(x) = 2 x + ₈₀₀₀^x f) C(x) = 1000 + 96x + 2x³²

Exercise 7.2. For each of the given cost and demand function nd the production level that will maximize prots

a) C(x) = 680 + 4x + 0.01x², p(x) = 12 b) C(x) = 680 + 4x + 0.01x², p(x) = 12 − ₅₀₀^x

c) C(x) = 1200 + 25x + 0.0001x², p(x) = 55 − ₁₀₀₀^x

d) C(x) = 900 + 110x − 0.1x²+ 0.02x³, p(x) = 260 − 0.1x e) C(x) = 1450 + 36x − x²+ 0.001x³, p(x) = 60 − 0.01x f) C(x) = 10 000 + 28x − 0.01x²− 0.002x³, p(x) = 90 − 0.02x

Exercise 7.3. For each of the given cost function nd the production level at which the marginal cost starts to increase

a) C(x) = 0.001x³ − 0.3x²+ 6x + 900 b) C(x) = 0.0002x³− 0.25x² + 4x + 1500

Exercise 7.4. A baseball team plays in a stadium that holds 55 000 spectators. With tickets prices at $10, the average attendance has been 27 000. When ticket prices were lowered to $8, the average attendance rose to 33 000.

a) Find the demand function, assuming that it is linear, b) how should tickets prices be set to maximize revenue?

Exercise 7.5. During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $10 and his sales averaged 20 per day. When he increased the price by $1, he found that he lost two sales per day.

a) Find the demand function, assuming that it is linear,

b) if the material for each necklace costs Terry $6, what should the selling price be to maximize prots?

Exercise 7.6. A manufacturer has been selling 1000 television sets a week at $450 each. A market survey indicates that for each $10 rebate oered to the buyer, the number of sets sold will increase by 100 per weak.

a) Find the demand function,

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Exercise 7.7. The manager of a 100−units apartment complex knows from experience that all units will be occupied if the rent is $400 per month. A market survey suggests that, on the average, one additional unit will remain vacant for each $5 increase in rent. What rent should the manager charge to maximize revenue?

Exercise 7.8. Find the elasticity of the given function:

a) y = 3x − 6 b) y = 1 + 2x − x² c) y = 2x²+ 3x − 2 d) y = 120 − 0.4x² e) y = e^−x f) y = x ln x

g) y = x − 6 for x = 10 h) y = 1 + 2x + ¹₂x² for x = 1

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Sheet 8. Indenite Integrals

Exercise 8.1. Find the antiderivative of f (x) = ln x

x , x > 0 such that the point A(1, −1) belongs to its graph.

Exercise 8.2. Using basic properties of integral nd:

a)Z

(x³− 3x²+ 2x) dx b)Z

(x²− 1)³

x dx c) Z

x

1 − xdx d)Z x⁴ x²+ 1dx e)Z

3√³

x²+ 1

x³ − 2x√ x

dx f) Z x√³

x +√⁴ x

x² dx g)Z

x²−√ x

√3

x dx h)Z

√4

3^xdx

i)Z q xp

x√

xdx j) Z

2^x− 5^x

10^x dx k) Z

e^−2x− 4

e^−x+ 2dx l)Z

e^3x− 1 e^x− 1dx m) Z

cos 2x

cos x − sin xdx n)Z

sin² x

2dx o)Z

ctg²xdx p)Z

dx sin²x cos²x Exercise 8.3. Using the change of variable method (substitution method) nd:

a) Z xdx

1 + x² b)Z

xdx

(x²+ 3)⁶ c)Z e^3xdx

1 + e^6x d) Z

x³dx q

(1 − x²)³ e) Z

x√

x − 3dx f) Z √

3x + 1dx g)Z x√

1 + x²dx h) Z e^x¹ x²dx i) Z

dx x√

x²− 2 j) Z

√xdx

x²− 9 k) Z

x³dx

√1 − x⁸ l) Z

e^−4xdx

√4 + e^−4x

m) Z

sin xdx

3 + 2 cos x n)Z

sin x cos xdx o)Z

cos ln x

x dx p) Z

xe^−x²dx

Exercise 8.4. Find using the integration by parts:

a) Z

x cos xdx b)Z

x²e^xdx c) Z

e^xcos xdx d)Z

x sin x cos xdx e) Z

x ln²xdx f) Z ln xdx

x² g) Z xdx

sin²x h)Z (x − 1) e^x x² dx i) Z

x²sin xdx j) Z

e^2xsin xdx k) Z xdx

cos²x l)Z

xe^−3xdx

Exercise 8.5. Find:

Z Z Z Z

cos xdx

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Sheet 9. Denite Integrals

Exercise 9.1. Compute the integrals:

a)

2

Z

0

dx

x²+ 4 b)

1

Z

−1

√ dx

4 − x² c)

1

Z

0

xe^−xdx d)

π

Z

0

x²cos xdx

e)

π 2

Z

−π 2

cos³xdx f)

π 4

Z

0

x dx

cos²x g)

2

Z

1

x²dx h)

e

Z

1

ln x x dx Exercise 9.2. Find the area between the curves:

a) y = x³− 2x²− 3x, x = −1, x = 2 and OX axis b) y = 1

1 + x², x = −1, x = 1 and OX axis c) y = x², y² = x

d) y = x³, y = 4x

e) y = 10^x, y = 100, y = 10, x = 0 f) y = e^x, y = e^−x, x = 1

g) y = x²− 4, y = 4 − x² h) y = 1

1 + x², y = x² 2 i) y = a

x², x = a, x = 2a, y = 0 (a > 0)

Exercise 9.3. Find the average value of f on the given interval:

a) f(x) = sin³x, x ∈ [0, π] b) g(x) = e^x, x ∈ [−2, 2] c) h(x) = x

√1 − x², x ∈h 0,

√ 2 2

i

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Sheet 10. Improper Integrals. The Gamma and Beta Functions

Exercise 10.1. Compute the integral:

a)

1

Z

0

√xdx

1 − x² b)

3

Z

2

√xdx

x²− 4 c)

π

Z2

0

tg xdx d)

∞

Z

0

dx 1 + x² e)

∞

Z

√ 3

dx x²+ 9

f)

∞

Z

3

dx

x² g)

∞

Z

0

xe^−x²dx h)

∞

Z

0

e^−xsin xdx i)

0

Z

−∞

e^−xdx j)

−1

Z

−∞

dx x³

k)

∞

Z

−∞

(arctg x)² 1 + x² dx l)

∞

Z

−∞

xdx

x⁴+ 1 m)

2

Z

−1

dx

x n)

1

Z

−1

xdx ln x² Exercise 10.2. Find the area between the curves:

a) y = e^−x and OX and OY axis b) y = x

1 + x⁴ and OX c) y = 1

p3 _x

3 − 1, y = 0, x = 0, x = 3 d) y = 1

|x − 1|, y = 0, x = 0, x = 2 e) y = 8

x²+ 4, y = 0, x = 0 f) y = ln x, y = 0, x = 0, x = e g) y = q³

(x + 1)², y = 0, x = 0 h) y = 1

x³, x = 1 and OX and OY axis

Exercise 10.3. Is the area between the curves y = 2^x, y = 1

x − ¹₂ and OX axis nite?

Exercise 10.4. Compute:

a)

∞

Z

0

xe⁻^x²dx b)

∞

Z

0

x⁵²e^−xdx c)

∞

Z

0

x⁶e^−xdx

d)

∞

Z

0

x√

xe^−3xdx e)

∞

Z

0

x⁵e^−4xdx f)

∞

Z

0

x√³

xe^−2xdx

g)

1

Z

x¹² (1 − x)³² dx h)

1

Z

x¹² (1 − x)¹² dx i)

1

Z

x⁶(1 − x)⁴dx

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Sheet 11. Matrices

Exercise 11.1. Find the matrices A + B, 2A, 2A − B i A − αB, where α ∈ R and:

a) A =

"

3 1

0 −2

# B =

"

−2 4 3 5

#

b) A =







0 7 1 1 0 4

−2 ¹₂ 3





 B =







−2 1 3 0 7 4 1 3 2







c) A =

"

1 −3 0

3 4 −2

# B =

"

2 2 √

2 1 5 −√

3

#

d) A =

"

α 1 − α

−3 4

# B =

"

−α 7 + 2α

0 3α

#

Exercise 11.2. Find if exist: A · B, A · B^T, B · A, B · A^T, where:

a) A =

"

1 −3 0

3 4 −2

# B =







−2 1 3 0 7 4 1 3 2





 b) A =

"

3 1

0 −2

# B =

"

−2 4 3 5

#

Exercise 11.3. Solve the following equations or set of equations:

a) 2

"

1 −4 −2

3 7 0

#

+ X =

"

3 −5 2 1 −11 3

#

b)











X + Y =

"

7 0

1 −2

#

2X + 3Y =

"

1 4

5 −1

#

Exercise 11.4. Compute the determinants:

a)

7 0

1 −1

b)

1 3 −2

2 4 5

−1 0 −2

c)

1 0 1 7

0 −3 2 2

0 0 4 −4

0 0 0 5

Exercise 11.5. Compute the determinants reducing to the triangular matrix:

a)

−1 −2 4 1

−3 1 3 1

0 2 1 4

1 −1 2 0

b)

0 1 3 −2

1 2 −1 4

−1 −3 −5 0

1 3 −2 1

Exercise 11.6. Compute the determinants using cofactor expansions (a, b, c, d ∈ R):

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Exercise 11.7. Solve the equations:

a)

2 x + 2 −1

1 1 −2

5 −3 x

= 0 b)

1 + x 1 1 1

1 1 − x 1 1

1 1 1 + x 1

1 1 1 1 − x

= 0

Exercise 11.8. Find the inverse matrix if exists:

a)

"

1 3 2 5

#

(from the denition)

b)







1 −1 0

2 3 1

1 1 1





 c)







1 2 3 0 1 5 1 2 3





 d)







2 0 0 4 0 0 0 1 0 2 0 0

−1 0 1 0





 Exercise 11.9. Solve the equations:

a) X ·

"

−1 1

3 −4

#

=

"

−2 −1

3 4

#

b) 3 · X +

"

1 3

−2 1

#

=

"

5 6 7 8

#

· X

c)







1 1 −1

4 5 −4

−2 −3 3





· Y =







1 2

0 −1

2 1





 Exercise 11.10. Reduce to the base form:

a)







1 2 5 2 4 10 3 6 15





 b)







1 −1 0 2 1

3 1 1 3 2

−1 −3 −1 1 0





 c)







2 1 1 1 1 3 1 1 1 1 4 1 1 1 1 5 1 2 3 4 1 1 1 1







d)







1 3 5 −1

2 −1 −3 4

5 1 −1 7

7 7 9 1







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Sheet 12. Systems of Linear Equations

Exercise 12.1. Solve the following systems:

(a) by means of Cramer's Rule,

(b) by means of elementary operations.

a)







x₁ − 3x₂ + 5x₃ = −4 2x₁ + 5x₂ − x₃ = 3

− x₁ − x₂ + 3x₃ = −4

b)







− x₁ + 2x₂ − x₃ = 2 3x₁ − x₂ + x₃ = ¹₂ 2x₁ + 8x₂ − 3x₃ = 12

c)







5x₁ − 3x₂ + 7x₃ = 0

− 4x₁ + x₂ − 5x₃ = 0 x₁ − x₂ + x₃ = 0

d)











x₂ − 3x₃ + 4x₄ = 0

x₁ − 2x₃ = 0

3x1 + 2x2 − 5x4 = 2

4x1 − 5x3 = 0

e)







2x − y + z = 2

3x + 2y + 2z = −2

x − 2y + z = 1

f)











x + 2y − 3z = 0

4x + 8y − 7z + t = 1

x + 2y − z + t = 1

− x + y + 4z + 6t = 0

g)







1 1 −1

1 −3 2

−1 2 −1











 x₁ x₂ x₃





=







−2 0 1





 h)







x₁ − 2x₂ = −2

2x₂ + x₃ = 1

x₁ − x₃ = 1

i)







x₁ + 3x₂ − x₃ = 8 x₁ + x₂ − 3x₃ = 2 2x₂ + x₃ = 5

j)







x₁ + x₂ + x₃ = 0 2x₁ − x₂ − x₃ = −3 x₁ − − x₂ + x₃ = 0 Exercise 12.2. Solve the following systems:

a)

"

2 5 −7

3 −8 5

#





 x₁ x₂ x₃





=

"

1 2

#

b)

"

1 −2 5

1 2 −3

#





 x₁ x₂ x₃





=

"

0 0

#

c)





1 ⁴₃ −3 3 2 −5 3 4 −9









 x₁ x₂ x





=





 3 0 9





 d)







x₁ − 2x₂ = 2

− 2x₁ + 4x₂ + x₃ = 3

− x + 2x + x = 1

(22)

Exercise 12.3. Find the general solutions of the following systems. Indicate two distinct particular solutions:

a)







2x₁ − x₂ + x₃ − x₄ = 0

− x₁ + 3x₂ − x₃ + 2x₄ = 0

x₁ + 2x₂ + x₄ = 0

b)











− x₁ + 2x₃ + x₄ = 0

2x₁ − x₂ + 2x₃ − x₄ = 0

x₁ + 2x₂ − x₃ = 0

2x₁ + x₂ + 3x₃ = 0

c)







x₁ + x₂ + 2x₃ + 3x₄ = 0 2x1 − 3x2 + 4x3 + x4 = 0 4x1 − x2 + 8x3 + 7x4 = 0

d)







x₁ − x₂ + 2x₃ = −3 x2 + x3 = −2 x1 − 2x2 + x3 = −1 Exercise 12.4. Find one of the basis solutions of the following systems:

a)

( x1 − x2 + x3 − x4 = −1

2x₁ + x₂ − 3x₃ + x₄ = 5 b)







x₁ − 2x₂ = 1

2x₂ + x₃ = −1

x₁ + x₃ = 0

c)







x₁ − x₂ + x₃ + 2x₄ = 0

− x₁ + 2x₂ + x₃ = 1

− 2x₂ + 3x₄ = −2

d)







− 2x1 + x3 + x4 = 5

x1 + x2 − x3 = −2

3x₁ + 2x₃ + x₅ = −2

Exercise 12.5. Find all basis solutions of the systems:

a)

( x₁ − 2x₂ + 3x₄ = 2

− 2x₁ + 4x₂ + x₃ + x₄ = 3 b)







x₁ + x₂ + x₃ = 1

− 2x₁ + 2x₂ + 3x₃ = −1

− x₁ + 3x₂ + 4x₃ = 0

c)











− 5x₂ + x₅ = 3

+ 6x₂ + x₄ = 3

x1 − 7x2 = 2

8x2 + x3 = 1

d)







x₁ + x₂ + x₃ = 3

− 2x₁ + 2x₂ + 3x₃ = 3

− x₁ + 3x₂ + 4x₃ = 6

Exercise 12.6. Solve the system of equations by means of elementary operations on rows. What are the basis variables and the free variables of the found general solution. Indicate two distinct particular solutions of the system, one of them should be the basis solution. Find the conditions for the general solution to be nonnegative

 

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Exercise 12.7. Solve the system of equations







6x₁ + 3x₂ − x₃ + x₄ = 3 x₂ + x₃ − x₄ = 1 3x₁ + 3x₂ + x₃ − x₄ = 3

with respect to the indicated variables, characterize the set of nonnegative solutions.

(a) The basis variables x1, x₂, (b) the basis variables x3, x₄,

(c) the basis variables x2, x₃.

(24)

Sheet 13. Systems of Linear Inequalities

Exercise 13.1. Mark in the coordinate system sets of solutions to the following systems of inequalities:

a)











x₁ + x₂ ≥ 1 3x₁ + 2x₂ ≤ 6 x₁ − 2x₂ ≤ 4 3x₁ + 2x₂ ≥ 6

x₁ ≤ 6

b)







1 −2

−1 −1

1 0

0 −1







"

x1

x2

#

≤





 4 2 6 0





 c)











x₁ − 2x₂ ≤ 4 x₁ + x₂ ≥ 5 x₁ + x₂ ≤ 5

x₁ ≤ 6

x₂ ≤ 3

d)







1 −2

−1 −1

1 0

0 1







"

x₁ x₂

#

≤





 4 2 6 3





 e)











x₁ + x₂ ≥ 2 x₁ − 2x₂ ≤ 2

x₁ ≥ 0

x2 ≥ 0

f)







x₁ + x₂ ≤ 4 x₁ − x₂ ≥ 3 x₂ ≥ 1

Exercise 13.2. Using the geometrical method solve the system of inequalities and nd the set of its' nonnegative solutions:

a)







2x₁ + x₂ ≤ 7

− 2x₁ + x₂ ≤ 3 2x₁ − 5x₂ ≤ 1

b)







y − x ≥ −1

y − 3x ≤ 1

y x ≤ −4

c)

( − 2x + y ≤ 3

y − 5x ≤ 1

Exercise 13.3. Using the graphical and algebraical methods solve the following systems of inequalities:

a)











x1 + x2 ≤ 10 x1 − x2 ≤ 6

x₁ ≥ 0

x₂ ≥ 0

b)







2 4

3 3

5 1

−1 0

0 −1







"

x1

x₂

#

≤





 180 180 200 0 0







Exercise 13.4. Find the analytic formula for the sets of solutions to the following systems:









5x₁ + 3x₂ ≤ 150

6x₁ + 4x₂ + 3x₃ ≤ 120 5x₂ + 4x₃ ≤ 200

x₁ ≥ 0

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Exercise 13.5. Find all solutions and all nonnegative solutions to the system







x ≥ −2

x − y ≤ 3 x + y ≤ 7

applying the pure algebraic method and next conrm obtained result by means of graphical method.

(26)

Sheet 14. Multivariable Functions

Exercise 14.1. Find domain of the function z = f(x, y). Mark it in coordinate system.

a) f (x, y) = 1

px²+ y² b) f (x, y) = 1 x − y c) f (x, y) =√

xy d) f (x, y) =√

x²− 1 e) f (x, y) = px²− y² f) f (x, y) =√

x +√ y

g) f (x, y) = ln (4 + 4x − y²) h) f (x, y) = px²+ y²− 1 + ln (4 − x²− y²) i) f (x, y) = arcsin^x_y

Exercise 14.2. Find domains and level sets of the functions:

a) f (x, y) = x²+ y² b) f (x, y) = y − x² c) f (x, y) = y x² + y² d) f (x, y) = xy e) f (x, y) = p9 − x²− y² f) f (x, y) = y² g) f (x, y) = 1 − 1

2x −1 3y

Exercise 14.3. Show that the following limit does not exist:

a) lim

(x, y)→(0, 0)

xy

x²+ y² b) lim

(x, y)→(0, 0)

x²

x²+ y² c) lim

(x, y)→(0, 0)

2x²+ y²

x²− y² d) lim

(x, y)→(0, 1)

x⁶ y²− 1 Exercise 14.4. Show that:

a) lim

x→0 y→1

1

x + y² = 1 b) lim

x→0 y→0

x⁴− y⁴

x²+ y² = 0 c) lim

x→0 y→2

q

x²+ (y − 2)²+ 1 − 1 x²+ (y − 2)² = 1

2 d) lim

x→0 y→0

x³

x²+ y² = 0 e) lim

x→0 y→0

e^x²^+y² − 1 x²+ y² = 1

Exercise 14.5. Decide, if it is possible to dene function f to be continuous at (0, 0):

a) f (x, y) = p9 + x²+ y²− 3

x²+ y² b) f (x, y) = x sin 1

x²+ y² c) f (x, y) = (1 + x²+ y²)^x2+y2¹ d) f (x, y) = e^x+y− 1

x + y e) f (x, y) = sin 1

x²+ y² f) f (x, y) = x² x²+ y² g) f (x, y) = x⁴

x²+ y²

(27)

Exercise 14.6. Find the rst-order partial derivatives of the given functions:

a) f(x, y) = x³y + 2xy b) f(x, y) = e^x(cos x + x sin y) c) f(x, y) = y

x² + y² d) f(x, y) = x − y x + y

e) z = x^y f) z = e^xy²

g) z = ln

x +px²− y²

h) z = arctany x

i) f(x, y, z) = x²y²z⁴+ 3xy j) f(x, y, z) = x⁵y¹⁰− x³sin z + y²e^x k) f(x, y, z) = ln (x + y + z) l) f(x, y, z) = sin (x²+ y²+ z²) m) u = e^x(x²+ y²+ z²) n) u = e^{x sin yz}

Exercise 14.7. Find the partial elasticity with respect to each variable:

a) f(x, y) = x³y + 2xy b) f(x, y) = px²+ y⁴+ 2x sin y c) f(x, y) = e^xy² Exercise 14.8. Find the second-order derivatives of the given functions:

a) f(x, y) = x³+ xy²− 5xy³+ y⁵ b) f(x, y) = xy + x² y³ c) f(x, y) = x^y d) f(x, y) = e^x^y e) z = lnx

y f) z = arctan xy

g) f(x, y, z) = e^xyz h) u = e^3x+4ycos 5z

Exercise 14.9. Let z = xe^y + ye^x. Verify that the function z satises the equation:

∂³z

∂x³ + ∂³z

∂y³ = x ∂³z

∂x∂y² + y ∂³z

∂x²∂y.

Exercise 14.10. Find the directional derivatives of the function z = x²y + y² at the point P0(1, 1) in the direction given by angles α = ¹₃π, β = ¹₆π.

Exercise 14.11. Find the directional derivative of the function u = xy²z³ w at the point P0(3, 2, 1) in the direction from P0 to P1(5, 4, 2).

(28)

Exercise 14.12. Find the gradient vector for each of the given function at the point P0: a) f(x, y) = x²y³ − x sin y, P₀(−2, 0)

b) f(x, y) = x√ y + y

√x, P₀ ¹₄, 9 c) z = px²− y², P0(−5, 3) d) z = ln(x²+ y²), P0(3, −4)

e) f(x, y, z) = x³+ 3xyz + yz³, P₀(5, −2, 1) f) f(x, y, z) = (3x²y + z⁴)¹⁰, P₀(−1, 0, 1) g) u = x⁵y¹⁰− x³sin z + y²e^x, P₀(−1, 1, 0)

Exercise 14.13. Find the directional derivatives of the given functions at P0 in the direction u a) z = x²+ y², P0(−3, 4), u = ¹²₁₃,₁₃⁵

b) z = sin x cos y, P₀(0, π), u = h

−¹₂,

√3 2

i c) f(x, y) = arctan xy, P₀(1, 1), u = [1, 1]

d) f(x, y, z) = xy²+ z²− xyz, P₀(1, 1, 2), u = [1, 2, 1]

e) f(x, y, z) = z − x

z + y, P0(1, 0, −3), u = −⁶₇,³₇, −²₇ f) u = ln (x²+ y²+ z²), P₀(1, 2, 1), u = [2, 4, 4]

g) u = e^xyz, P₀(−1, 1, −1), u =h

1 2, −³₄,

√3 4

i

Exercise 14.14. Verify, if for the function f(x, y) = ln √ x +√

y

the following equality holds [x, y] ◦ grad f (x, y) = ¹₂.

Exercise 14.15. Find the vector b = ¹₂det Hf (−1, 1) grad f (1, −1)for f(x, y) = y x².

Faculty of Management  Mathematics  Exercises Contents