1 Arithmetic calculations (calculator is not allowed)

Transcription

1 1 ARITHMETIC CALCULATIONS (CALCULATOR IS NOT ALLOWED) 1 Arithmetic calculations (calculator is not allowed) 1.1 Check the result Problem 1.1. Problem 1.2. Problem 1.3. Problem = : : 1 1 = ( ) (2 : ) 9 32 = 1 4 Problem 1.5. ( ) : : ( ) : 3 3 = 9 ( ) ( )( ) = Problem 1.6. ( ) (5 : ) : ( ) : 0.1 = 1 Problem 1.7. ( ( )/ ( ) ) : ( ) : = 1 Problem 1.8. Problem 1.9. ( ) 2 = 2 ( )( ) =

2 1.2 Calculate 2 ALGEBRAIC CALCULATIONS 1.2 Calculate Problem ( (1 1 4 : ) : ( ) ) Problem ( ) ( ( : 0.003) : 1 ) 15 (3 3 : /0.8) : Problem (10 : : 10)( ) ( ) Problem (( ) : ) ( ) Algebraic calculations 2.1 Factorize Example 2.1. [ ] 2x x 3 = x( 2 x)( 2 + x) Problem a a 2 b + 20ab 2 Problem 2.2. a 2 (x 1) b 2 (x 1) Problem 2.3. (x + y) 2 4xy 2

3 2.1 Factorize 2 ALGEBRAIC CALCULATIONS Problem 2.4. (a + b) 3 a(a + b) 2 Problem 2.5. x 3 + 3x 2 9x 27 Problem 2.6. a 5 a 3 a Problem 2.7. x n x n y 2 Problem 2.8. a n+1 + a n Problem 2.9. Problem Problem a b a2 b ab2 c 3 d c6 d 8 2a a 2 a 3 Problem (n 2 2nx) 2 + 2(n 2 x 2nx 2 ) + x 2 Problem a 2 n 2 + 2np p 2 Problem xy xz (y 2 2yz + z 2 ) 3

4 2.1 Factorize 2 ALGEBRAIC CALCULATIONS Problem a 2 a 2 n + (n 2)(an a) 2 Problem c c c 8 + 8c 10 Problem x 6n + 12x 4n + 48x 2n + 64 Problem x 8 2x 7 + x 6 x 5 + 2x 4 x 3 Problem (4a + 3b) 2 16(a b) 2 Problem xyz + x 2 y 2 + 3x 4 y 5 + 3x 3 y 4 z xy z Problem x 3 + x Problem x 4 4 Problem (x + y + z) 3 x 3 y 3 z 3 Problem (a b) 3 + (b c) 3 + (c a) 3 Problem x 3 + 8x x

5 2.1 Factorize 2 ALGEBRAIC CALCULATIONS Problem x 4 12x x 2 60x Problem (ax by) 2 + (bx + ay) 2 Problem xyz 2 (x y)x 3 z 4 Problem (x + 1) 4 1 Problem x Problem (n x)(5n 2 4x 2 ) (3x 2 4n 2 )(x n) Problem x 11 48x 3 6x x Problem a 6 a 6 z 4 + 3a 4 z 2 a 4 z 6 + 3a 2 z 4 + z 6 Problem x 3 + (a 1)x + a Problem x Problem x 4 + x

6 2.2 Simplify 2 ALGEBRAIC CALCULATIONS Problem (a + 1) Problem (a + n) 6 + (a n) 6 Problem x 4 + x 2 + 2x Simplify Problem Problem ab bc c 2 + ac ab + ac ab ac b 2 + bc + bc ab + ac bc a 2 Problem x x x2 + x 1 x 3 x 2 + x 1 + x2 x 1 x 3 + x 2 + x + 1 2x3 x 4 1 Problem Problem Problem n 2 + nx + x 2 n 3 + x 3 : n 3 x 3 n 2 nx + x 2 a ( + 1) : (1 3a2 a 1 1 a ) 2 3 5x 3 x + y (x + y 5x x y) (a + n 2 3n n2 (3n + a) ) : ( 1 2 2a a n 2 3an) 6

7 2.2 Simplify 2 ALGEBRAIC CALCULATIONS Problem Problem ( 1 + x 1 x 1 x 1 + x + 4x2 x 2 1 ) : ( 1 x 3 + x 2 1 x x 2 1) Problem ( 2x2 5nx + 2n 2 n 2x (2n + x)) ( n x 4x n+1 ) Problem ( x4 x x 4 + x 5 ) : (x3 + x + 1 x + 1 x 3 ) Problem ( 4xn 4 x 2n 2 + x2n 2 x n + 1 ) : ( 3x 2n 2 12x n 4 x 3n + 5x 2n 2x n 10 x4n+1 + 5x 3n+1 ) x n+2 4 Problem (x n 1 7 xn 3 + x ) : ( 6x 2n 24 n x 2n+3 + 6x n+3 + 9x 2x 3 3x n + 6 : 4 x n+2 + 3x ) 2 Problem ( (a+x)2 4)( (a x)2 + 4) : (a 6 x 6 ) ax ax (a 2 x ax 2 ) : (((a + x) 2 ax)((a x) 2 + ax)) a ax a+x a ax a x Problem Problem ( a + 2b ab 4 2a + b a 4 b ) : (b2 + c 2 b 2 c 2 ( 1 b 2 1 c 2 ) ( 1 a 2 1 c 2 )a2 + c 2 a 2 c 2 ) ( 8x 2 y 2y x x x) : ( 2y x) ( 3 a 2 3 ab + 3 b 2 )( 3 a + 3 b) 7

8 2.2 Simplify 2 ALGEBRAIC CALCULATIONS Problem Problem a + x a x 4a 2 a x + a + x a 2 x 2 ( m + n)( 4 m 4 n)( 4 m + 4 n) Problem Problem Problem Problem Problem x + y + x y x + y x y Problem a b a + b Problem Problem

9 3 EQUATIONS Problem Problem Problem x x x x x 11/16 1 (ab) mn m+n a 2m+n b m+2n a m+2n b 2m+n x y x y x 3 y x 1/3 y 1/3 3 Equations 3.1 Solve sets Problem x 3y = 15 5x + 6y = 27 Problem x 4y + 1 = 0 31x 5y + 16 = 0 Problem 3.3. Problem 3.4. Problem 3.5. ax 3y = 4 x y = 4 3 2x + ay = 8 3x 5y = 6 x y = 2 2x 2y = a 9

10 3.1 Solve sets 3 EQUATIONS Problem 3.6. x + y = 1 y = 2(x + 1) Problem (1 + y) = ( 3 + x) 3(y + 1) = x Problem 3.8. Problem x+y 2 1 x y 2 = x+y x y 2 = 1 (x 2)(y + 6) = xy + 13 (y 2)(x + 4) = xy 13 Problem Problem x+y x y+1 = 1 5 x+y 1 3 x y+1 = 1 x 2 y 2 = 3(x + y) x + y = 4(x y) + 1 Problem x + y + z = 6 x + y z = 10 x y + z = 0 Problem ax + by + cz = d a 2 x + b 2 y + c 2 z = d 2 a 3 x + b 3 y + c 3 z = d 3 10

11 3.2 Solve equations 3 EQUATIONS Problem a b x + b c y + c a z a 2 b 2 + b2 c 2 + c2 a 2 x y z (a b) 2 x + (b c)2 y = a + b + c (a c)2 z = 2(ab + ac + bc) = 2(a c)b 3.2 Solve equations Problem (2x + 1) 2 8x 4x 2 1 = 3(2x 1) 7(2x + 1) Problem x x 2 9 = 1 x 2 + 3x 3 6x + 2x 2 Problem Problem x 1 14x 2 + 7x x 2 3 = 2x + 1 6x 2 3x x 3 x + 1 = 8 x x x x 2 Problem x x + x = 3 2 2x 2 + 6x Problem x 3 + 3x 2 + 5x + 3 x 2 + 3x + 2 = 2x2 + x + 3 2x + 1 Problem x 21 2x 2 + 5x x 3 x + 4 = x + 4 2x 3 Problem x 2 5ax 2a 2 = 0 Problem x 2 2nx + 4n 4 = 0 11

12 3.2 Solve equations 3 EQUATIONS Problem Problem Problem Problem Problem x + 1 x = 2a2 + b 2 a 2 b 2 x a x b + x b x a = 2.5 ax 2 x 1 = (a + 1)2 1 x a + 1 x b = 1 a + 1 b (x a) 2 + x(x a) + x 2 (x a) 2 x(x a) + x = Problem Problem x + 1 a + 1 b = 1 x + a + b x x x = 1 Problem x 4 5x = 0 Problem x 4 + a 2 = x 2 + 4a 2 x 2 Problem m 2 n 2 x 4 (m 4 + n 4 )x 2 + m 2 n 2 = 0 Problem x 4 2(a 2 + b 2 )x 2 + (a 2 b 2 ) 2 = 0 12

13 3.2 Solve equations 3 EQUATIONS Problem (x 2 8) 2 + 4(x 2 8) 5 = 0 Problem (x + 1 x )2 4.5(x + 1 x ) + 5 = 0 Problem Problem x 2 + 2x x 2 + 2x + 2 = 18 x 2 + 2x + 1 (x + 3) 3 (x + 1) 3 = 56 Problem x 4 = 2(2 + 3)x Problem (x 2 6x) 2 2(x 3) 2 = 81 Problem (x + 5) 4 = 13(x + 5) 2 x 2 36x 4 Problem x 3 + 2x 2 = 2x 3 Problem x x 3 = 32 Problem x 5 x 3 + 4x 2 = 4x Problem x 6 64 = 0 13

14 3.2 Solve equations 3 EQUATIONS Problem Problem Problem Problem Problem x x 2 = 1 x x = 2.8x x 2 7 x 2 9 = x x = x 2 x 3(x 2 1) + 2x 3(1 x 4 ) = 1 x(1 + x 2 ) Problem Problem Problem x 2 + 2x 8 15 x 2 + 2x 3 = 2 x 2 x x 2 x + 1 x2 x + 2 x 2 x 2 = 1 7(x + 1 x ) 2(x2 + 1 x 2 ) = 9 Problem Problem Problem x 4 3 = 1106 x 4 2 4x x 2 + x x x 2 5x + 3 = 3 2 x2 2 = x 14

15 3.2 Solve equations 3 EQUATIONS Problem x 1 x + 4 = 6 Problem x + 3 = (3x + 1)(x 1) Problem x2 + 5 = 2 3 x + 1 Problem x = 3 Problem x 11 x 2 = x + 3 Problem x x 4 = 2 Problem Problem Problem x 2 2 x = 2 x 10 x + 10 x x 10 x = x 2 4x = 3 x 2 4x Problem x 3 x = 3 x Problem x + 3 x = 3( x 3 x) 15

16 4 EXPONENTS AND LOGS Problem x + 2 = 3 3x + 2 Problem x x x + 1 x + 7 = 4 Problem x + a + 3 x + a + 1 = 3 x + a Exponents and logs Useful formulae: y = a x x = log a y a x a y = a x+y a x b x = (ab) x log(ab) = log a + log b log a b = log c b log c a x y = e y ln x 4.1 Find x without calculator Problem 4.1. x = 10 log 10 3 log 10 2 Problem 4.2. x = 36 log 6 2 Problem 4.3. x = log 9 7 Problem 4.4. log x = 3 16

17 4.2 Solve equations 4 EXPONENTS AND LOGS Problem 4.5. log x n = n Problem 4.6. x = a 2+log a b Problem 4.7. log 2 x = 4 Problem 4.8. x = log 2 2 ( 1 8 ) Problem 4.9. log x (2 2 3) = 2 Problem log ( 2+1) ( ) = x 4.2 Solve equations Problem (x 2)(x 3) = 1 Problem /x = 100 x Problem x2 +x 0.5 = 4 2 Problem x 5 x = 0.1(10 x 1 ) 5 Problem x x = 30 17

18 4.2 Solve equations 4 EXPONENTS AND LOGS Problem x = 9 2 x 2 Problem x 2 +x 1 = 17 2 x 2 +x 1 Problem ( 3) 10x 2+ 10x 29 = 27 Problem x+2 x x 1+ x 2 2 = 6 Problem ( x + 2) 10x2 3x 1 = 1 Problem (0.4) x 1 = (6.25) 6x 5 Problem log x 1 (x 2 5x + 10) = 2 Problem log 2 log 3 log 4 x = 0 Problem log 5 log 10 x = 0 Problem log x 5x = logx 5 Problem log a (1 + log b (1 + log c (1 + log p x))) = 0 18

19 5 TRIGONOMETRY Problem log 2 (4 x + 4) = x + log 2 (2 x+1 3) Problem ln x 5 ln x+1 = 3 5 ln x 1 ln x Problem log 5 (4x 6) log 5 (2x 2) 2 = 0 Problem ( 4 + x ( 15) + 4 x 15) = 8 Problem ( log a x + log x a) 10 = 0 Problem log 3 x + log x x log 1/3 x = 6 Problem log x log 2x 64 = 3 Problem log x 9x 2 log 2 3 x = 4 Problem log 5 (x 2) + log 5 (x3 2) + log 0.2 (x 2) = 4 5 Trigonometry Useful formlulae: sin(π/2 x) = cos x sin(x + y) = sin x cos y + sin y cos x 19

20 5.1 Prove 5 TRIGONOMETRY 5.1 Prove cos(x + y) = cos x cos y sin x sin y tan x = sin x cos x Problem 5.1. tan x + tan y tan x tan y = sin(x + y) sin(x y) Problem 5.2. sin 6x cos 6x = 1 tan 3x Problem 5.3. tan(π/8 + x) + tan(π/8 x) 1 tan(π/8 + x) tan(π/8 x) = 1 Problem sin 2 x cos 2 x = cos 4x Problem cos 2 x cos 2x = 1 Problem 5.6. tan x 1 + tan x + tan x 1 tan x = tan 2x Problem sin 2 (π/4 x) + sin 2x = 1 Problem 5.8. sin 2x + 2 sin( 5π x) cos(5π + x) = Problem 5.9. tan( π 4 + x 2 ) 1 sin x cos x = 1 Problem sin 6x cos(6x π) + sin 2x cos 2x = 2 20

21 5.1 Prove 5 TRIGONOMETRY Problem cos 2x + sin 2x tan x = 1 Problem sin 4x 2 cos 2 2x + 1 = 2 sin(4x π/4) Problem sin 2 (x π/2) cos 2 (y 3π/2) = cos(x + y) cos(x y) Problem sin 2x + sin 4x + sin 6x = 4 sin 3x cos 2x cos x Problem cos x sin x sin 2x = cos x cos 2x Problem sin x + cos x + tan x = 2 2 cos 2 x 2 sin(π/4 + x) cos x Problem tan a + tan b + tan c tan a tan b tan c = sin(a + b + c) cos a cos b cos c Problem sin(a + b) 2 cos a sin b 2 cos a cos b cos(a + b) = tan(a b) Problem tan(a b) + tan b tan(a + b) tan b = cos(a + b) cos(a b) 21

22 5.2 Solve equations 5 TRIGONOMETRY 5.2 Solve equations Problem sin x cos 2x = 0 Problem cos x = sin 2x cos x Problem sin x sin 2 x = 2 Problem cos x = sin x Problem sin 3 2x = sin 2x Problem Problem tan 2 x + 1 tan 2 x = 2 cos 4x = cos x Problem sin 2 x + sin 2 2x + sin 2 3x = 1.5 Problem cos 2 x + cos 2 2x + cos 2 3x = 1 Problem cos x + sin x = 3 Problem sin x + 3 cos x = 2 22

23 6 DIFFERENTIATION Problem sin a cos(a + x) = cos a sin(a + x) Problem sin 4 x + cos 4 x = sin 2x Problem sin x + sin 2x + sin 3x + sin 4x = 0 Problem sin x + sin 2x + sin 3x = cos x + cos 2x + cos 3x Problem sin(x π/3) = cos(x + π/6) Problem sin 6 x + cos 6 x = Differentiation Notation: function y(x), first derivative y (x) or dy/dx, second derivative (derivative of derivative) y (x) or d 2 y/dx 2. Useful formulae: (y(x) + z(x)) = y (x) + z (x) (yz) = y z + yz Chain rule: let z = z(y) and y = y(x), so that z = z(y(x)), then z x = z y y x. Example 6.1. [ ] z = (ln x) 2 : z = y 2, y = ln x z y = 2y = 2 ln x y x = 1 x 23

24 6.1 Find first and second derivatives 6 DIFFERENTIATION z x = 2 x ln x Basic functions and their derivatives: y = x a, y = ax a 1 y = ln x, y = 1/x y = e x, y = sin x, y = e x y = cos x y = cos x, y = sin x y = arcsin x, y 1 = 1 x Find first and second derivatives Problem 6.1. y = x3 3 2x2 + 4x 5 Problem 6.2. y = 1 x + 1 x 2 1 3x 3 Problem 6.3. y = x(x 3 x + 1) Problem 6.4. y = ( a x) 2 Problem 6.5. y = (x 2 3x + 3)(x 2 + 2x + 1) Problem 6.6. y = x + 1 x 1 Problem 6.7. y = x x

25 6.1 Find first and second derivatives 6 DIFFERENTIATION Problem 6.8. y = 6 3 x Problem 6.9. y = 8 4 x 6 3 x Problem y = ( x ) 3 Problem Problem y = x 1 x 1 + x 2 y = x sin x Problem y = x cos x Problem y = sin 2 x + sin x Problem y = sin 2 x 3 Problem y = cos x 1 sin x Problem y = sin x + sin 1 x Problem y = 1 x 2 + arcsin x 25

26 6.1 Find first and second derivatives 6 DIFFERENTIATION Problem y = ln x 2 x Problem y = x ln x 1 Problem y = ln(1 + cos x) Problem Problem y = 1 ln x + ln x x n y = x n ln x ln 2 x Problem y = ln(x + a 2 + x 2 ) Problem y = 2 3 x 4 ln 2 + x Problem Problem y = ln x 2 1 x 2 y = 2 x + x 2 Problem y = xe x Problem y = ae x/a + xe x/a 26

27 7 INTEGRALS Problem y = x x Problem y = cos x sin x Problem y = (x x ) x 7 Integrals 7.1 Indefinite integrals If y = z then ydx = z + C, C = const. Substitution: if y = f(x) then g(y)dy = g(y(x))y dx. Integration by parts: if u(x) and v(x) are two functions then vu dx = vu uv dx. Attention: (1/x)dx = ln x. Problem 7.1. xdx Problem 7.2. m xn dx Problem 7.3. dx x 2 Problem x dx Problem 7.5. a x e x dx 27

28 7.1 Indefinite integrals 7 INTEGRALS Problem 7.6. dx 2 x Problem 7.7. (1 2u)du Problem 7.8. ( x + 1)(x x + 1)dx Problem 7.9. x x 3 e x + x 2 dx x 3 Problem ( 1 z ) 2 dz z Problem (1 x) 2 x x dx Problem x 3 dx x Problem dx 3 3x 2 Problem x 2 3 x 2 x dx Problem cos 2 x 1 + cos 2x dx Problem cos 2x cos 2 x sin x dx 28

29 7.1 Indefinite integrals 7 INTEGRALS Problem tan 2 xdx Problem sin 2 x 2 dx Problem (1 + 2x 2 ) x 2 (1 + x 2 ) dx Problem (1 + x) 2 x(1 + x 2 ) dx Problem dx cos 2x + sin 2 x Problem (arcsin x + arccos x)dx Problem (x + 1) 15 dx Problem dx (a + bx) c, c 1 Problem (8 3x)6 dx Problem x x 2 + 1dx Problem x 2 5 x 3 + 2dx 29

30 7.2 Definite integrals 7 INTEGRALS Problem sin 3 x cos xdx Problem sin xdx cos 2 x Problem cos 3 x sin 2xdx Problem ln x x dx Problem (arctan x) 2 dx 1 + x 2 Problem dx arcsin x 3 1 x 2 Problem e x sin e x dx Problem e x dx e x + 1 Problem tan xdx 7.2 Definite integrals If y = z then b ydx = z(b) z(a). a Problem π/2 sin xdx 0 30

31 7.2 Definite integrals 7 INTEGRALS Problem e x dx 0 Problem (1 + x)dx 0 Problem / 3 dx 1 + x 2 Problem π/4 sin 2xdx 0 Problem π/2 cos xdx 0 Problem π/4 tan xdx 0 Problem /2 1/2 dx 1 x 2 Problem dx 1 + x 2 Problem x dx 0 Problem x ln xdx 0 31

32 8 SERIES (PROGRESSIONS) Problem π/2 0 dx a 2 sin 2 x + b 2 cos 2 x, a, b > 0 Problem π 0 x sin xdx Problem ln 2 xe x dx 0 Problem xf (x)dx 8 Series (progressions) 8.1 Arithmetic Arithmetic progression (series): a n+1 = a n + d, a n = a 1 + (n 1)d, 0 S n = a a n = (a 1 + a n )n 2 = na 1 + n(n 1)d 2 32

33 8.2 Geometric 8 SERIES (PROGRESSIONS) Problem 8.1. Given a1 = 1.6, d = 0.2, find a 23. Problem 8.2. Given a 1 = 5.2, d = 0.4, find S 43. Problem 8.3. Given a 1 = a, a n = 9a + 8b, find d and S 9. Problem 8.4. Given d = 1 + q, a n = q, find a 1 and S 28. Problem 8.5. Find n k=1 k. Problem 8.6. Given 5a a 5 = 0 and S 4 = 14, find a 1 and d. Problem 8.7. Given a n = 55, a 2 + a 5 = 32.5, S 15 = 412.5, find a 1, d, and n. Problem 8.8. Given a a 2 12 = 1170, a 7 + a 15 = 60, find a 1 and d. Problem 8.9. Solve x = 117 Problem (x + 1) + (x + 4) + (x + 7) (x + 28) = 155 Problem Find sum of the first n terms: x 1 x + x 3 x + x 5 x +... Problem Calculate Geometric Geometric progression (series): a n+1 = a n q, q 1; a n = a 1 q n 1, S n = a a n = a 1(q n 1) q 1 33

34 8.3 Geometric with q < 1 8 SERIES (PROGRESSIONS) Problem Given a 3 = 135, S 3 = 195, find a 1 and q. Problem Given a 1 = 2.5, q = 1.5, find S 5. Problem Given q = 3/4, a 3 = , find a 1 and S 5. Problem Given q = 2, S 12 = 4095, find a 1 and a 12. Problem Given q = 2, a n = 96, S n = 189, find n. Problem Given progression: 4, -1, 1/4,..., find S 6. Problem Given a 1 = 3, q = 1/2, a n = 3/64, find n and S n. Problem Given a 1 and q, find a 1 a 2... a n. 8.3 Geometric with q < 1 In this case S = S n = a 1 1 q. Problem Given a 1 = 1, S = 2, find q. Problem Given S 4 = , S = 36, find a 1. Problem Given a 1 = 66, S = 110, find q. Problem =? Problem (2 + 2) + ( 2 + 1) + ( ) +... =? Problem For which x the infinite sum exists? a + x a x + a x a + x + (a x a + x ) =? Problem =? 34

35 8.3 Geometric with q < 1 8 SERIES (PROGRESSIONS) The source: V.A. Bachurin, Zadachi po elementarnoi matematike i nachalam matematicheskogo analiza (in Russian). 35