. sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y = e x, y = e x 6 sinhx) coshx) 4 y-axis x-axis : y = cosh x, y = s

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Download ". sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y = e x, y = e x 6 sinhx) coshx) 4 y-axis x-axis : y = cosh x, y = s"

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1 [] sinh x = ex e x, cosh x = ex + e x ) sinh cosh 4 hyperbolic) hyperbola) = 3 cosh x cosh x) = e x + e x = cosh x )

2 . sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y = e x, y = e x 6 sinhx) coshx) 4 y-axis x-axis : y = cosh x, y = sinh x sin x, cos x tan x = sin x cos x, cos x cot x = sin x, sec x = cos x, cosec x = sin x sinh x, cosh x tanh x = sinh x cosh x = ex e x cosh x, coth x = e x + e x sinh x = ex + e x e x e, x sech x = cosh x =, cosech x = e x + e x sinh x = e x e x 4) cosh x, sinh x tanh x tanh x, coth x

3 . 3 4 tanhx) cothx) y-axis x-axis : y = tanh x, y = coth x tan ) cot ) tg, ctg sinh, cosh, tanh, coth sh, ch, th, cth [] ) cosh x sinh x =, tanh x = cosh x 5) n cosh n x = cosh x) n, sinh n x = sinh x) n, tanh n x = tanh x) n 5) ) cosh x 5) cos x + sin x =, + tan x = cos x

4 x, y) = cos t, sin t) x + y = x, y) t x, y) = cosh t, sinh t) 6) 5) x y = x > 0) t 6) x, y) x y = x a y = a > 0, b > 0) 7) b 3) x, y Ax + Bxy + Cy + Dx + Ey + F = 0 3 a > 0, b > 0) ellipse): x a + y b = parabola): y = ax

5 4. 5 x /a -y /b = y/b=±x/a y-axis -a 0 a x-axis 3: x /a y /b = hyperbola): x a y b = 7) x, y) = a cosh t, b sinh t) x, y) = a cosh t, b sinh t) 4 sinx + y) = sin x cos y + cos x sin y, cosx + y) = cos x cos y sin x sin y 8) sinhx + y) = sinh x cosh y + cosh x sinh y, coshx + y) = cosh x cosh y + sinh x sinh y 9)

6 5. 6 9) cosh x cosh y + sinh x sinh y = ex + e x e y + e y + ex e x e y e y = 4 ex e y + e x e y + e x e y + e x e y ) + 4 ex e y e x e y e x e y + e x e y ) = ex+y + e x y ) = coshx + y) tan x, tanh x 8), 9) tanx + y) = tan x + tan y tanh x + tanh y, tanhx + y) = tan x tan y + tanh x tanh y 5 sin x) = cos x, cos x) = sin x, tan x) = cos x 0) sinh x) = cosh x, cosh x) = sinh x, tanh x) = cosh x ) ) e x ) = e x, e x ) = e x tanh x tanh x) = ) sinh x = sinh x) cosh x cosh x) sinh x cosh x cosh x = cosh x sinh x cosh x = cosh x

7 6. 7 0) sin x dx = cos x + C, cos x dx = sin x + C ) sinh x dx = cosh x + C, cosh x dx = sinh x + C ) tan x f x) fx) dx = log fx) + C tan x dx = sin x cos x) cos x dx = dx = log cos x + C cos x tanh x tanh x dx = sinh x cosh x) cosh x dx = dx = log cosh x + C cosh x cosh x > 0 tanh x dx = logcosh x) + C 3) 6 arcsin, arccos, arctan )

8 6. 8 y = sin x π/ x π/ x = arcsin y y ) y = cos x 0 x π x = arccos y y ) y = tan x π/ < x < π/ x = arctan y < y < ) arcsin y arc y arcsin x x = sin y, x = cos y, x = tan y ) sin y sin y) ) [] sin y y = sinh x < x < < y < x = arcsinh y ) x = sinh y ) arc arc x, y) = cosh t, sinh t) 0 t a l l = a 0 x t)) + y t)) dt = a 0 sinh t + cosh t dt = a 0 cosh t dt a [] ) x = arcsinh y y = sinh x x = arcsinh y y = sinh x 4) sin y

9 x=arcsinhy) x=arccoshy) x-axis y-axis 4: x = arcsinh y, x = arccosh y y = cosh x x 0 x = arccosh y x = cosh y y ) y = tanh x < x < < y < x = arctanh y = tanh y < y < 4 x=arctanhy) x-axis y-axis 5: x = arctanh y y = sinh x y = ex e x

10 6. 0 e x ye x = 0, e x y) = y + y e x = ex + e x = cosh x < 0 e x y = y + ) arcsinh y = sin y = log y + y + < y < ) 4) y = cosh x y = ex + e x e x ye x + = 0, e x y) = y x 0, y y e x = ex e x = sinh x 0 e x y = y

11 6. ) arccosh y = cosh y = log y + y y ) 5) y = tanh x y = ex e x e x + e x = ex e x +, ex y) = + y < y < arctanh y = tanh y = log + y y < y < ) 6) 4), 5), 6) dx dy = dy dx x = arcsinh y arcsinh y) = dx dy = dy dx = = sinh x) cosh x 5) cosh x = sinh x + = y + arcsinh y) = y + < y < ) 7) x = arccosh y arccosh y) = cosh x) = sinh x

12 6. x 0 5) sinh x = cosh x = y arccosh y) = y y > ) 8) x = arctanh y arctanh y) = tanh x) = cosh x 5) arctanh y) = y < y < ) 9) 7), 8), 9) arcsin y) = y arccos y) = y arctan y) = < y < ), < y < ), + y < y < ) a > 0 dx x + a = a arctan x + C < x < ) 0) a dx x a = a arctanh x a + C a < x < a), a arctanh a x + C x < a, x > a) ) dx a x = arcsin x + C a < x < a) ) a

13 7. 3 dx x + a = arcsinh x + C < x < ) 3) a dx x a = arccosh x a + C x > a), arccosh x ) + C x < a), a 4) / ), / ) x = at, x = at ) x < a x/a < arctanh x ) = a x/a) a = a x a ) x > a a/x < arctanh a ) = x a/x) ) a x ) = a x a 7 e iθ = cos θ + i sin θ 5) e e x+iy = e x e iy = e x cos y + ie x sin y 6)

14 7. 4 e z e w = e z+w 5) e ix = cos x + i sin x, e ix = cos x i sin x cos x, sin x cosh x, sinh x cos x = eix + e ix, sin x = eix e ix i cos x, sin x x z = x + iy cos z = eiz + e iz, sin z = eiz e iz i 7) ) z cosh z, sinh z cosh z = ez + e z, sinh z = ez e z 8) tan z, tanh z cos iz = e z + e z = cosh z, 9) cosh iz = eiz + e iz = cos z, 30) sin iz = e z e z = i sinh z, i 3) sinh iz = eiz e iz = i sin z 3) cos z cosh iz cosh z cos iz sin z sinh iz sinh z sin iz z, w sinz + w) = sin z cos w + cos z sin w, cosz + w) = cos z cos w sin z sin w 33)

15 7. 5 sinhz + w) = i siniz + iw) = sin iz cos iw + cos iz sin iw) i = sinh z cosh w + cosh z sinh w, coshz + w) = cosiz + iw) = cos iz cos iw sin iz sin iw = cosh z cosh w + sinh z sinh w 33) 4 9) cosh z, sinh z x, y cosh z = ez + e z = ex+iy + e x iy = ex cos y + ie x sin y + e x cos y ie x sin y) 34) = ex + e x cos y + i ex e x sin y = cosh x cos y + i sinh x sin y, 35) sinh z = ez e z = ex+iy e x iy = ex cos y + ie x sin y e x cos y + ie x sin y) 36) = ex e x cos y + i ex + e x sin y = sinh x cos y + i cosh x sin y 37) cos z, sin z 33) 9), 3) cos z = cosx + iy) = cos x cos iy sin x sin iy

16 8. 6 = cos x cosh y i sin x sinh y, 38) sin z = sinx + iy) = sin x cos iy + cos x sin iy = sin x cosh y + i cos x sinh y 39) 8 ) sin x, cos x sin x = x! x3 3! + x5 5! x7 + < x < ), 40) 7! cos x = x! + x4 4! x6 + < x < ) 4) 6! e x, e x e x = + x! + x! + x3 + < x < ), 4) 3! e x = x! + x! x3 + < x < ), 43) 3! sinh x, cosh x sinh x = x! + x3 3! + x5 5! + x7 + < x < ), 44) 7! cosh x = + x! + x4 4! + x6 + < x < ) 45) 6! 40), 4) ix 7 9), 3) tan x tan x = sin x cos x = sin x cos x)

17 8. 7 π/ < x < π/ 0 cos x < cos x) = + cos x) + cos x) + cos x) 3 + tan x = sin x{ + cos x) + cos x) + } 40), 4) tan x ) ) ) = x x3 6 + x5 0 x + x4 x x ) { = x x3 6 + x5 0 + x + 4 ) } x = x ) x ) x = x + x x5 + tanh x 9), 3) tan ix = sin ix cos ix = i sinh x cosh x = i tanh x tanh x = i tan ix = i ix ix3 3 + i ) 5 x5 + = x x x5 + tan x arctan y arctan y) = + y = y + y 4 y 6 + < y < )

18 y arctan 0 = 0 arctan y = y y3 3 + y5 5 y7 + < y < ) 46) 7 arctanh y) = y = + y + y 4 + y 6 + < y < ) arctanh y = y + y3 3 + y5 5 + y7 + < y < ) 47) 7 / x x = x) / = / ) n n=0 n ) x n < x < ) 48) ) n / n ) ) 3 ) n ) = ) n n = 3 n ) 4 n) n)n ) 4 m = n > 0) ), m!! = n )n 3) 3 m = n > 0) ), m = 0, ) n 0 ) n / n ) = n )!! n)!!

19 ) x = n=0 n )!! n)!! x n < x < ) 49) y = n=0 n )!! n)!! y n < y < ) 0 y arcsin y = n=0 n )!! n)!! y n+ n + < y < ) 50) = n n )!! ) y n < y < ) + y n)!! n=0 0 y 7) n n )!! arcsinh y = ) n=0 n)!! y n+ n + < y < ) 5) arccosh y arccosh y y y = ) y = /t x = arccosh /t t 0 < t ) 5) arccosh t = log t + t = log + t t = log + ) t log t 5)

20 8. 0 y = /t) 8) arccosh t ) = 49) t = n=0 y ) = t n )!! n)!! t n 0 < t < ) /t ) arccosh ) = t t n )!! t n 0 < t < ) n= n)!! arccosh ) t + log t = n= n )!! n)!! t n 0 < t < ) +0 t t = t 0 < t < ) t arccosh t + log t lim arccosh ) x +0 x + log x 5) lim arccosh ) x +0 x + log x = lim log + ) x = log x +0 arccosh t + log t log = n= n )!! n)!! t n n 0 < t < ) y arccosh y = log y + log n= n )!! n)!! n y n y > ) 53) y = )

21 9. 9 y = cosh x ) ρ A B y = yx) 6) B A T x+ x) -T x) mg 0,-) x x+ x 6: y = yx) x, yx)) 0 T x) T x) y = yx), y x)) αx) T x) = αx), y x)) 54) x x + x x + x T x + x)

22 9. x T x) mg0, ) m ) 3 T x + x) T x) + ρg l0, ) = 0 55) l ρ m = ρ l 55) x x 0 l x x) + y) x x 0 l x + y x)) 54) = + ) y x T x + x) T x)) x = x αx + x) αx), αx + x)y x + x) αx)y x)) αx + x) αx) =, αx + x)y x + x) αx)y ) x) x x α x), αx)y x)) ) = α x), α x)y x) + αx)y x)) α x), α x)y x) + αx)y x)) + ρg + y x)) 0, ) = 0, 0) 56) 56) x α x) = 0

23 9. 3 αx) α 0 > 0) 56) y α 0 y x) ρg + y x)) = 0 y = yx) y = k + y ) k = ρg ) > 0 α 0 57) 57) C, C ) y = k coshkx + C ) + C 58) 57) 58) 58) 57) ) 58) y = sinhkx + C ), y = k coshkx + C ) 5) + y ) = + {sinhkx + C )} = coshkx + C ) y 57) 58) y = cosh x x y /k y = cosh x y = Y Y 3)

24 [] 999) [] I 99)

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