I & II
ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............ 1........ 3.................. 5.................... 10...................... 11 10................ 13....................... 13................... 14................. 15.................. 16 11................. 20................. 20 iii 4 35 13..................... 35................... 36................... 37................. 38 14...................... 39...................... 39............ 42............. 44 15........... 45........... 45................... 47.................... 48 16................ 51................. 51................... 53................. 55
iii alpha A α beta B β gamma Γ γ delta δ epsilon E ϵ, ε zeta Z ζ eta H η theta Θ θ, ϑ iota I ι kappa K κ lambda Λ λ mu M µ nu N ν omicron O o xi Ξ ξ pi Π π, ϖ rho P ρ, ϱ sigma Σ σ, ς tau T τ upsilon Υ υ phi Φ ϕ, φ chi X χ psi Ψ ψ omega Ω ω
1 3 Ÿ 9. a, b, a < b a b [a, b] = {x a x b : (a, b) = {x a < x < b : [a, b) = {x a x < b : (a, b] = {x a < x b : [a, + ) = {x a x : (a, + ) = {x a < x : (, b) = {x x < b : (, b] = {x x b : = (, ) D domain x D D xindependent variable explanator variable dependent variable explained variable f x = f(x) D domain { = f(x), x D f range D x a 0, a 1, a 2,, a n n f(x) = a 0 + a 1 x + a 2 x 2 + + a n x n n n P (x) Q(x) Q(x) x f(x) = P (x) Q(x)
Ÿ 9 2 a > 0 f(x) = a x exponential function R (0, + ) a e f(x) = e x e = ( 1 + 1 n 2.71828 n + n) Napier's constant π a n = (1 + 1 n )n n a n 1 2 2 2.25 4 2.44140625 8 2.565784514 16 2.637928497 32 2.676990129 64 2.697344953 128 2.70773902 256 2.712991624 512 2.715632 1) a = 1 f(x) = a x 1 2) a > 1 3) a < 1 75 14 0 x 0 x 0 x a = 1 a > 1 a < 1 f(x) = a x log f(x) = e kx, k = log a a > 0 f(x) = log a x logarithmic function = a x x = log a I & II 2
Ÿ 9 3 (0, + ) R = (, + ) a e f(x) = log e x = ln x = log x log 10 ln = log e = log common logarithm natural logarithm 1) 0 < a < 1 2) a > 1 f(x) = log a x log a x = log x log a f(x) = k log x, k = 1 log e a θ 0 30 60 90 180 360 0 π 6 π 3 π 2 π 2π sin θ 1 1 1 θ 0 cos θ 1 x tan θ = sin θ cos θ 1 x = f(x) x x = g() g f f 1 f(x) = 3x + 2 = 3x + 2 x x g() x = 2 3 g() = 2 3 I & II 3
Ÿ 9 4 f 1 (x) = x 2 3 // f(x) = e x = e x x = a x x = log a x = log e f 1 (x) = log x = ln x // Sin 1 (x) sin x sin 1 x [ 1, 1] sin x [ π/2, π/2] Sin 1 (x) cos x tan x Cos 1 (x) Tan 1 (x) = f(x) = f 1 (x) = x sin x cos x tan x Sin 1 (x) Cos 1 (x) Tan 1 (x) x = f(x) z z = g() f(x) g() z = g(f(x)) (g f)(x) z = g(f(x)) = (g f)(x) = f(x) z = g() x z z = (g f)(x) I & II 4
Ÿ 9 5 f(x) = e x, g() = sin (g f)(x) = g(f(x)) = g(e x ) = sin(e x ) (f g)(x) = f(g(x)) = f(sin x) = e sin x e sin x exp(sin x) x = f(x) explicit function F (x, ) = 0 implicit function = f(x) f(x) = F (x, ) = 0 9.1 9.1 = f(x) x a f(x) A x a f(x) A A f(x) A x a f(x) = A x = a + h x a h 0 f(a + h) = A h 0 = x 2 x 2 4 x = 2 x 2 x a f(x) x = a f(x) x = a A f(a) ε δ ε > 0 δ > 0 0 < x a < δ x f(x) R < ε f(x) x a R f(x) = f(x) = R + x a I & II 5
Ÿ 9 6 ε > 0 δ > 0 0 < a x < δ x f(x) L < ε f(x) x a L f(x) = f(x) = L x a ε > 0 δ > 0 0 < x a < δ x f(x) M < ε f(x) x a M f(x) = M (i) x = a ( ) ( ) ( ) n xn = x x = x = a n (ii) n f(x) = c 0 + c 1 x + c 2 x 2 + + c n x n f(x) = {c 0 + c 1 x + c 2 x 2 + + c n x n (iii) (iv) = c 0 + c 1 x + c 2 x 2 + + c n x n = c 0 + c 1 a + c 2 a 2 + + c n a n = f(a) x 2 + 4x 5 (x 2 + 4x 5) x 2 x 2 + 2x 3 = x 2 x 2 (x2 + 2x 3) = 22 + 4 2 5 2 2 + 2 2 3 = 7 5 x 2 + 4x 5 x 1 x 2 + 2x 3 = (x 1)(x + 5) x 1 (x 1)(x + 3) x + 5 = x 1 x + 3 = 1 + 5 1 + 3 = 3 2 9.1 x 1 (1) f(x) = x3 1 x 1 (2) g(x) = { x 2 + x + 1 x 1 1 x = 1 (1) f(x) x = 1 x 1 (x 1) f(x) = x 2 + x + 1 I & II 6
Ÿ 9 7 x 3 1 f(x) = x 1 x 1 x 1 = (x 2 + x + 1) = 3 x 1 f(x) g(x) x = 1 f(x) = 1 3 x = 1 f(x) 3 0 1 x 0 1 x (2) f(x) = x3 1 x 1 g(x) = { x 2 + x + 1 x 1 1 x = 1 x = 1 g(x) x 1 g(x) 3 // g(x) = 3, g(1) = 1 x 1 9.2 9.2 = f(x) x f(x) A x f(x) A f(x) A x + f(x) = A x + x + x ε > 0 N x > N x f(x) L < ε ε δ x + f(x) f(x) = L x + ε > 0, N such that x N = f(x) L < ε ε > 0, N such that x N = f(x) L < ε I & II 7
Ÿ 9 8 f(x) = L x (i) (ii) 1 x + x = 0 3x 2 + 2x + 1 3 + 2 1 x + 2x 2 + x 5 = x + 1 x 2 x + 2 + 1 x 5 1 x 2 x 2 3 + 0 + 0 = x + 2 + 0 0 1 x + x = 0 (iii) = 3 2 ( x 2 ( x + x x) = 2 + x x)( x 2 + x + x) x + x + x2 + x + x x 2 + x x 2 = x + x2 + x + x = x + = x + x x2 + x + x 1 1 + 1x x + 1 = 1 1 + 1 = 1 2 ( 1 + 1 x ( = 1 + x + x) 1 x = e x x) e 84 2 Napier ( e = 1 + 1 ) n n + n 9.3 x a f(x) f(x) f(x) + x a f(x) = + I & II 8
Ÿ 9 9 + x a f(x) f(x) x a + ε δ f(x) = + M > 0, δ such that 0 < x a < δ = f(x) > M f(x) = +, f(x) = + + x a f(x) f(x) x a ε δ f(x) = M > 0, δ such that 0 < x a < δ = f(x) < M f(x) =, f(x) = + (i) f(x) = 1 x 0+ + x 0 x (ii) f(x) = 1 x 0 + x2 (iii) f(x) = log x x 0+ log x (0, + ) (iv) f(x) = sin( 1 ) x 0 + 1 f(x) 1 x x a 9.1 f(x) g(x) ( ) (1) f(x) ± g(x) ( ) (2) k f(x) = k f(x) ( ) (3) f(x) g(x) f(x) = A, = f(x) ± g(x) k = f(x) g(x) f(x) f(x) (4) g(x) = B 0 g(x) g(x) = B 86 25 I & II 9
Ÿ 9 10 θ sin θ = 1 θ 0 θ 9.4 f(x) D x = a f(x) = f(a) f(x) x = a f(x) D f(x) D f(x) (a, b) a < c < b c f(x) f(x) = x c x c f(c) f(x) x = c ε δ ε > 0, δ such that x c < δ = f(x) f(c) < ε f(x) (a, b) f(x) f(x) [a, b] (a, b) f(x) = f(a) f(x) = f(b) x a x b f(x) [a, b] f(x) = f(a) f(x) = f(b) 5 x a x b x n n e x sin x (, + ) (i) f(x) = x x = 0 x (ii) f(x) = 1 x = 2 x 2 9.2 f(x) = x 2 = f(x) D a x a D a + x D = f(a + x) f(a) a x = f(x) x 0 f(a + x) f(a) = x (a + x) a f(a + x) f(a) = x I & II 10
Ÿ 9 11 a x 9.5 x 0 x = f(a + x) f(a) x 0 x f(x) x = a f f(a + x) f(a) (a) = x 0 x f (a) x = a f(x) 9.3 = f(x) x = a (1) x (2) x 2 (3) x 3 (4) k (5) 1 x 9.6 D f(x) D x f (x) x f(x) f (x),, d, df(x) ε > 0, f(a + x) f(a) δ such that x < δ = f (a) < ε x 9.2 f(x) x = a f(x) D D x = a x h f(a + h) f(a) (9.1) = f (a) + ε h h 0 f (a) ε 0 h 0 (9.1) h 0 f(a + h) f(a) = h (f (a) + ε) 0 I & II 11
Ÿ 9 12 f(x) x = a // { h 0 f(a + h) f(a) = 0 f(a + h) = f(a) h 0 9.3 f(x) g(x) D (1) d { f(x) ± g(x) = d { f(x) ± d { g(x) (2) d { k f(x) = k d { f(x) k (3) d { f(x) g(x) = d { f(x) g(x) + f(x) d { g(x) (4) d { f(x) = f (x)g(x) f(x)g (x) g(x) g 2 g(x) 0 (x) 9.4 9.3 (1) f(x) = x 3 2x 2 + 3 (2) f(x) = (x + 1)(x 2 3) (3) f(x) = x2 + 1 x 2 9.4 n (x n ) = nx n 1 n 0 n n = 0 x 0 = 1 f(x) = 1 f(a + h) f(a) (9.2) h d {x 0 n = k = 1 1 h = d { 1 = 0 d {x k = k x k 1 = 0 d {x k+1 = d { x k x 9.3 (3) = d {x k x + x k d { x n 0 = k x k 1 x + x k 1 = (k + 1)x k d {x n = nx n 1 I & II 12
Ÿ 10 13 n < 0 n n = m m d {x n = d {x m = d { 1 x m = (1) x m 1 (x m ) (x m ) 2 9.3 (4) // = mxm 1 (x m ) 2 = mx m 1 2m = mx m 1 = nx n 1 n = m d {x n = nx n 1 n 74 20 n x n n x n, d {x n = nx n 1 84 24 11.3 n Ÿ 10. 9 = f(x) f (a) 2 P(a, f(a)) P(b, f(b)) f(b) f(a) = x b a PQ b a Q P f (a) PQ P f (a) PT PT P = f(x) P Q P 0 a b x 0 a b x I & II 13