評量4.31

  1. 2x(2x)+x2(2x)ln2
  2. 4√x ln4( (1/2)x(-1/2) )
  3. 切縣斜率 m=f'(0)=e0=1，所以切線為 y=x+1
    ∵ y=ex切切線於(0,1)，且y"=ex>0凹向上
    所以圖形在切線上方，所以ex≧x+1
  4. 0.4578;3.3132
  5. f(0) + (f'(0)/2)x + (f"(0)/2!)x2 = 1+ x + (1/2)x2
  6. -1 / (z(lnz)2)
  7. e-x / (1 - e-x)
  8. y / [ x(1+3y3) ]

評量4.32

  1. 2/3
  2. 1-x2+x4-x6+x8-…
  3. ∵ (tan-1x)' = 1/(1+x2) = 1-x2+x4-x6+x8-…
    ∴∫(tan-1x)'dx = ∫(1/(1+x2)dx =∫(1-x2+x4-x6+x8-…)dx
    ∴ tan-1x = x - x3/3 + x5/5 - x7/7 + …
  4. 1
  5. -1 / (x+1)
  6. 1 + x/2 - x2/8 + x3/16
  7. 1 - x2/2

評量4.33

  1. 第一部分:
    1/2n + 1/(2n+1) + 1/(2n+2) + … + 1/(2n+1-1) > 1/2n+1 + 1/2n+1 + 1/2n+1 + … + 1/2n+1
    = 2n/2n+1 = 1/2
    第二部分:
    1 + 1/2 + 1/3 + 1/4 + … = 1 + (1/2+1/3) + (1/4 + … + 1/7) + (1/8 + … + 1/15) + … >
    1 + 1/2 + 1/2 + 1/2 + … → ∞ 所以發散。
  2. b0 = anx0n + an-1x0n-1 + … + a1x0 + a0
  3. a = f"(x0) ; b = 0 ; c = f(4)(x0)/12 ; d = 0
  4. 用Maple執行 : evalf((-1)^(1/3), 20);
    得到: .50000000000000000001 + .86602540378443864676I
    把答案自乘三遍檢驗之。
  5. 1 - e-1
  6. (e1 - e-1)/2
  7. 用Maple執行 : q := convert(series(sqrt(1+x^2), x=1, 8), polynom);
    或 : P := series(sqrt(1+x^2), x, 8);
           p := convert(P, polynom);

評量4.34

  1. = 1 + (-x) + x2 + (-x)3 + x4
                      = 1 - (1/3)x - (1/9)x2 - (5/81)x3 - (10/243)x4
  2. = 4(1/2) + x4(-1/2) + x24(-3/2) + x34(-5/2) + x44(-7/2)
                      =     2 + (1/4)x - (1/64)x2 + (1/512)x3 - (5/16384)x4
  3. = (1/2)(1/2 - 1)(1/2 - 2)(1/2 - 3)…(1/2 - k +1) / k! = [(-1)(k+1)(2k-1)!!] / [2kk!(2k-1)]
    (要整理到最後才可以!!)
  4. -1/6
  5. 1
  6. x + x2
  7. 1 + x + (1/2)x2
  8. eix- = ei(n+1)x - 1
    ∴ =
  9. -24

評量4.35

  1. 2/3
  2. f"(x)
  3. 1
  4. 0
  5. 不存在
  6. ea
  7. 不存在
  8. 0