✔ 最佳答案
1.Limit(x->0)[√(5x+4) - 2]'/x'= Limit(x->0)[5/2√(5x+4)] / 1= 5/2√4= 5/4= (D)
2.Limit(x->0)(x^2-x-sin(x))'/(2x)'= [2x-1-cos(x)]/2= (-1-1)/2= -1= (B)
3.y = limit(x->oo)[(x^2-3) / (2x-4)]= limit(x->oo)[(x/2 + 1) + 1/(2x-4)]= x/2 + 1 + 1/oo= x/2 + 1 + 0=> y = x/2 + 1 = (C)
4. w'(3) = -1/2√(4-z)= -1/2√1= -1/2= (w-2)/(z-3)=> w = (z+7)/2 = (C)
5.y' = sec(x)*tan(x)= (sec(x))'*tan(x) + sec(x)*(tan(x))'= sec(x)*tan(x)^2 + sec(x)^3= (D)
6.y^2 + x^2 = y^4 - 2x2yy' + 2x = 4y^3*y' - 2y'(-2) = (x+1)/y(2y^2-1)= (-2+1)/1*(2*1-1)= -1= (B)
7. y(x) = x^3 - 3x^2 - 1y(3) = -1y^(-1) = x,y互換x = y^3 - 3y^2 - 11 = 3y^2*y' - 6yy'y'(3)= 1/3y(y-2)= 1/9*(3-2)= 1/9= (C)
8.y = x^(x+1)ln(y) = (x+1)*ln(x)y'/y = ln(x) + (x+1)*1/xy ' =y[ln(x) + (x+1)*1/x]= x^(x+1)*[ln(x) + (1+1/x)]= (B)
9.∫(0~π/2)sin(t)*cos(t)*dt= ∫sin(t)*d(sin(t))= (sin(t))^2 / 2= (sin90)^2 / 2 - (sin0)^2 / 2= 1/2= (C)
10.Σ(k=1~n)(n + 2k)= Σn + Σ2k= n^2 + 2n(n+1)/2= n^2 + n^2 + n= 2n^2 + n= (A)
2015-07-02 07:32:24 補充：
11.y = ∫(5~x)3t*sin(t)dt, dy(-pi/2)/dx = ?
dy/dx = d/dx{∫(5~x)3tsin(t)dt}
= 3x*sin(x) - d常數/dx
= -3*pi/2 * sin(-pi/2)
= 3*pi/2
= (C)
12.∫(0~4)√(2x+1)dx
= ∫√(2x+1)*d(2x+1)/2
= 1/2*∫√y*dy
= 1/2*2/3*y^1.5
= 1/3 * (2x+1)^1.5
= 1/3 * (27-1)
= 26/3
= (A)
2015-07-02 07:40:56 補充：
13.y = ∫(x+4)dx/(x^2+5x-6)
=∫(x+4)dx/(x+6)(x-1)
=∫(a/(x+6)+b/(x-1))dx
x+4 = a(x-1) + b(x+6)
=> a = 2/7, b = 5/7
=> y =∫2dx/7(x+6) + ∫5dx/7(x-1)
= 2/7 * ln|x+6| + 5/7 * ln|x-1|
= (A)
2015-07-02 07:42:36 補充：
14.∫(0~60)tan(x)*dx
= ∫sin(x)dx/cos(x)
= -∫d(cos(x))/cos(x)
= -ln(cos(x))
= -[ln(cos60) - ln(cos0)]
= -(ln(1/2) - ln(1))
= ln(2)
= (B)
2015-07-02 07:49:55 補充：
15.∫(0~180)sin(x/2)^3*dx
= -∫sin(x/2)^2*d(cos(x/2))
= ∫(cos(x/2)^2 - 1)d(cos(x/2))
= ∫(y^2 - 1)dy
= y^3/3 - y
= cos(x/2)^3 / 3 - cos(x/2)
= cos(90)^3/3 - cos(90) - cos(0)^3/3 + cos(0)
= 0 - 0 - 1/3 + 1
= 2/3
= (A)
16.∫(-1~1)|x-1|dx
= |x^2/2 - x|
= |1/2 * (1-1) - (1+1)|
= 2
= (D)
2015-07-02 08:02:30 補充：
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