P165
Q1:用MATLAB软件求下列数列极限：
(1)lim(?2)?3n(?2)n?1nn???3n?1
>> syms n
>> limit(((-2)^n+3*n)/((-2)^(n+1)+3^(n+1)),n,inf) ans = 0
(4)lim(n?2?2n?1?n??n)
>> syms n
>> limit(sqrt(n+2)-2*sqrt(n+1)+sqrt(n),n,inf) ans = 0
Q2:用MATLAB软件求下列函数极限：
3(1)lim1?x?1xx?0
>> syms x
>> limit((((1+x)^(1/3)-1)/x),x,0) ans = 1/3
1??3x?x2??x?x??e?(4)xlim???2????6x?1?
?>> syms x
>> limit(((x^3-x^2+x/2).*exp(1/x)-sqrt(x^6+1)),x,inf) ans = 1/6
Q3:求下列函数的导数。
(1)y?x?x?x
>> syms x
>> f=sqrt(x+sqrt(x+sqrt(x))); >> diff(f,x) ans =
1/2/(x+(x+x^(1/2))^(1/2))^(1/2)*(1+1/2/(x+x^(1/2))^(1/2)*(1+1/2/x^(1/2))) (2)y?x?2?3?x?4?x?1?5
>> syms x
>> f=sqrt(x+2)*(3-x)^4/(x+1)^5; >> diff(f,x) ans =
1/2/(x+2)^(1/2)*(3-x)^4/(1+x)^5-4*(x+2)^(1/2)*(3-x)^3/(1+x)^5-5*(x+2)^(1/2)*(3-x)^4/(1+x)^6 (3)y?1?sinx1?cosx
>> syms x
>> f=(1+sin(x))/(1+cos(x)); >> diff(f,x) ans =
cos(x)/(1+cos(x))+(1+sin(x))/(1+cos(x))^2*sin(x)
(4)y?xcos2xcos3x >> syms x
>> f=x.*cos(2*x).*cos(3*x); >> diff(f,x) ans =
cos(2*x)*cos(3*x)-2*x*sin(2*x)*cos(3*x)-3*x*cos(2*x)*sin(3*x)
Q4:求高阶导数。
（1）已知y?xsinbx，求y(3)。 （2）求y?x4cos7x的40阶导数。 （3）已知y?(1) >> syms x b >> f=x*sin(b*x); >> diff(f,x,3) ans =
-3*sin(b*x)*b^2-x*cos(b*x)*b^3 (2)
>> x=sym('x') x = x
>> diff(x^4*cos(7*x),40)
ans =
5816200368074729538861238724268369360*cos(7*x)-4401448927191687218597694169716603840*x*sin(7*x)-1216189835145071468296731283737482640*x^2*cos(7*x)+145526988820777782531232803182262880*x^3*sin(7*x)+6366805760909027985741435139224001*x^4*cos(7*x)
(3)
>> x=sym('x');
>> diff(sqrt(x*sin(sqrt(3^(exp(x)-log(x))))),3) ans =
3/8/(x*sin((3^(exp(x)-log(x)))^(1/2)))^(5/2)*(sin((3^(exp(x)-log(x)))^(1/2))+1/2*x*cos((3^(exp(x)-log(x)))^(1/2))*(3^(exp(x)-log(x)))^(1/2)*(exp(x)-1/x)*log(3))^3-3/4/(x*sin((3^(exp(x)-log(x)))^(1/2)))^(3/2)*(sin((3^(exp(x)-log(x)))^(1/2))+1/2*x*cos((3^(exp(x)-log(x)))^(1/2))*(3^(exp(x)-log(x)))^(1/2)*(exp(x)-1/x)*log(3))*(cos((3^(exp(x)-log(x)))^(1/2))*(3^(exp(x)-log(x)))^(1/2)*(ex
xsin3e?lnxx，求y(3)。
p(x)-1/x)*log(3)-1/4*x*sin((3^(exp(x)-log(x)))^(1/2))*3^(exp(x)-log(x))*(exp(x)-1/x)^2*log(3)^2+1/4*x*cos((3^(exp(x)-log(x)))^(1/2))*(3^(exp(x)-log(x)))^(1/2)*(exp(x)-1/x)^2*log(3)^2+1/2*x*cos((3^(exp(x)-log(x)))^(1/2))*(3^(exp(x)-log(x)))^(1/2)*(exp(x)+1/x^2)*log(3))+1/2/(x*sin((3^(exp(x)-log(x)))^(1/2)))^(1/2)*(-3/4*sin((3^(exp(x)-log(x)))^(1/2))*3^(exp(x)-log(x))*(exp(x)-1/x)^2*log(3)^2+3/4*cos((3^(exp(x)-log(x)))^(1/2))*(3^(exp(x)-log(x)))^(1/2)*(exp(x)-1/x)^2*log(3)^2+3/2*cos((3^(exp(x)-log(x)))^(1/2))*(3^(exp(x)-log(x)))^(1/2)*(exp(x)+1/x^2)*log(3)-1/8*x*cos((3^(exp(x)-log(x)))^(1/2))*(3^(exp(x)-log(x)))^(3/2)*(exp(x)-1/x)^3*log(3)^3-3/8*x*sin((3^(exp(x)-log(x)))^(1/2))*3^(exp(x)-log(x))*(exp(x)-1/x)^3*log(3)^3-3/4*x*sin((3^(exp(x)-log(x)))^(1/2))*3^(exp(x)-log(x))*(exp(x)-1/x)*log(3)^2*(exp(x)+1/x^2)+1/8*x*cos((3^(exp(x)-log(x)))^(1/2))*(3^(exp(x)-log(x)))^(1/2)*(exp(x)-1/x)^3*log(3)^3+3/4*x*cos((3^(exp(x)-log(x)))^(1/2))*(3^(exp(x)-log(x)))^(1/2)*(exp(x)-1/x)*log(3)^2*(exp(x)+1/x^2)+1/2*x*cos((3^(exp(x)-log(x)))^(1/2))*(3^(exp(x)-log(x)))^(1/2)*(exp(x)-2/x^3)*log(3))
Q8:求由方程y>> syms x y
5?2y?x?3x?0所确定的隐函数的导数
7dydx。
>> F=y^5+2*y-x-3*x^7; >> dF_dx=diff(F,x); >> dF_dy=diff(F,y); >> pretty(-dF_dx/dF_dy)
6 1 + 21 x --------- 4 5 y + 2 即dydx?1?21x465y?2。
代入x=0，即
15y?224。
Q9:求下列函数的
?z?x22，
?z?y22和
?z?x?y2：
(1)z?sin(xy)?cos(xy) >> syms x y
>> z=sin(x*y)+cos(x*y).^2; >> diff(z,x,2) ans =
-sin(x*y)*y^2+2*sin(x*y)^2*y^2-2*cos(x*y)^2*y^2
>> diff(z,y,2) ans =
-sin(x*y)*x^2+2*sin(x*y)^2*x^2-2*cos(x*y)^2*x^2
>> diff(diff(z,x),y) ans =
-sin(x*y)*x*y+cos(x*y)+2*sin(x*y)^2*x*y-2*cos(x*y)^2*x*y-2*cos(x*y)*sin(x*y) (4)z?e11?(?)xy
>> syms x y
>> z=exp(-(1/x+1/y)); >> diff(z,x,2)
ans =
-2/x^3*exp(-1/x-1/y)+1/x^4*exp(-1/x-1/y)
>> diff(z,y,2) ans =
-2/y^3*exp(-1/x-1/y)+1/y^4*exp(-1/x-1/y)
>> diff(diff(z,x),y) ans =
1/x^2/y^2*exp(-1/x-1/y)
Q12:求下列函数的极值。
2(1)f(x)?x3(x?8)
2