1. 求下列函数极限（Find the limits of the following functions）
(1) limsinx
x?0x
>> syms x >> f=sin(x)/x; >> limit(f,x,0) ans = 1
(2) lim(1?)
x??1xx >> syms x
>> f=(1+1/x)^x; >> limit(f,x,inf) ans = exp(1)
(3) lim(1?)
x???axx >> syms a x >> f=(1+a/x)^x; >> limit(f,x,inf) ans = exp(a)
(4) lim(1?)
x???1xx
>> syms x >> f=(1+1/x)^x; >> limit(f,x,-inf) ans = exp(1) (5)lim1?cosx 2x?0x >> syms x
>> f=(1-cos(x))/(x^2); >> limit(f,x,0) ans = 1/2 (6) limnn???n
>> syms n >> f=n^(1/n); >> limit(f,n,inf) ans = 1
?(cosx)x (7)lim?x?0>> syms x
>> f=(cos(x^0.5))^(pi/x); >> limit(f,x,0,'right') ans = 1/exp(pi/2)
2.求下列函数的导数或偏导数( Find the derivatives of the following functions) (1)f(x)?ax2?bx?c,g(x)?ex?xsinx 2 >> syms a b c x
>> f=(a*x*x+b*x+c)^0.5;
>> g=(exp(x^2)+x*sin(x))^0.5; >> f1=diff(f,x) f1 =
(b + 2*a*x)/(2*(a*x^2 + b*x + c)^(1/2)) >> f2=diff(g,x) f2 =
(sin(x) + 2*x*exp(x^2) + x*cos(x))/(2*(exp(x^2) + x*sin(x))^(1/2)) (2)f(x)?ln(x3) >> syms x >> f=log(x^3); >> diff(f,x) ans = 3/x
?x(3)Find the 3rd derivative of f(x). f(x)?xe
2>> syms x
>> f=x*exp(-x^2); >> diff(f,x,3) ans =
(24*x^2)/exp(x^2) - 6/exp(x^2) - (8*x^4)/exp(x^2)
?23(x?2x2y2?3y?5) (4) )Find 2?x>> syms x y
>> f=x^3-2*x*x*y*y+3*y-5; >> diff(f,x,2) ans =
6*x - 4*y^2
?2(5)Find (x3?2x2y2?3y?5).
?x?y >> syms x y
>> f=x^3-2*x*x*y*y+3*y-5; >> diff(diff(f,y),x) ans = (-8)*x*y
3. 求下列函数的不定积分或定积分（Find indefinite integrals or definite integrals of the functions）
(1)(sinx?2cos3x??1?e?x)dx x>> syms x
>> f=sin(x)-2*cos(3*x)+1/x+exp(-x); >> int(f,x) ans =
log(x) - (2*sin(3*x))/3 - cos(x) - 1/exp(x)
xx (2)esin(e)dx
?
>> syms x
>> f=exp(x)*sin(exp(x)); >> int(f,x) ans =
-cos(exp(x))
(3)
?x2x?46dx
>> syms x
>> f=(x^2)/((x^6+4)^0.5); >> int(f,x)
Warning: Explicit integral could not be found. ans =
int(x^2/(x^6 + 4)^(1/2), x) (4)cos3xcos5xdx >> syms x
>> f=cos(3*x)*cos(5*x); >> int(f,x) ans =
sin(2*x)/4 + sin(8*x)/16 (5)
??x2?a2dx x>> syms x
>> f=((x^2-a^2)^0.5)/x; >> int(f,x) ans =