xCOMMAND WINDOWS
3. Câu 1 điểm
Dạng 1: Tính giới hạn
9n
lim
x →∞ n!
syms n
limit((9^n)/factorial(n),inf)
1
x→∞ n + ( −1) n
syms n
limit(1/(n + (-1)^n), inf)
lim n 2 + 1 − 3 n3 + 1
syms n
limit(sqrt(n^2+1)-(n^3+1)^(1/3),
inf)
2n + 3n
lim n n
x→∞ 2 − 3
syms n
limit((2^n + 3^n)/(2^n - 3^n), inf)
2n3 + 3n 2 − ln 9 n
lim
x→∞
3ln 7 n − n3
syms n
limit((2*n^3 + 3*n^2 - (log(n))^9)/
(3*(log(n))^7 - n^3), Inf)
x −1
lim n
x →1
x −1
syms m n x
limit((x^(1/m) - 1)/(x^(1/n) - 1), x, 1)
lim
x→∞
m
lim
x →1
x + x −1 −1
x2 − 1
syms x
limit((sqrt(x) + sqrt(x-1) - 1)/sqrt(x^2
- 1), 1)
2 − 2cos x
limπ
π − 4x
x→
syms x
limit((sqrt(2) - 2*cos(x))/(pi - 4*x),
pi/4)
πx
x−a
lim tan
sin
x →a
2a
2
syms x a
limit((tan(pi*x/(2*a)))*sin((x-a)/2), x,
a)
4
lim
x→0
log a x ( 1 + x )
x
2 x +1
x−3
lim
÷
x→∞ x + 2
syms x
limit(((x - 3)/(x + 2))^(2*x + 1), inf)
1x
lim x a − 1÷
x→∞
lim ( 2 + x )
syms x a
limit(x*(a^(1/x) - 1), Inf)
syms x
limit((2 + x)^(1/x), x, 0, 'left')
limit((2 + x)^(1/x), x, 0, 'right')
1
x
x→±0
tan ( 4 x − π )
lim
π
π
x→ ± 0
2x −
4
2
x
1x 1
lim e + ÷
x →0
x
syms x
limit(abs(tan(4*x - pi))/(2*x - pi/2), x,
pi/4, 'right')
limit(abs(tan(4*x - pi))/(2*x - pi/2), x,
pi/4, 'left')
syms x
limit((exp(1/x) + 1/x)^x, x, 0)
2x − x2
lim
x →2 x − 2
syms x
limit((2^x - x^2)/(x - 2), 2)
tan(2 x) − 3arcsin(4 x)
lim
x→0 sin(5 x ) − 6arctan(7 x )
syms x
limit((tan(2*x) - 3*asin(4*x))/(sin(5*x)
- 6*atan(7*x)), 0)
lim
esinx + ln ( 1 − x ) − 1
syms x
limit((exp(x) + log(1 - x) - 1)/(asin(x)
- sin(x)))
lim
e x + ln ( 1 − sinx ) − 1
x →0
x →0
arcsin x − sinx
3
8 − x4 − 2
(1+ x)
lim
x→0
1
x
−e
sin 2 x + x
1 + x cos x − 1 + 2 x
lim
x →0
ln(1 + x) − x
lim ( coslnx )
x→0
1
1−cosx
2x2 + 3 2
lim 2
÷x
x→∞ 2 x − 1
syms x
limit((exp(x) + log(1 - sin(x)))/((8 x^4)^(1/3) - 2))
syms x
limit(((1 + x)^(1/x) - exp(1))/
((sin(x))^2 + x))
syms x
limit((1 + x*cos(x) - sqrt(1 + 2*x))/
(log(1 + x) - x))
syms x
limit((cos(log(x)))^(1/(1 - cos(x))))
syms x
limit(((2*x^2 + 3)/(2*x^2 - 1))*x^2,
inf)
tan
x
lim 2 − ÷
x→∞
a
πx
2a
syms x a
limit((2 - x/a)^tan((pi*x)/(2*a)), x, a)
Dạng 2: Tính đạo hàm
Subs(f(x), a): Tính giá trị hàm số tại a
Diff(f(x), n): Tính đạo hàm cấp n của f(x)
1
f ( x) =
2
sin x − cos x
f ( x) =
, f '''(0)
sin x + cos x
(
)
1
x −1
+ 1÷, f '' ( 1)
x
π
3
syms x
subs(diff((sqrt(x) 1)*(1/sqrt(x) + 1), 2), 1)
syms x
subs(diff((sin(x) - cos(x))/
(sin(x) + cos(x)), 3), 0)
3
2
x
f ( x ) = e cos , f '(0)
3
syms x
subs(diff(exp(pi/3)*(cos(x/3))
^2), 0)
4
π x
f ( x ) = ln tan + ÷, f "(0)
4 2
syms x
subs(diff(log(tan(pi/4 + x/2)),
2), 0)
5
f ( x ) = x + x + x , f '(1)
6
f ( x ) = ( sin x )
7
f ( x ) = e 2 x sin 3 x, f '"(0)
8
f ( x ) = x 3 ln x, f 4 (1)
syms x
subs(diff(sqrt(x + sqrt(x +
sqrt(x)))), 1)
syms x
subs(diff((sin(x))^asin(x), 2),
1)
syms x
subs(diff(exp(2*x)*sin(3*x),
3), 0)
syms x
subs(diff(x^3*log(x), 4), 1)
9
f ( x ) = 2sin x cos(sin x), f "(0)
1
0
x = t ( t cos t − 2sin t ) π
, y' t = ÷
y
=
t
t
sin
t
+
2cos
t
(
) 4
arcsinx
, f "(1)
syms x
subs(diff(2^((sin(x)))*cos(sin(
x)), 2), 0)
syms t
xt = subs(diff(t*(t*cos(t) 2*sin(t))), pi/4)
yt = subs(diff(t*(t*sin(t) +
2*cos(t))), pi/4)
1
1
res = yt/xt
syms t
xt = diff(acos(1/sqrt(1 +
t^2)))
yt = diff(asin(t/sqrt(1 +
t^2)))
resp = (diff(y1/xt))/xt
1
x
=
arccos
1+ t2
, y"
t
y = arcsin
1+ t2
y xx " =
1
2
x = arctan t
, y"
2
y
=
ln
1
−
t
,
t
∈
−
1,1
(
)
(
)
1
3
ex
f ( x ) = 2 , f "( 1)
x
1
4
π
f ( x ) = ( x + sin x ) , f ' ÷
4
1
5
f ( x ) = ln x 2 + x 4 + 1 , f '(0)
1
6
1
7
f ( x ) = ( 2 x + 3) e − x , f "
1
8
)
(
(
f ( x) = ∫ e
0
−t 2
xt '
%
syms t
xt = diff(atan(t))
yt = diff(log(1 - t^2))
res = diff(yt/xt)/xt
syms x
subs(diff(exp(x)/x^2, 2), 1)
x
x
( yx ') t '
)
+ t dt
tại
x0 = 1
x
et
f ( x ) = ∫ dt , x0 = ln 2
t
1
syms x
subs(diff((x + sin(x))^x),
pi/4)
syms x
subs(diff(log(x^2 + sqrt(x^4
+ 1))), 0)
syms x
diff((2*x + 3)*exp(-x), 2)
syms x t
subs(int(exp(-t^2) + t, 0, x),
1)
syms x t
subs(int(exp(t)/t, 1, x), log(2))
x
%
et
ei ( x ) = ∫ dt
t
−∞
Dạng 3: Tính tích phân
Int(f, x): Tích phân của f theo x
Int(f(x), a, b): Tích phân của f(x) từ a tới b
1
2
cos
∫ xdx
+ x − 2 ) dx
syms x
int((cos(x))^2)
2
∫( x
3
∫ arctan xdx
syms x
int(atan(x))
4
∫x e
syms x
int((x^2)*exp(-x))
5
ln x
∫ x dx
syms x
int(log(x)/x)
6
2
syms x
int(x*log(x), 1, 2)
2
2 −x
dx
∫ x ln xdx
syms x
int(x^2 + x - 2)
1
7
1
∫ x arctan xdx
syms x
int(x*atan(x), 0, 1)
0
8
+∞
∫ xe
−x
1
x
syms x
int(x*exp(-x), 0, inf)
dx
0
9
∫
0
1− x
2
dx
syms x
int(x/sqrt(1 - x^2), 0, 1)
1
0
dx
∫0 a 2 + x 2
syms x a
int(1/(x^2 + a^2), 0, a)
1
1
+∞
syms x
int(1/(x^2 + a^2), 0, inf)
a
∫
0
1
2
+∞
dx
a2 + x2
∫e
− x2
dx
syms x
int(exp(-x^2), 0, inf)
0
1
3
π 2
∫
0
sin x
dx
x
syms x
int(sin(x)/x, 0, pi/2)
1
4
+∞
∫
0
1
5
0
dx
x3 + x + 1
∫ xe
−x
dx
syms x
int(1/(x^3 + x + 1), 0, inf)
syms x
int(x*exp(-x), -inf, 0)
−∞
Dạng 4: Vẽ miền D (không cần thiết tô màu)
1
∆ABC , A(1,1), B (2,3), C ( −1, 2) X = [1 2 -1], Y = [1 3 2]
2
−1 ≤ x ≤ 2,0 ≤ y ≤ e x
3
y = cos x, y = 0,0 ≤ x ≤ 2π
4
x 2 + y 2 = 2 x
2
2
x + y = 2 y
5
y = ln x, y = −1, x = e
6
x 2 + y 2 ≤ 2 x
2
2
x + y ≤ 2
7
x 2 + y 2 ≤ 2 y
0 ≤ x ≤ 3 y
8
y = sinh(x)
y = 0, x = 3
fill(X, Y, 'g') % D : vùng xanh
lá cây
x = -1: 1/9: 2
area(exp(x)) % D : vùng màu
xanh
x = 0: pi/100: 2*pi;
y = cos(x);
fill(x, y, 'g') % D : vùng xanh
lá cây
syms x y
ezplot(x^2 + y^2 == 2*x)
hold on
ezplot(x^2 + y^2 == 2*y)
% D : Giao điểm 2 đường tròn
ezplot(log(x) == y), hold on
syms x y
ezplot(x^2 + y^2 == 2*x)
hold on
ezplot(x^2 + y^2 == 2)
% D: Phần chung của 2 hình
tròn
syms x y, ezplot(x^2 + y^2
== 2*y), hold on
% D : Phần nằm dưới đường
thẳng và ở bên trong hình
tròn
syms x y, ezplot(y ==
sinh(x)), hold on
9
1
0
3
y = ,y = 4− x
x
y = arcsin x, x = 0, y =
π
2
Dạng 5 : Tính diện tích miền phẳng
1
y = sin x, y = 0,0 ≤ x ≤ 2π
2
y = x 2 − 2 x, y = 0,0 ≤ x ≤ 3
3
y=
x
, y = 0,0 ≤ x ≤ +∞
x +1
3
4
y 2 = 4 x, x 2 = 4 y
5
y = e x − 1, y = e3 x − 3, x = 0
6
x 2 + y 2 = 1, x 2 + y 2 − 2 y = 1
7
y = ln ( x + 2 ) , y = 2ln x, x =
8
x 2 + y 2 = 1, x 2 + y 2 + 2 y = 1
9
syms x
int(abs(sin(x)), 0, 2*pi)
syms x
int(abs(x^2 - 2*x),0, 3)
syms x
int(abs(sqrt(x)/(x^3 +1)), 0, inf)
x 2 + y 2 = 8, y 2 = 2 x
1
e
syms x
int(abs(sqrt(4*x) - x^2/4), 0, 4)
syms x
A = solve(exp(x) - 1 == exp(3*x) 3)
isreal(A(1,1)), isreal(A(2,1)),
isreal(A(3,1))
int(abs(exp(3*x) - exp(x) - 2), 0,
A(1,1))
syms x y
A = solve(x^2 + y^2 == 1, x^2 +
y^2 - 2*y ==1)
A = [A.x A.y]
int(abs(sqrt(1 - x^2) - sqrt(2 x^2) - 1), A(1,1), A(1,2))
syms x
xo = solve(log(x + 2) = 2*log(x))
int(abs(log(x + 2) - 2*log(x)),
1/exp(1), xo)
syms x y
A = solve(x^2 + y^2 == 1, x^2 +
y^2 + 2*y ==1)
A = [A.x A.y]
int(abs(sqrt(1 - x^2) - sqrt(2 x^2) + 1), A(1,1), A(1,2))
syms x y
A = solve(x^2 + y^2 == 8, y^2
== 2*x)
abs(int(abs(y^2/2 - sqrt(8 - y^2)),
2, -2))
1
0
syms x
solve(27/(x^2 + 9) = x^2/6)
int(abs(27/(x^2 + 9) - x^2/6), -3,
3)
27
x2
y= 2
,y =
x +9
6
Dạng 6 : Tính diện tích mặt cong
b
S x = 2π ∫ f ( x ) 1 + ( y ' ( x ) ) dx
2
a
1
x3
y = , x = 0, x = 1
3
syms x
2*pi*int(abs(x^3/3)*sqrt(1 +
(diff(x^3/3))^2), 0, 1)
2
y = x2 , y = x
3
y = x, y = 5 x + x 2
4
2 y = x 2 ,2 x = y 2
syms x
solve(x^2 - x == 0)
2*pi*int(abs(x^2-x)*sqrt(1 + (diff(x^2
- x))^2), 0, 1)
syms x
solve(x == 5*x + x^2)
2*pi*int(abs(x^2 + 4*x)*sqrt(1 +
(diff(x^2 + 4*x))^2), -4, 0)
syms x
5
x2 y 2
+
=1
4
9
6
Giải phương trình vi phân: dsolve
D2y = y”
y − xy ' = y ln
x
y
( 1 − y ) ( y '+ y ) = e− x , y (2) = 1
syms y(x)
So = dsolve(y x*diff(y) == y*log(x/y))
% Nghiệm y1 = x, y2 =
x*exp(C7*x), trong đó
C7 là hằng số nào đó
syms y(x)
So = dsolve((1 x)*(diff(y) + y) ==
y '− y cot x = sin x
5 y "− 6 y '+ 5 y = xe x
exp(-x), y(2) == 1)
syms y(x)
So = dsolve(diff(y) y*cot(x) == sin(x))
syms y(x)
dsolve('5*D2y - 6*Dy +
5*y == x*exp(x)')
Tìm tham số để hàm liên tục tại x =x0 và vẽ
đường cong minh họa (đánh dấu điểm đặc biệt
(x0,f(x0)))
x + 1, x ≤ 1
f ( x) =
, x0 = 1
2
3
−
ax
,
x
>
1
syms x a
solve(limit(x+1, x, 1,
'left') == limit(3 a*x^2, x, 1, 'right'), a)
%a=1
% Vẽ đồ thị
syms x
ezplot('3 - x^2', [1,
10])
hold on
ezplot('x + 1', [-10, 1])
axis([-10, 10, -10, 10])
text(1, 2, ' \leftarrow
(x0,(f(x0)) ')