Bài tập lớn môn giải tích 1

xCOMMAND WINDOWS Câu điểm Dạng 1: Tính giới hạn 9n lim x →∞ n! syms n limit((9^n)/factorial(n),inf) x→∞ n + ( −1) n syms n limit(1/(n + (-1)^n), inf) lim n + − n3 + syms n limit(sqrt(n^2+1)-(n^3+1)^(1/3), inf) 2n + 3n lim n n x→∞ − syms n limit((2^n + 3^n)/(2^n - 3^n), inf) 2n3 + 3n − ln n lim x→∞ 3ln 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 − syms x limit((sqrt(x) + sqrt(x-1) - 1)/sqrt(x^2 - 1), 1) − 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 syms x a limit((tan(pi*x/(2*a)))*sin((x-a)/2), x, a) lim x→0 log a x ( + x ) x x +1  x−3 lim  ÷ x→∞ x +   syms x limit(((x - 3)/(x + 2))^(2*x + 1), inf)  1x  lim x  a − 1÷ x→∞   lim ( + 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') x x→±0 tan ( x − π ) lim π π x→ ± 2x − x  1x  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 − 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 ( − x ) − syms x limit((exp(x) + log(1 - x) - 1)/(asin(x) - sin(x))) lim e x + ln ( − sinx ) − x →0 x →0 arcsin x − sinx − x4 − (1+ x) lim x→0 x −e sin x + x + x cos x − + x lim x →0 ln(1 + x) − x lim ( coslnx ) x→0 1−cosx  2x2 +  lim  ÷x x→∞ x −   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  − ÷ 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 a Diff(f(x), n): Tính đạo hàm cấp n f(x) f ( x) = sin x − cos x f ( x) = , f '''(0) sin x + cos x ( )   x −1  + 1÷, f '' ( 1)  x  π 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) x f ( x ) = e cos , f '(0) syms x subs(diff(exp(pi/3)*(cos(x/3)) ^2), 0) π x  f ( x ) = ln tan  + ÷, f "(0)  2 syms x subs(diff(log(tan(pi/4 + x/2)), 2), 0) f ( x ) = x + x + x , f '(1) f ( x ) = ( sin x ) f ( x ) = e x sin x, f '"(0) f ( x ) = x ln x, f (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) f ( x ) = 2sin x cos(sin x), f "(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 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  x = arccos  1+ t2  , y"  t  y = arcsin  1+ t2 y xx " =  x = arctan t , y"  y = ln − t , t ∈ − 1,1 ( ) ( )  ex f ( x ) = , f "( 1) x π  f ( x ) = ( x + sin x ) , f '  ÷ 4 f ( x ) = ln x + x + , f '(0) f ( x ) = ( x + 3) e − x , f " ) ( ( f ( x) = ∫ e −t 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 x0 = x et f ( x ) = ∫ dt , x0 = ln t 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 f theo x Int(f(x), a, b): Tích phân f(x) từ a tới b cos ∫ xdx + x − ) dx syms x int((cos(x))^2) ∫( x ∫ arctan xdx syms x int(atan(x)) ∫x e syms x int((x^2)*exp(-x)) ln x ∫ x dx syms x int(log(x)/x) syms x int(x*log(x), 1, 2) 2 −x dx ∫ x ln xdx syms x int(x^2 + x - 2) ∫ x arctan xdx syms x int(x*atan(x), 0, 1) +∞ ∫ xe −x x syms x int(x*exp(-x), 0, inf) dx ∫ 1− x dx syms x int(x/sqrt(1 - x^2), 0, 1) dx ∫0 a + x syms x a int(1/(x^2 + a^2), 0, a) 1 +∞ syms x int(1/(x^2 + a^2), 0, inf) a ∫ +∞ dx a2 + x2 ∫e − x2 dx syms x int(exp(-x^2), 0, inf) π ∫ sin x dx x syms x int(sin(x)/x, 0, pi/2) +∞ ∫ dx x3 + x + ∫ 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) ∆ABC , A(1,1), B (2,3), C ( −1, 2) X = [1 -1], Y = [1 2] −1 ≤ x ≤ 2,0 ≤ y ≤ e x y = cos x, y = 0,0 ≤ x ≤ 2π  x + y = x  2  x + y = y y = ln x, y = −1, x = e  x + y ≤ x  2  x + y ≤  x + y ≤ y  0 ≤ x ≤ y  y = sinh(x)   y = 0, x = fill(X, Y, 'g') % D : vùng xanh x = -1: 1/9: area(exp(x)) % D : vùng màu xanh x = 0: pi/100: 2*pi; y = cos(x); fill(x, y, 'g') % D : vùng xanh syms x y ezplot(x^2 + y^2 == 2*x) hold on ezplot(x^2 + y^2 == 2*y) % D : Giao điểm 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 hình trịn syms x y, ezplot(x^2 + y^2 == 2*y), hold on % D : Phần nằm đường thẳng bên hình trịn syms x y, ezplot(y == sinh(x)), hold on y = ,y = 4− x x y = arcsin x, x = 0, y = π Dạng : Tính diện tích miền phẳng y = sin x, y = 0,0 ≤ x ≤ 2π y = x − x, y = 0,0 ≤ x ≤ 3 y= x , y = 0,0 ≤ x ≤ +∞ x +1 y = x, x = y y = e x − 1, y = e3 x − 3, x = x + y = 1, x + y − y = y = ln ( x + ) , y = 2ln x, x = x + y = 1, x + y + y = 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 + y = 8, y = x e syms x int(abs(sqrt(4*x) - x^2/4), 0, 4) syms x A = solve(exp(x) - == 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)) syms x solve(27/(x^2 + 9) = x^2/6) int(abs(27/(x^2 + 9) - x^2/6), -3, 3) 27 x2 y= ,y = x +9 Dạng : Tính diện tích mặt cong b S x = 2π ∫ f ( x ) + ( y ' ( x ) ) dx a x3 y = , x = 0, x = syms x 2*pi*int(abs(x^3/3)*sqrt(1 + (diff(x^3/3))^2), 0, 1) y = x2 , y = x y = x, y = x + x y = x ,2 x = y 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 x2 y + =1 Giải phương trình vi phân: dsolve D2y = y” y − xy ' = y ln x y ( − y ) ( y '+ y ) = e− x , y (2) = syms y(x) So = dsolve(y x*diff(y) == y*log(x/y)) % Nghiệm y1 = x, y2 = x*exp(C7*x), C7 số syms y(x) So = dsolve((1 x)*(diff(y) + y) == y '− y cot x = sin x y "− y '+ 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 x =x0 vẽ đường cong minh họa (đánh dấu điểm đặc biệt (x0,f(x0)))  x + 1, x ≤ f ( x) =  , x0 = − ax , x >  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)) ') ... 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 -... 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) ,... −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) ∆ABC , A (1, 1), B (2,3), C ( ? ?1, 2) X = [1 -1] , Y = [1 2] ? ?1 ≤ x ≤ 2,0 ≤ y