Việc sở hữu đề thi Phương pháp tính sẽ giúp bạn nắm vững cấu trúc và nội dung thi, từ đó dễ dàng đạt điểm cao trong kì thi cuối kì ở Đại học Bách Khoa. Đề thi thường phản ánh chính xác những kiến thức trọng tâm mà giảng viên muốn nhấn mạnh, giúp bạn tập trung ôn luyện những phần quan trọng. Hơn nữa, khi làm quen với đề thi, bạn sẽ biết được các dạng bài tập thường xuất hiện và cách giải quyết chúng một cách hiệu quả. Ngoài ra, việc luyện tập với đề thi còn giúp bạn cải thiện kỹ năng quản lý thời gian, tránh được tình trạng lúng túng khi làm bài. Vì vậy, hãy mua và sử dụng đề thi Phương pháp tính một cách thông minh để tối ưu hóa kết quả học tập của mình.
Trang 1Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCM
Faculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
Subjectcode
Note:
- ONLY Paper materials and pocket calculators are used
- There are two parts: multiple choice (8 points) and essay (2 points)
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point)
Full name: Proctor 1:
ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θ
dt2 + g
Lsin θ = 0where g ≈ 9.8(m/s2) is the gravity constant, L is the pendulum length, and θ is the angle between thependulum and the vertical axis Given the step size h = 0.1 (second)
1 Given that θ(0) = π/6, θ′(0) = 0 and L = 1.2(m) Compute θ at t = 0.2 (sec) Using the modifiedEuler method
Trang 2Questions from 6 to 7.
Given the system of ordinary differential equations
(dx
dt = t + x + y + 3.4dy
dt = ty + 3.4where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1
6 Using the Euler method, approximate the solution at t = 1.3
A (2.6519,2.3696) B (2.8809,2.489) C (3.2335,2.6452) D (2.6045,2.3918) E (2.4441,1.8272)
7 Using the modified Euler method, approximate the solution at t = 1.1
A (1.0915, 0.914) B (1.3495,1.59) C (2.0327,1.1234) D (1.5643,1.633) E (1.8381,1.3964)
Questions from 8 to 10
Let the function f be defined on [0.4, 2.4], and I =
Z 2.4 0.4
f (x)dx
The graph of y = f (x) passes through Pk(xk, yk), where xk(k = 0, 1, 2, 3, 4) are equally spaced Supposethat x0 = 0.4, x4 = 2.4 Given that f1 be the polynomial interpolation of f at nodes x0, x1 and x2,meanwhile f2 be the polynomial interpolation of f at nodes x2, x3 and x4 The figures below show theareas of 6 regions
y P0
0.575 0.475
O
x y
x0 x1 x2 x3 x4
y = f1(x)
0.8
y = f2(x) 0.57
8 Let I1∗ be the approximate result of I by using the composite trapezoidal formula with 4 subintervals.Then the value of I1∗ is:
Questions 11 to 14 Given the equation x3− 9x − 2 = 0
11 Which interval below has only one real root of the equation?
A None of others B (1,1.9) C (1.9,2.4) D (-0.3,0) E (-0.7,-0.5)
12 Using the bisection method and the inteval (−1, 0), find the approximate solution x5
A None of others B -0.2343 C -0.2345 D -0.2342 E -0.2344
Trang 313 Using the Newton method, compute the solution x1 with the initial guest x0 = 0.
x(0)
y(0)
= 03
15 By the Jacobi method, which vector sequence below converges to the exact solution as n → ∞?
−0.5x(n−1)+ 0.5
B None of others C x(n)
y(n)
= 0.0336y(n−1)+ 10.8322x(n−1)+ 0.5
D x(n)
y(n)
=0.8322y(n−1)+ 30.0157x(n−1)0.5
E x(n)
y(n)
= 0.0336y(n−1)+ 10.0157x(n−1)+ 0.5
16 Using the Jacobi method, compute x(3) (the first coordinate of the approximate solution after threeiterations)
20 Given the boundary value problem y′′(x) − y(x) = ex, where the boundary condition is y(0) =
M, y(1) = 2 With the step size h = 0.25, solve this problem by the finite difference method
Trang 4Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCM
Faculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
Subjectcode
Note:
- ONLY Paper materials and pocket calculators are used
- There are two parts: multiple choice (8 points) and essay (2 points)
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point)
Full name: Proctor 1:
ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θ
dt2 + g
Lsin θ = 0where g ≈ 9.8(m/s2) is the gravity constant, L is the pendulum length, and θ is the angle between thependulum and the vertical axis Given the step size h = 0.1 (second)
1 Given that θ(0) = π/6, θ′(0) = 0 and L = 0.9(m) Compute θ at t = 0.2 (sec) Using the modifiedEuler method
Trang 5Questions from 6 to 7.
Given the system of ordinary differential equations
(dx
dt = t + x + y + 1.7dy
dt = ty + 1.7where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1
6 Using the Euler method, approximate the solution at t = 1.3
f (x)dx
The graph of y = f (x) passes through Pk(xk, yk), where xk(k = 0, 1, 2, 3, 4) are equally spaced Suppose
that x0 = 0.7, x4 = 2.7 Given that f1 be the polynomial interpolation of f at nodes x0, x1 and x2,
meanwhile f2 be the polynomial interpolation of f at nodes x2, x3 and x4 The figures below show the
O
x y
8 Let I1∗ be the approximate result of I by using the composite trapezoidal formula with 4 subintervals
Then the value of I1∗ is:
A 1.825 B None of others C 0.45625 D 1.275 E 0.9125
9 Let I2∗ be the approximate result of I by using the composite Simpson formula with 4 subintervals
Then the value of I2∗ is:
A 0.02 B 2.616 C None of others D 2.398 E 2.18
10 Using the center difference formula for 3 nodes x0, x1 and x2, approximate f′(x1)
A -0.9 B -0.45 C -0.225 D -0.675 E None of others
Questions 11 to 14 Given the equation x3− 9x − 2 = 0
11 Which interval below has only one real root of the equation?
A (1,1.7) B (-0.6,-0.5) C (1.9,2.2) D (-0.4,0) E None of others
12 Using the bisection method and the inteval (−1, 0), find the approximate solution x5
A -0.2345 B -0.2343 C None of others D -0.2342 E -0.2344
Trang 613 Using the Newton method, compute the solution x1 with the initial guest x0 = 0.
x(0)
y(0)
= −12
15 By the Jacobi method, which vector sequence below converges to the exact solution as n → ∞?
y(n)
= 0.6575y(n−1)+ 0.750.473x(n−1)+ 0.2857
E x(n)
y(n)
= 0.2788y(n−1)+ 30.473x(n−1)0.2857
16 Using the Jacobi method, compute x(3) (the first coordinate of the approximate solution after threeiterations)
20 Given the boundary value problem y′′(x) − y(x) = ex, where the boundary condition is y(0) =
M, y(1) = 2 With the step size h = 0.25, solve this problem by the finite difference method
Trang 7Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCM
Faculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
Subjectcode
Note:
- ONLY Paper materials and pocket calculators are used
- There are two parts: multiple choice (8 points) and essay (2 points)
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point)
Full name: Proctor 1:
ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θ
dt2 + g
Lsin θ = 0where g ≈ 9.8(m/s2) is the gravity constant, L is the pendulum length, and θ is the angle between thependulum and the vertical axis Given the step size h = 0.1 (second)
1 Given that θ(0) = π/6, θ′(0) = 0 and L = 0.7(m) Compute θ at t = 0.2 (sec) Using the modifiedEuler method
Trang 8Questions from 6 to 7.
Given the system of ordinary differential equations
(dx
dt = t + x + y + 3.4dy
dt = ty + 3.4where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1
6 Using the Euler method, approximate the solution at t = 1.3
f (x)dx
The graph of y = f (x) passes through Pk(xk, yk), where xk(k = 0, 1, 2, 3, 4) are equally spaced Suppose
that x0 = 0.7, x4 = 2.7 Given that f1 be the polynomial interpolation of f at nodes x0, x1 and x2,
meanwhile f2 be the polynomial interpolation of f at nodes x2, x3 and x4 The figures below show the
0.675
O
x y
8 Let I1∗ be the approximate result of I by using the composite trapezoidal formula with 4 subintervals
Then the value of I1∗ is:
A 0.49375 B 0.9875 C None of others D 1.3625 E 1.975
9 Let I2∗ be the approximate result of I by using the composite Simpson formula with 4 subintervals
Then the value of I2∗ is:
A 2.616 B 2.398 C None of others D 2.18 E 0.48
10 Using the center difference formula for 3 nodes x0, x1 and x2, approximate f′(x1)
A -0.45 B None of others C -0.9 D -0.675 E -0.225
Questions 11 to 14 Given the equation x3− 7x − 2 = 0
11 Which interval below has only one real root of the equation?
A (1,1.8) B None of others C (1.9,2.3) D (-0.5,0) E (-0.6,-0.5)
12 Using the bisection method and the inteval (−1, 0), find the approximate solution x5
A None of others B -0.2967 C -0.297 D -0.2968 E -0.2969
Trang 913 Using the Newton method, compute the solution x1 with the initial guest x0 = 0.
x(0)
y(0)
= 03
15 By the Jacobi method, which vector sequence below converges to the exact solution as n → ∞?
B None of others C x(n)
y(n)
=
0.5323y(n−1)+ 10.8862x(n−1)+ 0.3333
y(n)
=
0.5323y(n−1)+ 10.2828x(n−1)+ 0.3333
16 Using the Jacobi method, compute x(3) (the first coordinate of the approximate solution after threeiterations)
20 Given the boundary value problem y′′(x) − y(x) = ex, where the boundary condition is y(0) =
M, y(1) = 2 With the step size h = 0.25, solve this problem by the finite difference method
Trang 10Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCM
Faculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
Subjectcode
Note:
- ONLY Paper materials and pocket calculators are used
- There are two parts: multiple choice (8 points) and essay (2 points)
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point)
Full name: Proctor 1:
ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θ
dt2 + g
Lsin θ = 0where g ≈ 9.8(m/s2) is the gravity constant, L is the pendulum length, and θ is the angle between thependulum and the vertical axis Given the step size h = 0.1 (second)
1 Given that θ(0) = π/6, θ′(0) = 0 and L = 1.2(m) Compute θ at t = 0.2 (sec) Using the modifiedEuler method
Trang 11Questions from 6 to 7.
Given the system of ordinary differential equations
(dx
dt = t + x + y + 3.3dy
dt = ty + 3.3where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1
6 Using the Euler method, approximate the solution at t = 1.3
A (3.2276,2.724) B (2.8646,2.6766) C (2.5286,2.1188) D (3.1474,2.7308) E (2.4078,1.7936)
7 Using the modified Euler method, approximate the solution at t = 1.1
f (x)dx
The graph of y = f (x) passes through Pk(xk, yk), where xk(k = 0, 1, 2, 3, 4) are equally spaced Supposethat x0 = 0.4, x4 = 2.4 Given that f1 be the polynomial interpolation of f at nodes x0, x1 and x2,meanwhile f2 be the polynomial interpolation of f at nodes x2, x3 and x4 The figures below show theareas of 6 regions
y P0
O
x y
x0 x1 x2 x3 x4
y = f1(x)
0.9
y = f2(x) 0.87
8 Let I1∗ be the approximate result of I by using the composite trapezoidal formula with 4 subintervals.Then the value of I1∗ is:
Questions 11 to 14 Given the equation x3− 7x − 2 = 0
11 Which interval below has only one real root of the equation?
A (1,1.6) B (-0.6,-0.5) C (1.9,2.1) D None of others E (-0.4,0)
12 Using the bisection method and the inteval (−1, 0), find the approximate solution x5
A -0.2967 B None of others C -0.2969 D -0.2968 E -0.297
Trang 1213 Using the Newton method, compute the solution x1 with the initial guest x0 = 0.
−3x + 7y = 1 and the initial vector
x(0)
y(0)
= 01
15 By the Jacobi method, which vector sequence below converges to the exact solution as n → ∞?
B x(n)
y(n)
=
0.5767y(n−1)+ 0.250.6827x(n−1)+ 0.1429
C None of
E x(n)
y(n)
= −0.625y(n−1)+ 0.250.4286x(n−1)+ 0.1429
16 Using the Jacobi method, compute x(3) (the first coordinate of the approximate solution after threeiterations)
20 Given the boundary value problem y′′(x) − y(x) = ex, where the boundary condition is y(0) =
M, y(1) = 2 With the step size h = 0.25, solve this problem by the finite difference method
Trang 13Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCM
Faculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
Subjectcode
Note:
- ONLY Paper materials and pocket calculators are used
- There are two parts: multiple choice (8 points) and essay (2 points)
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point)
Full name: Proctor 1:
ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θ
dt2 + g
Lsin θ = 0where g ≈ 9.8(m/s2) is the gravity constant, L is the pendulum length, and θ is the angle between thependulum and the vertical axis Given the step size h = 0.1 (second)
1 Given that θ(0) = π/6, θ′(0) = 0 and L = 0.7(m) Compute θ at t = 0.2 (sec) Using the modifiedEuler method
A 1.1427 B 1.2241 C 0.3857 D 0.6573 E 0.8411
2 Which statement below is true about the rate change of θ at t = 0.2 (Using the modified Eulermethod and the given data in question 1)
A It is decreasing with a rate at 1.8517 (m/s) B None of others C It is decreasing with
a rate at 1.3131 (m/s) D It is decreasing with a rate at 2.0837 (m/s) E It is decreasingwith a rate at2.0441 (m/s)
Trang 14Questions from 6 to 7.
Given the system of ordinary differential equations
(dx
dt = t + x + y + 2.4dy
dt = ty + 2.4where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1
6 Using the Euler method, approximate the solution at t = 1.3
f (x)dx
The graph of y = f (x) passes through Pk(xk, yk), where xk(k = 0, 1, 2, 3, 4) are equally spaced Supposethat x0 = 0.5, x4 = 2.5 Given that f1 be the polynomial interpolation of f at nodes x0, x1 and x2,meanwhile f2 be the polynomial interpolation of f at nodes x2, x3 and x4 The figures below show theareas of 6 regions
y P0
0.625
O
x y
Questions 11 to 14 Given the equation x3− 8x − 2 = 0
11 Which interval below has only one real root of the equation?
A (-0.8,-0.5) B (-0.4,0) C None of others D (1,2) E (1.9,2.5)
12 Using the bisection method and the inteval (−1, 0), find the approximate solution x5
A -0.2655 B None of others C -0.2654 D -0.2657 E -0.2656
Trang 1513 Using the Newton method, compute the solution x1 with the initial guest x0 = 0.
−3
15 By the Jacobi method, which vector sequence below converges to the exact solution as n → ∞?
B None of others C x(n)
y(n)
=
0.606y(n−1)+ 30.9962x(n−1)0.2857
E x(n)
16 Using the Jacobi method, compute x(3) (the first coordinate of the approximate solution after threeiterations)
20 Given the boundary value problem y′′(x) − y(x) = ex, where the boundary condition is y(0) =
M, y(1) = 2 With the step size h = 0.25, solve this problem by the finite difference method
Trang 16Lecturer: Date Approved by: Date
HCM City University of Technology - VNUHCM
Faculty of Applied Science
Final test Semester/YearDate 125/12/20232023-2024Subject Numerical methods
Subjectcode
Note:
- ONLY Paper materials and pocket calculators are used
- There are two parts: multiple choice (8 points) and essay (2 points)
- The final results are rounded to 4 decimal places Each incorrect answer will get a penalty of 0.05 (point)
Full name: Proctor 1:
ID Student: Proctor 2: Questions 1 to 2 Under ideal conditions, the motion of a simple pendulum is described by a differentialequation of second order:
d2θ
dt2 + g
Lsin θ = 0where g ≈ 9.8(m/s2) is the gravity constant, L is the pendulum length, and θ is the angle between thependulum and the vertical axis Given the step size h = 0.1 (second)
1 Given that θ(0) = π/6, θ′(0) = 0 and L = 0.6(m) Compute θ at t = 0.2 (sec) Using the modifiedEuler method
A 1.0978 B 0.3632 C 0.8832 D 1.1234 E 1.2818
2 Which statement below is true about the rate change of θ at t = 0.2 (Using the modified Eulermethod and the given data in question 1)
A It is decreasing with a rate at 2.3008 (m/s) B It is decreasing with a rate at 1.8834 (m/s)
C None of others D It is decreasing with a rate at1.6202 (m/s) E It is decreasing with arate at 1.5146 (m/s)
Trang 17Questions from 6 to 7.
Given the system of ordinary differential equations
(dx
dt = t + x + y + 1.3dy
dt = ty + 1.3where x(1) = 0.5, y(1) = 0.5 The step size is h = 0.1
6 Using the Euler method, approximate the solution at t = 1.3
f (x)dx
The graph of y = f (x) passes through Pk(xk, yk), where xk(k = 0, 1, 2, 3, 4) are equally spaced Suppose
that x0 = 0.4, x4 = 2.4 Given that f1 be the polynomial interpolation of f at nodes x0, x1 and x2,
meanwhile f2 be the polynomial interpolation of f at nodes x2, x3 and x4 The figures below show the
areas of 6 regions
y P0
O
x y
x0 x1 x2 x3 x4
y = f1(x)
0.83
y = f2(x) 0.78
8 Let I1∗ be the approximate result of I by using the composite trapezoidal formula with 4 subintervals
Then the value of I1∗ is:
A None of others B 1.0125 C 1.4625 D 0.50625 E 2.025
9 Let I2∗ be the approximate result of I by using the composite Simpson formula with 4 subintervals
Then the value of I2∗ is:
A 0.05 B 1.771 C 1.61 D None of others E 1.932
10 Using the center difference formula for 3 nodes x0, x1 and x2, approximate f′(x1)
A -0.2 B -0.8 C -0.4 D -0.6 E None of others
Questions 11 to 14 Given the equation x3− 10x − 2 = 0
11 Which interval below has only one real root of the equation?
A (-0.5,0) B (-0.8,-0.5) C None of others D (1.9,2.1) E (1,1.6)
12 Using the bisection method and the inteval (−1, 0), find the approximate solution x5
A -0.2029 B -0.2031 C None of others D -0.203 E -0.2032