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Examples of Exam Problem Solving

Lecture



1. Obtain a solution to the equation f ( x) = x 3 + x 2 -9x + 9 = 0 by dividing the segment in half with an accuracy of 0.05. Isolation interval (-4, -3.8)

We verify that this segment is an isolation interval: f (-4) = - 3, f (-3.8) = 2.768.

Examples of Exam Problem Solving

Consequently,

Examples of Exam Problem Solving

This segment is an isolation interval.

Calculations

k

a

b

c

f (c)

f (a)

ba

0

-four

-3.8

-3.9

-3

-0.009

0.2

one

-3.9

-3.8

-3.85

-0.009

1.405875

0.1

2

-3.9

-3.85

-3.875

-0.009

0.705078

0.05

3

-3.9

-3.875

-3.8875

-0.009

0.349705

0.025

Calculation formulas:

Examples of Exam Problem Solving

Examples of Exam Problem Solving

Examples of Exam Problem Solving

Answer: x = -3.8875

2. Obtain a solution to the equation f ( x) = x 3 + x 2 -9x + 9 = 0 using a simple iteration method with an accuracy of 0.001. Isolation interval (-5, -3).

Similarly, we prove that the interval is an isolation interval.

f (-5) = - 46, f (-3) = 18

Examples of Exam Problem Solving

Calculation formulas:

Examples of Exam Problem Solving

Examples of Exam Problem Solving

Examples of Exam Problem Solving

Examples of Exam Problem Solving

k

x

f (x)

accuracy

0

-four

-3

one

-3.91

-0.29837

0.09

2

-3.90105

-0.03925

0.008951

3

-3.89987

-0.00529

0.001178

four

-3.89971

-0.00072

0.000159

Answer: X = -3.8997

3. Obtain a solution to the equation f ( x) = x 3 + x 2 -9x + 9 = 0 by the Newton method with an accuracy of 0.001. Isolation interval (-5, -3).

Calculation formulas:

Examples of Exam Problem Solving

f (-5) = -46, f // (-5) = 6 * (- 5) + 2 = -28

Consequently, Examples of Exam Problem Solving

k

x

f (x)

f '(x)

accuracy

0

-five

-46

56

one

-4.17857

-8.89217

35.02423

0.821429

2

-3.92469

-0.72721

29.36009

0.253886

3

-3.89992

-0.00659

28.82821

0.024769

four

-3.89969

-5.6E-07

28.82332

0.000229

Answer: x = -3.89969

4. Solve a system of linear equations by simple iteration with an accuracy of 0.05:

Examples of Exam Problem Solving

Check the condition of diagonal dominance:

Examples of Exam Problem Solving

Solve the system of equations for x i

Examples of Exam Problem Solving

Examples of Exam Problem Solving

k

x1

x2

x3

accuracy

0

0

0

0

one

-0.3333333

-0.125

-one

one

2

-0.0138889

-0.33333

-1.07639

0.319444

3

-0.0115741

-0.26302

-1.05787

0.070313

four

-0.0099344

-0.26013

-1.04577

0.012105

5. Solve the system of linear equations by the Gauss – Seidel method with an accuracy of 0.05:

Examples of Exam Problem Solving

Similarly, we check the condition of diagonal dominance.

Solve the system of equations for x i

Examples of Exam Problem Solving

Examples of Exam Problem Solving

k

x1

x2

x3

accuracy

0

0

0

0

one

-0.3333333

-0.20833

-1.09028

1.090278

2

0.0069444

-0.25955

-1.0421

0.340278

3

-0.0148052

-0.25896

-1.04563

0.02175

6. For the table given function:

x -2 one 1.5 2
f 0.1 -0.2 0.5 1.2

calculate the value of the function at the point z = 1.2, using linear interpolation formulas.

We define the interval to which z belongs: [1,1.5].

Calculation formulas:

Examples of Exam Problem Solving

f (z) = - 0.2 + (0.5 + 0.2) / (1.5-1) * (1.2-1) = 0.2375.

7. For a table-specific function:

x -2 one 1.5 2
f 0.1 -0.2 0.5 1.2

write the basic polynomials and calculate the value of the Lagrange polynomial at the point z = 1.2.

n = 3.

Examples of Exam Problem Solving

Examples of Exam Problem Solving

8. Calculate the integral by the trapezoid method for the function given in the table:

X

-one

-0.5

0

one

f

2

3

four

4.5

In this task, x changes with a constant step of 0.5

Formula trapezoid:

Examples of Exam Problem Solving

If the step is not constant, for example:

X

-one

-0.6

0

0.8

f

2

3

four

4.5

then it is necessary to use the general trapezium formula:

Examples of Exam Problem Solving

Examples of Exam Problem Solving

Similarly for left and right rectangle formulas:

Examples of Exam Problem Solving

9. The method of least squares .

Examples of Exam Problem Solving

Z

Z 1

Z 2

...

Z n

y

Y 1

Y 2

...

Y n

Examples of Exam Problem Solving

created: 2014-10-03
updated: 2021-05-01
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Numerical methods

Terms: Numerical methods