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Undergraduate practice chapters

1. Undergraduate · Chapter 401

   Newton root iteration · Number whose root is sought A=4; Positive starting guess b=3.5

   Perform exactly n=4 updates. Find x₄, its equation residual x₄²−A, and its absolute error relative to √A. Givens: Number whose root is sought A=4; Positive starting guess b=3.5.

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2. Undergraduate · Chapter 402

   Two-state Markov chains · Transition 1 → 2: p=0.5; Transition 2 → 1: q=0.1

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.5; Transition 2 → 1: q=0.1.

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3. Undergraduate · Chapter 403

   Orthogonal projections · Vector horizontal a=3; Vector vertical b=-2

   Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=3; Vector vertical b=-2.

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4. Undergraduate · Chapter 404

   Diagonal eigenmodes · First eigenvalue a=1.5; Second eigenvalue b=-2

   Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=1.5; Second eigenvalue b=-2.

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5. Undergraduate · Chapter 405

   Matrix conditioning · Small diagonal entry a=0.55; Final data error b=0.3

   Use the current perturbation δ=0.18, so Δb=(0,0.18). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.55; Final data error b=0.3.

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6. Undergraduate · Chapter 406

   Logistic differential equations · Capacity K=4; Growth rate r=0.2

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=4; Growth rate r=0.2.

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7. Undergraduate · Chapter 407

   Harmonic oscillators · Angular frequency ω=1.5; Initial displacement b=-1.5

   Evaluate at time t=6π/5, retaining π during calculation. Find displacement, velocity, and conserved energy (y′²+ω²y²)/2. Givens: Angular frequency ω=1.5; Initial displacement b=-1.5.

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8. Undergraduate · Chapter 408

   Euler stability · Decay rate a=0.75; Step size h=0.9

   Use n=6 updates, corresponding to time t=nh=5.4. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=0.75; Step size h=0.9.

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9. Undergraduate · Chapter 409

   Newton root iteration · Number whose root is sought A=5; Positive starting guess b=4

   Perform exactly n=4 updates. Find x₄, its equation residual x₄²−A, and its absolute error relative to √A. Givens: Number whose root is sought A=5; Positive starting guess b=4.

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10. Undergraduate · Chapter 410

    Orthogonal projections · Vector horizontal a=-3; Vector vertical b=-1

    Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=-3; Vector vertical b=-1.

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11. Undergraduate · Chapter 411

    Diagonal eigenmodes · First eigenvalue a=2; Second eigenvalue b=-1.5

    Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=2; Second eigenvalue b=-1.5.

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12. Undergraduate · Chapter 412

    Matrix conditioning · Small diagonal entry a=0.6; Final data error b=0.4

    Use the current perturbation δ=0.24, so Δb=(0,0.24). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.6; Final data error b=0.4.

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13. Undergraduate · Chapter 413

    Logistic differential equations · Capacity K=4.5; Growth rate r=0.4

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=4.5; Growth rate r=0.4.

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14. Undergraduate · Chapter 414

    Harmonic oscillators · Angular frequency ω=2; Initial displacement b=-1

    Evaluate at time t=6π/5, retaining π during calculation. Find displacement, velocity, and conserved energy (y′²+ω²y²)/2. Givens: Angular frequency ω=2; Initial displacement b=-1.

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15. Undergraduate · Chapter 415

    Newton root iteration · Number whose root is sought A=6; Positive starting guess b=4.5

    Perform exactly n=4 updates. Find x₄, its equation residual x₄²−A, and its absolute error relative to √A. Givens: Number whose root is sought A=6; Positive starting guess b=4.5.

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16. Undergraduate · Chapter 416

    Two-state Markov chains · Transition 1 → 2: p=0.7; Transition 2 → 1: q=0.3

    Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.7; Transition 2 → 1: q=0.3.

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17. Undergraduate · Chapter 417

    Orthogonal projections · Vector horizontal a=-2.5; Vector vertical b=-0.5

    Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=-2.5; Vector vertical b=-0.5.

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18. Undergraduate · Chapter 418

    Diagonal eigenmodes · First eigenvalue a=2.5; Second eigenvalue b=-1

    Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=2.5; Second eigenvalue b=-1.

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19. Undergraduate · Chapter 419

    Matrix conditioning · Small diagonal entry a=0.65; Final data error b=0.5

    Use the current perturbation δ=0.3, so Δb=(0,0.3). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.65; Final data error b=0.5.

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20. Undergraduate · Chapter 420

    Logistic differential equations · Capacity K=5; Growth rate r=0.6

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=5; Growth rate r=0.6.

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