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Undergraduate practice chapters

1. Undergraduate · Chapter 41

   Two-state Markov chains · Transition 1 → 2: p=0.6; Transition 2 → 1: q=0.6

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.6; Transition 2 → 1: q=0.6.

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2. Undergraduate · Chapter 42

   Orthogonal projections · Vector horizontal a=0.5; 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=0.5; Vector vertical b=0.5.

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3. Undergraduate · Chapter 43

   Diagonal eigenmodes · First eigenvalue a=1.5; 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=1.5; Second eigenvalue b=1.5.

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4. Undergraduate · Chapter 44

   Matrix conditioning · Small diagonal entry a=0.35; Final data error b=0.7

   Use the current perturbation δ=0.42, so Δb=(0,0.42). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.35; Final data error b=0.7.

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5. Undergraduate · Chapter 45

   Directional derivatives · x² coefficient a=0.5; y² coefficient b=1

   Set θ=6π/5 radians. Find the directional derivative and the largest possible unit-direction derivative at this point. Givens: x² coefficient a=0.5; y² coefficient b=1.

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6. Undergraduate · Chapter 46

   Logistic differential equations · Capacity K=4; Growth rate r=1.4

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=4; Growth rate r=1.4.

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7. Undergraduate · Chapter 47

   Euler stability · Decay rate a=2; Step size h=0.7

   Use n=6 updates, corresponding to time t=nh=4.2. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=2; Step size h=0.7.

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8. Undergraduate · Chapter 48

   Newton root iteration · Number whose root is sought A=8; 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=8; Positive starting guess b=4.

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9. Undergraduate · Chapter 49

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

   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=0.7.

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10. Undergraduate · Chapter 50

    Uniform convolution · First uniform width a=0.5; Second uniform width b=1

    Evaluate at sum value s=0.9, three fifths of the support length. Find the density f_S(s), cumulative probability P(S≤s), and expected sum. Givens: First uniform width a=0.5; Second uniform width b=1.

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11. Undergraduate · Chapter 51

    Orthogonal projections · Vector horizontal a=1; 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=1; Vector vertical b=1.

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12. Undergraduate · Chapter 52

    Matrix conditioning · Small diagonal entry a=0.4; Final data error b=0.8

    Use the current perturbation δ=0.48, so Δb=(0,0.48). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.4; Final data error b=0.8.

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13. Undergraduate · Chapter 53

    Directional derivatives · x² coefficient a=1; y² coefficient b=1.5

    Set θ=6π/5 radians. Find the directional derivative and the largest possible unit-direction derivative at this point. Givens: x² coefficient a=1; y² coefficient b=1.5.

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14. Undergraduate · Chapter 54

    Logistic differential equations · Capacity K=4.5; Growth rate r=1.6

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=4.5; Growth rate r=1.6.

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15. Undergraduate · Chapter 55

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

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16. Undergraduate · Chapter 56

    Euler stability · Decay rate a=0.5; Step size h=0.2

    Use n=6 updates, corresponding to time t=nh=1.2. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=0.5; Step size h=0.2.

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17. Undergraduate · Chapter 57

    Newton root iteration · Number whose root is sought A=9; 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=9; Positive starting guess b=4.5.

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18. Undergraduate · Chapter 58

    Two-state Markov chains · Transition 1 → 2: p=0.8; Transition 2 → 1: q=0.8

    Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.8; Transition 2 → 1: q=0.8.

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19. Undergraduate · Chapter 59

    Orthogonal projections · Vector horizontal a=1.5; Vector vertical b=1.5

    Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=1.5; Vector vertical b=1.5.

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20. Undergraduate · Chapter 60

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

    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=2.5; Second eigenvalue b=2.5.

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