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Undergraduate practice chapters

1. Undergraduate · Chapter 241

   Logistic differential equations · Capacity K=4.5; Growth rate r=2

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=4.5; Growth rate r=2.

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2. Undergraduate · Chapter 242

   Harmonic oscillators · Angular frequency ω=3; Initial displacement b=-2

   Evaluate at time t=6π/5, retaining π during calculation. Find displacement, velocity, and conserved energy (y′²+ω²y²)/2. Givens: Angular frequency ω=3; Initial displacement b=-2.

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3. Undergraduate · Chapter 243

   Euler stability · Decay rate a=0.75; Step size h=0.6

   Use n=6 updates, corresponding to time t=nh=3.6. 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=0.6.

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4. Undergraduate · Chapter 244

   Two-state Markov chains · Transition 1 → 2: p=0.3; 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=0.3; Transition 2 → 1: q=0.6.

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5. Undergraduate · Chapter 245

   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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6. Undergraduate · Chapter 246

   Diagonal eigenmodes · First eigenvalue a=2.5; 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=2.5; Second eigenvalue b=-2.

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7. Undergraduate · Chapter 247

   Matrix conditioning · Small diagonal entry a=0.55; Final data error b=0.2

   Use the current perturbation δ=0.12, so Δb=(0,0.12). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.55; Final data error b=0.2.

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8. Undergraduate · Chapter 248

   Logistic differential equations · Capacity K=5; Growth rate r=0.2

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=5; Growth rate r=0.2.

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9. Undergraduate · Chapter 249

   Harmonic oscillators · Angular frequency ω=0.5; Initial displacement b=0.5

   Evaluate at time t=6π/5, retaining π during calculation. Find displacement, velocity, and conserved energy (y′²+ω²y²)/2. Givens: Angular frequency ω=0.5; Initial displacement b=0.5.

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10. Undergraduate · Chapter 250

    Euler stability · Decay rate a=1; 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=1; Step size h=0.7.

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11. Undergraduate · Chapter 251

    Newton root iteration · Number whose root is sought A=1; Positive starting guess b=1.5

    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=1; Positive starting guess b=1.5.

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12. Undergraduate · Chapter 252

    Two-state Markov chains · Transition 1 → 2: p=0.4; 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=0.4; Transition 2 → 1: q=0.7.

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13. Undergraduate · Chapter 253

    Orthogonal projections · Vector horizontal a=0; 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=0; Vector vertical b=1.

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14. Undergraduate · Chapter 254

    Diagonal eigenmodes · First eigenvalue a=3; 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=3; Second eigenvalue b=-1.5.

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15. Undergraduate · Chapter 255

    Matrix conditioning · Small diagonal entry a=0.6; 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=0.6; Final data error b=0.3.

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16. Undergraduate · Chapter 256

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

    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=0.5.

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17. Undergraduate · Chapter 257

    Logistic differential equations · Capacity K=5.5; Growth rate r=0.4

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=5.5; Growth rate r=0.4.

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18. Undergraduate · Chapter 258

    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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19. Undergraduate · Chapter 259

    Euler stability · Decay rate a=1.25; Step size h=0.8

    Use n=6 updates, corresponding to time t=nh=4.8. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=1.25; Step size h=0.8.

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20. Undergraduate · Chapter 260

    Newton root iteration · Number whose root is sought A=2; Positive starting guess b=2

    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=2; Positive starting guess b=2.

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