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Undergraduate practice chapters

1. Undergraduate · Chapter 181

   Directional derivatives · x² coefficient a=2.5; 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=2.5; y² coefficient b=1.

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2. Undergraduate · Chapter 182

   Logistic differential equations · Capacity K=1; Growth rate r=0.6

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=1; Growth rate r=0.6.

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3. Undergraduate · Chapter 183

   Harmonic oscillators · Angular frequency ω=2.5; 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=2.5; Initial displacement b=1.5.

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4. Undergraduate · Chapter 184

   Euler stability · Decay rate a=0.75; Step size h=0.5

   Use n=6 updates, corresponding to time t=nh=3. 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=0.5.

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5. Undergraduate · Chapter 185

   Newton root iteration · Number whose root is sought A=9; Positive starting guess b=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=5.

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6. Undergraduate · Chapter 186

   Two-state Markov chains · Transition 1 → 2: p=0.5; 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=0.5; Transition 2 → 1: q=0.7.

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7. Undergraduate · Chapter 187

   Orthogonal projections · Vector horizontal a=2.5; Vector vertical b=3

   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=3.

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8. Undergraduate · Chapter 188

   Diagonal eigenmodes · First eigenvalue a=-1; Second eigenvalue b=0

   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; Second eigenvalue b=0.

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9. Undergraduate · Chapter 189

   Matrix conditioning · Small diagonal entry a=0.2; 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=0.2; Final data error b=0.5.

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10. Undergraduate · Chapter 190

    Directional derivatives · x² coefficient a=3; 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=3; y² coefficient b=1.5.

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11. Undergraduate · Chapter 191

    Logistic differential equations · Capacity K=1.5; Growth rate r=0.8

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=1.5; Growth rate r=0.8.

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12. Undergraduate · Chapter 192

    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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13. Undergraduate · Chapter 193

    Euler stability · Decay rate a=1; 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=1; Step size h=0.6.

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14. Undergraduate · Chapter 194

    Newton root iteration · Number whose root is sought A=10; Positive starting guess b=5.5

    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=10; Positive starting guess b=5.5.

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15. Undergraduate · Chapter 195

    Two-state Markov chains · Transition 1 → 2: p=0.6; 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=0.6; Transition 2 → 1: q=0.8.

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16. Undergraduate · Chapter 196

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

    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=-3.

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17. Undergraduate · Chapter 197

    Diagonal eigenmodes · First eigenvalue a=-0.5; Second eigenvalue b=0.5

    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=-0.5; Second eigenvalue b=0.5.

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18. Undergraduate · Chapter 198

    Matrix conditioning · Small diagonal entry a=0.25; Final data error b=0.6

    Use the current perturbation δ=0.36, so Δb=(0,0.36). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.25; Final data error b=0.6.

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19. Undergraduate · Chapter 199

    Directional derivatives · x² coefficient a=0.5; y² coefficient b=2.5

    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=2.5.

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20. Undergraduate · Chapter 200

    Logistic differential equations · Capacity K=2; Growth rate r=1

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=2; Growth rate r=1.

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